Quantum Chemistry Calculations of Circularly Polarized Luminescence (CPL): From Spectral Modeling to Molecular Design

Abstract

Circularly polarized luminescence (CPL)the emission of circularly polarized light from luminescent chiral nonracemic matterhas garnered unprecedented attention in the past decade. Once a niche technique used for the characterization of excited states, CPL has evolved to a powerful and widespread tool for developing functional materials with multiple applications. The development of novel CPL emitters is costly and time-consuming because the key CPL quantities (dissymmetry factor, g _lum, and CPL brightness, B _CPL) often elude simple structure-to-property relationships based on existing knowledge. Today, research in the field is aided by quantum chemistry calculations which offer insight into CPL properties and serve as a predictive tool for the rational design of efficient CPL-active materials. The present review is divided into three sections: (1) a comprehensive presentation of the theoretical foundation of CPL calculations, electronic structure description, environment effects, vibronic modulation, band shape broadening, and aggregate simulation; (2) an extensive literature survey, organized according to a structural criterion; and (3) a critical reassessment of literature data, accompanied by a statistical analysis, aimed at offering the best practices for accurate CPL calculations and identifying the key structural and electronic features that enable the simulation-guided design of novel CPL emitters.

1. Introduction

Upon excitation, a chiral nonracemic system may emit light with different intensities for the left (I _L) and right (I _R) circularly polarized components. This phenomenon, called circularly polarized luminescence (CPL) or less commonly circularly polarized emission, can be thought of as the emission counterpart of electronic circular dichroism (ECD). More specifically, it can also be referred to as circularly polarized fluorescence or phosphorescence, depending on the related nonpolarized phenomenon. CPL has evolved in the last 10 years from being considered to be a rather specialized topic to be a technique now routinely employed by hundreds of research groups worldwide. Indeed, in the past decade, the number of manuscripts reporting CPL studies has increased by approximately 5 times. This growth was led by the promise of applications of CPL-active systems in several technologies on one hand − and by the commercial availability of a few options of instrumentation capable of reliable measurements on the other hand − as well as by the possibility of building relatively cheap equipment in the laboratory. −

CPL is commonly observed from (small) organic molecules in solution, − in particular helicenes and helicenoids, − 1,1′-binaphthyls and other biaryls, − BODIPY derivatives, pyrene intramolecular dimers or oligomers, and cyclophanes. Small organic molecules offer the advantages of low cost, ease of functionalization and facile modulation, and control of optical and chiroptical properties through organic synthesis. Beyond purely organic molecules, chiral metal coordination compounds such as Ru^II, Ir^III, Pt^II, and Cr^III complexes are often investigated. − Among coordination compounds, trivalent lanthanide (Ln^III) complexes have found a special place since the dawn of CPL, given the very high CPL performances they can achieve, thanks to the magnetic allowed character of certain f–f transitions. ,− Another relevant case is given by aggregated organic molecules. Very high CPL activity is obtained by small organic molecules or polymers forming aggregates in solutions (due to concentration effects or to the presence of a “poor” solvent) or in thin films. − This family includes multicomponent systems where a stereochemically ordered aggregation of achiral compounds is driven by the presence of chiral dopants. , In these cases, it is fundamental that the monomers increase, or at least retain, their luminescence upon aggregation (aggregation-induced emission, AIE, or aggregation-enhanced emission), instead of undergoing aggregation-caused quenching (ACQ), which prevents any CPL activity. CPL is nowadays also commonly reported for other systems, such as nanoparticles, − quantum and carbon dots (both functionalized with chiral molecules and intrinsically chiral), − perovskites, − and in general (nano)-structured materials including up-converting systems. − Moreover, CPL can be generated by extrinsic mechanisms. An example is the case of CPL stemming from a coupling between nonparallel/nonorthogonal linear birefringence and linear fluorescence anisotropy in self-assembled domains. , This rather rare phenomenon leads to a so-called nonreciprocal emission, which is circular polarization of the emitted light that is opposite for the two faces of the film. Another example of extrinsic mechanism-generating CPL is the Bragg circular phenomenon, displayed by luminescent chiral nematic liquid crystals (CNLC) or CNLC doped with luminescent compounds. − In this case, their periodic helical structure allows for a polarization-selective reflection potentially reaching extremely high dissymmetry factors.

As already mentioned, the interest in CPL-active systems has been stirred by the promise of new possibilities in chiral optoelectronics and photonics as well as in bioimaging and microscopy and in security inks. , In fact, in the last few decades several new techniques and technologies based on CPL have emerged. A particularly promising application of chiral emitters is in organic light-emitting diodes (OLEDs), where CPL emitters can be used in the active layer (CP-OLEDs). ,− In such devices, a net circular polarization of the electroluminescence is observed. CP-OLEDs are expected to increase the efficiency of displays, where circular polarizers are used as antiglare elements, which currently cut out more than 50% of the emission from nonpolarized LEDs. Besides electroluminescence, emission can be excited by an electrochemical reaction in a process termed electrochemiluminescence (ECL). CPL-active systems can in principle emit CP-ECL under these conditions. The most exploited ECL-active compound is [Ru­(bpy)₃]²⁺ (bpy: 2,2′-bipyridine), which emits red light upon electrochemical oxidation in the presence of an amine. In 1987, Gillard, Dekkers, and co-workers reported CP-ECL from an enantiopure Ru-based complex, , and more recently, CP-ECL activity was also reported from a pyrene-based organic compound, exploiting the strong polarization of the excimer emission. , Finally, since the early days of the technique, several examples of CPL probes for bioanalytes of relevant interest have been reported. − CPL is very sensitive to the conformational change of the luminescent probe; therefore, the CPL signal modification upon the probe–analyte interaction is usually more pronounced than in the case of total luminescence. In the last few years, two new techniques capable of substantially improving the possibilities in the field were reported: CPL laser scanning confocal microscopy (CPL-LSCM) and two-photon CPL. CPL-LSCM allows one to track the CPL of a fluorophore with micrometer resolution (e.g., within a cell or a complex material) and to differentiate between areas with different CPL signals or to exploit the so-called enantioselective differential chiral contrast in the case of enantiomer segregation. , Two-photon CPL joins the advantages of CPL with those of two photon emission (i.e., lower-energy excitation and the possibility for higher resolution due to the emission quadratic dependence on the excitation intensity). Another breakthrough will likely happen in the near future by merging the two-photon CPL with CPL-LSCM, allowing for an unprecedented high-resolution tracking of enantio- or diastereo-selective processes. To this end, systems with a large two-photon absorption cross section, good quantum yield, and high dissymmetry factors are necessary; therefore, a rational molecular design is highly beneficial.

By analogy to other chiroptical spectroscopies, CPL activity is commonly quantified by the dissymmetry factor (g _lum), which measures the relative excess of a circularly polarized component over the other

$$g_{\mathrm{lum}}=\frac{\Delta I}{I}=2\frac{I_L-I_R}{I_L+I_R}$$ 1

where ΔI and I are the difference and the semisum of the intensity of emitted left- and right-polarized light components I _L and I _R, respectively.

As extensively discussed from a theoretical point of view in Section , a simplified expression can be used to relate the phenomenological expression for g _lum in eq to the electric (μ _n0 ) and magnetic ( m _0n ) transition dipole moments between emitting excited states (n) and the ground state (0) as

$$g_{\mathrm{lum}}=4\frac{R_{n0}}{D_{n0}}=4\frac{{\mathit{\mu }}_{\mathit{n}\mathbf{0}}·{\mathbf{m}}_{\mathbf{0}\mathit{n}}}{{|{\mathit{\mu }}_{\mathit{n}\mathbf{0}}|}^{\mathbf{2}}+{|{\mathbf{m}}_{\mathbf{0}\mathit{n}}|}^{\mathbf{2}}}=4\frac{|{\mathit{\mu }}_{\mathit{n}\mathbf{0}}|·|{\mathbf{m}}_{\mathbf{0}\mathit{n}}|}{{|{\mathit{\mu }}_{\mathit{n}\mathbf{0}}|}^{\mathbf{2}}+{|{\mathbf{m}}_{\mathbf{0}\mathit{n}}|}^{\mathbf{2}}}\cos {\theta }_{\mathbf{\mu m}}$$ 2

where R _n0 is the rotatory or rotational strength, D _n0 is the dipolar strength, and θ _μm is the angle between μ _n0 and m _0n vectors:

$$R_{n0}={\mathit{\mu }}_{\mathit{n}\mathbf{0}}·{\mathbf{m}}_{\mathbf{0}\mathit{n}}$$ 3

$$D_{n0}={|{\mathit{\mu }}_{\mathit{n}\mathbf{0}}|}^{\mathbf{2}}+{|{\mathbf{m}}_{\mathbf{0}\mathit{n}}|}^{\mathbf{2}}\approx {|{\mathit{\mu }}_{\mathit{n}\mathbf{0}}|}^{\mathbf{2}}$$ 4

In the limit of |μ _n0 | ≫ |m _0n |, which is valid for almost any CPL-emitting system except lanthanide complexes, eq is reduced to

$$g_{\mathrm{lum}}\approx 4\frac{|{\mathbf{m}}_{\mathbf{0}\mathit{n}}|}{|{\mathit{\mu }}_{\mathit{n}\mathbf{0}}|}\cos {\theta }_{\mathbf{\mu m}}$$ 5

This expression is valid under the assumption of an orientational relaxation of systems and the same line shape function for ΔI and I, as discussed in Section . In the very large majority of cases, emission to the ground state occurs from the first excited state of a given multiplicity (e.g., singlet S₁ or triplet T₁), after full internal conversion processes from highest excited states (S _n or T _n ). This well-known Kaska’s rule implies that in the equations above n = 1, and the relevant transition dipoles are μ ₁₀ and m ₀₁ . Of course, anti-Kasha behavior is also possible in some cases. One may therefore rewrite eq as

$$g_{\mathrm{lum}}\approx 4\frac{|{\mathbf{m}}_{\mathbf{01}}|}{|{\mathit{\mu }}_{\mathbf{10}}|}\cos {\theta }_{\mathbf{\mu m}}$$ 6

This latter equation parallels that of absorption dissymmetry factor g _abs in ECD, which is expressed for each possible S₀–S _n transition:

$$g_{\mathrm{abs}}\approx 4\frac{|{\mathbf{m}}_{\mathit{n}\mathbf{0}}|}{|{\mathit{\mu }}_{\mathbf{0}\mathit{n}}|}\cos {\theta }_{\mathbf{\mu m}}$$ 7

For allowed transitions, such as π–π* ones, the electric transition dipoles may be up to 4 orders of magnitude larger than the magnetic ones. For this reason, g _lum factors on the order of 10^–4–10^–2 are commonly observed for organic molecules in good solvents. A different situation may be observed for certain magnetically allowed transitions of lanthanides, where |μ ₁₀ | ≈ |m ₀₁ |. A notable case is the ⁵D₀ → ⁷F₁ transition of Eu^III (approximately at 595 nm), for which g _lum factors close to the theoretical maximum (±2) are reported. Finally, large g _lum factors, on the order of 0.1–1, are also reported for certain aggregates of organic compounds. In those cases, such high figures are probably justified by long-range excitonic coupling within supramolecular ordered structures possibly formed by twisted fibrils organized in the cylinder blue phase. State-of-the-art spectrofluoropolarimeters based on a photoelastic modulator (PEM) can measure g _lum down to 10^–4–10^–5. On the other hand, measurements performed with static optics (quarter wave plate coupled with a linear polarizer) can be performed only with compounds with g _lum above 10^–2. The latter type of revelation is much cheaper than that using PEM-based instruments and can be implemented with standard laboratory equipment or even hand-held instrumentation.

Another parameter used to quantify the performance of CPL emitters beyond the dissymmetry factor is the so-called CPL brightness (B _CPL). By considering the extinction coefficient at the excitation wavelength (ε_λ) and the quantum yield (Φ_f), this quantity is proportional to the differential photon flux of left/right polarized emitted light. It is defined as

$$B_{\mathrm{CPL}}=\frac{1}{2}{\epsilon }_{\lambda }{\mathrm{\Phi }}_{\mathrm{f}}{g}_{\mathrm{lum}}$$ 8

B _CPL is proportional to the CPL signal which is actually measured. It is therefore useful to consider such metrics for applications where the feasibility of the measurement depends critically on the total signal output leading to circularly polarized emission. State-of-the-art spectrofluoropolarimeters should be capable of measuring B _CPL > 0.1 M^–1 cm^–1. This quantity has an essential practical function, as it indicates the feasibility of the CPL signal collection under the chosen conditions. In the case of compounds featuring multiple bands with opposite sign, B _CPL has to be evaluated for the wavelength interval of interest (a, b) as

$$B_{\mathrm{CPL}}=\frac{1}{2}{\epsilon }_{\lambda }{\mathrm{\Phi }}_{\mathrm{f}}\frac{{\int }_{{\lambda }_a}^{{\lambda }_b}\Delta I(\lambda )d\lambda }{{\int }_{{\lambda }_{in}}^{{\lambda }_{end}}I(\lambda )d\lambda }$$ 9

while the integral in the denominator is evaluated on the overall emission, over which the quantum yield is measured. Such an expression is useful when the spectral window in which the CPL signal is evaluated is selected through color filters and a camera, as in CPL imaging techniques. An alternative measure of CPL efficiency is the dimensionless circular polarization luminosity index (Λ_CPL) which is analogous to B _CPL but with ε_λ replaced by the oscillator strength.

The emission dissymmetry factor g _lum and the quantum yield Φ_f are in general statistically uncorrelated with each other. Maximizing parameters such as g _lum and B _CPL is fundamental to proceeding further to the applications mentioned above and to moving them from the proof-of-concept stage to fully developed technologies. This can be achieved only through a rational molecular design. ^, , Until recent years, the field has moved forward mainly through trial and error at the cost of an enormous synthetic effort. On the other hand, a more rational approach may be offered by modern computational chemistry tools. Indeed, starting from the first ab initio calculations appeared in 2010–2011, , in the last 10 years there has been an increasing interest in the quantum chemistry modeling of CPL properties of various compounds; especially noteworthy is the explosion seen since 2020 (Figure ). This observation provided the primary impetus for this review. In fact, the only two reviews on CPL calculations available in the literature, dated 2016 by Longhi and co-workers and 2018 by Chen and co-workers, appeared at very early stages of development in the field and could cover only a few literature reports. More recent contributions, focused on some methodological aspects, did not offer a literature coverage. A survey of the literatureas of July 2025counts more than 260 papers devoted to quantum mechanical simulations of CPL spectra and related parameters, a very large majority of which have not been accounted for in any review. They will be systematically surveyed in Section .

1.

Papers on CPL calculations over time.

As will be outlined in Section , a basic CPL calculation requires, at least, a knowledge of the structure at the minimum of the excited emitting state(s) and hence the evaluation of vertical transition energies and the transition dipole vectors μ ₁₀ and m ₀₁ to calculate the g _lum factor according to eq . Despite the tremendous development of computational techniques, such tasks are extremely difficult or even unfeasible to perform on complex systems. For instance, open-shell systems containing heavy metal atoms, such as lanthanides, require methods to account for large spin–orbit coupling, relativistic effects, and the multiconfiguration character in the f–f transition (Section ). For different reasons, CPL calculations of aggregate compounds with long-range chirality are also extremely challenging due to structure complexity, including the possible existence of multiple aggregation pathways, and the large molecular size of extended aggregate structures (Section ).

On the other hand, ab initio simulations of CPL features have been applied to hundreds of organic molecules in solution, and especially small organic molecules (SOMs, Section ), often offering exceptional accuracy when compared to experimental data. Helicenes, multiple helicenes, and helicenoids are by far the most commonly encountered SOMs (Section ). Very well represented are also 1,1′-binaphthyls and other biaryls (Section ), [2.2]­paracyclophane derivatives (Section ), and tetra-coordinated boron compounds, including BODIPY (Section ). Successful examples of calculations also cover CPL-active systems endowed with properties relevant for applications in electronic devices, sensors, and photochromic materials, such as excimer emission, thermally activated delayed fluorescence (TADF), and excited-state intramolecular proton transfer (ESIPT).

Section lays the theoretical groundwork for understanding CPL before delving into the literature review presented in Section . It begins with a concise overview of the typical workflow involved in CPL calculations (Section ) followed by a detailed examination of the theoretical principles underpinning CPL phenomena (Section ). Subsequently, the section discusses various electronic structure methods for modeling transition energies and dipole moments (Section ), which serve as foundational elements referenced throughout the subsequent survey. The discussion progresses to encompass environmental effects (Section ), band shape simulation (Section ), and aggregate systems (Section ). The last part of this section (Section ) will explore the significance of different parameters and pieces of information which can be extracted from CPL simulations, also in comparison with experimental data. The reader who is not interested in theoretical background may skip Section .

In Section , we will first summarize and discuss the efficiency of various calculation methodologies for the prediction of CPL properties for specific or broad families of compounds (Section ). Then, we will critically assess the possibility offered by computational chemistry toward the design of more efficient CPL emitters (Section ).

2. Methods

In this section, after a brief outline of the various steps required to simulate a CPL signal, we present all of the related methodological aspects, starting from the general theoretical quantum mechanical framework to different aspects of calculation, including electronic structure description, vibronic modulation, line shape description, supramolecular excitonic effects, aggregate description, and the environment polarization response. Readers who do not wish to delve into the theory may skip this section, in particular the paragraphs marked with an asterisk (*).

2.1. Outline of CPL Calculations

In a nutshell, the calculation of the main CPL quantities (R _i0, g _lum, and λ_em, emission wavelength) of a chiral compound requires the calculation of the first (lowest energy) vertical transition from the first excited state (S₁) geometry. The sequence of steps, depicted in Figure a, is the following:

• 1) Geometry optimization of the ground state (GS). This step should be also coupled with a frequency calculation (at the same level of theory) to confirm that the optimized geometry corresponds to a true local minimum on the potential energy surface (PES) (i.e., its associated frequencies are all real). For closed-shell systems, the GS corresponds to a singlet state (S₀). This step furnishes, obviously, the GS geometry to be used subsequently to compute the absorption and ECD spectra and to include vibronic effects (in the harmonic approximation) by coupling it with the corresponding quantities of the S₁ state. In the case of flexible systems or if a sampling strategy is adopted to simulate the spectral broadening (Section ), a better description of the ECD spectrum can be obtained by sampling the GS conformational ensemble using a classical molecular dynamics simulation.
• 2) Calculation of vertical electronic transitions from GS geometry. This step provides transition energies or wavelengths and dipolar (oscillator) strengths, rotational strengths, and g _abs values for each computed transition. One can use a phenomenological band shape (usually Gaussian) or one simulated by adopting different strategies to obtain full absorption and ECD spectra. The simulation of the electric and transition dipole moments μ _0i and m _i0 and the angle θ _μm between them provide an estimate of their relative orientation, aiding the interpretation of experimental data. Furthermore, molecular orbitals (MO), natural transition orbitals (NTO), and transition density plots may be visualized to gain a perception of the character of each transition or may even be quantitatively analyzed by different metrics. −
• 3) Geometry optimization and frequency calculations of the first excited electronic state (ES). The obtained geometry can be compared to the GS one. In the case of close similarity, one may expect small Stokes shifts, the same sign between the first calculated ECD band and CPL band, and consistent g _abs and g _lum values. Exceptions to this common outcome include a large structural rearrangement in the excited state (e.g., due to excimer formation) or the presence of two different ES minima. According to Kasha’s rule, the emitting state of closed-shell systems is either S₁ (singlet) or T₁ (triplet). Usually, the same level of theory is used in the geometry optimization of the S₁ state and for the evaluation of its emission properties (next steps); however, this is not always computationally feasible, and mixed approaches have been largely adopted in the literature: for instance, multireference methods (Section ) for electronic structure simulations are usually coupled with less expensive (TD-)­DFT methods (Section ) for geometry optimization and frequency determination.
• 4) Calculation of vertical de-excitation from ES minimum geometry. Although a minimum number of states (roots) is included in the calculations to ensure proper diagonalization, it is obvious that only the first one corresponds to the true S₁–S₀ or T₁–S₀ emission. Analogous to step 2), this step provides the emission energy or wavelength and hence the calculated Stokes shift, dipolar strength, rotational strength, and g _lum value. Again, μ _i0 and m _0i moments and angle θ _μm can be estimated, and MOs, NTOs, and transition densities can be plotted. Although less common than for Abs and ECD, even in the case of CPL one can adopt a phenomenological band shape or simulate it by theoretical means to obtain emission and CPL spectra. Obviously, the same strategies presented in step 2) can be employed to characterize the electronic structure rearrangement during the emissive phenomena.

2.

Schematic description of the common steps used for CPL calculations: (a) gas-phase, electronic-only calculation; (b) gas-phase, vibronic calculation; and (c) electronic-only calculation including environment.

The above sequence has many variants, the two most important ones consisting of the explicit simulation of the band shape by including shape modulation due to vibronic effects, homogeneous and inhomogeneous broadening, and the inclusion of embedding effects (like the solvent or protein matrix); they can also be combined with each other. Some of them have been included in Figure for a direct comparison, but all of the different aspects will be discussed with large details in devoted sections (Sections and ). Here we just stress one point: if the solvent effects are included by using a continuum dielectric model, then once the excited state structure is optimized, the emission simulation requires a two-step calculation, as described in section Section .

2.2. Theoretical Framework (*)

The theory of light absorption and optical activity phenomena, including ECD and optical rotation (OR), is commonly based on the semiclassical treatment of light–matter interaction. Spontaneous emission phenomena, such as fluorescence and CPL, cannot be treated quantitatively using the same semiclassical approach and require the methods and formalism of quantum electrodynamics (QED). A general QED-based treatment of CPL was proposed by Riehl and Richardson in 1976, , which we summarize here (Figure ).

3.

Schematic representation of QED perturbative treatment of spontaneous emission phenomena.

The interaction Hamiltonian of the molecular system with the electromagnetic field is governed by the couplings due to the electric polarization P and magnetization M densities with the quantized transverse electric E and magnetic B fields of the radiation:

$${\mathrm{Ĥ}}^{\prime }=-\mathbf{P̂}·{\mathbf{Ê}}_{\perp }-\mathbf{M̂}·{\mathbf{B̂}}_{\perp }$$ 10

Electric polarization and magnetization densities can be expanded in terms of molecular multipole moments. If we limit the expansion to the electric quadrupolar term, then the interaction Hamiltonian reduces to

$${\mathrm{Ĥ}}^{\prime }=-\mathit{\mu ̂}·{\mathbf{Ê}}_{\perp }-\mathbf{m̂}·{\mathbf{B̂}}_{\perp }+\frac{\mathbf{1}}{\mathbf{6}}\mathit{Q̂}:\mathrm{\nabla }{\mathbf{Ê}}_{\perp }+···$$ 11

$$\mathit{\mu ̂}=\sum _i{\mathbf{r̂}}_{\mathit{i}}$$ 12

$$\mathbf{m̂}=-\frac{\mathrm{i}}{2}\sum _i{\mathbf{r̂}}_{\mathit{i}}\times {\mathrm{\nabla ̂}}_{\mathit{i}}$$ 13

The transverse (with respect to the wave vector propagation) fields Ê _⊥ and B̂ _⊥ are expressed in terms of photon creation and annihilation operators for different modes. In QED, the light quantization modes describe the allowed energy states of electromagnetic waves within a given space constrained by boundary conditions (like walls of a cavity) or simply by the properties of the free space. Each mode can either be “empty” (no photons present) or “occupied” (one or more photons present), and the energy in each mode is quantized in discrete packets (E = nℏω, where n is the number of photons). Modes determine how molecules interact with light: for instance, in molecular spectroscopy a molecule can absorb photons only from modes whose energy matches the molecule’s electronic or vibrational transitions, or in confined spaces (such as nanocavities or plasmonic systems) the interaction between molecules and specific electromagnetic modes can lead to phenomena like strong coupling, which modifies molecular properties. The final expressions take the form

$${\mathbf{E}}_{\perp }(\mathit{r})=\mathrm{i}\sum _l^{\mathrm{mod}\mathrm{e}\mathrm{s}}\sum _{\sigma =X,Y}{\left(\frac{{\omega }_l}{2V_q}\right)}^{1/2}{ê}_{l,\sigma }[a_{l,\sigma }{e}^{i{\mathit{k}}_l·\mathit{r}}-{a^{†}}_{l,\sigma }{e}^{i{\mathit{k}}_l·\mathit{r}}]$$ 14

$${\mathbf{B}}_{\perp }(\mathit{r})=\frac{\mathrm{i}}{c}\sum _l^{\mathrm{mod}\mathrm{e}\mathrm{s}}\sum _{\sigma =X,Y}{\left(\frac{{\omega }_l}{2V_q}\right)}^{1/2}({ê}_{l,\sigma }\times {\mathrm{k̂}}_l)[a_{l,\sigma }{e}^{i{\mathit{k}}_l·\mathit{r}}-{a^{†}}_{l,\sigma }{e}^{i{\mathit{k}}_l·\mathit{r}}]$$ 15

In eqs and , l runs over the field modes, a _l, σ and a ^† _l,σ are, respectively, the creation and annihilation operators for photons of wave vector k _l of frequency ω _l , and ê _l,σ is the unit vector of the plane of light polarization σ (that can be X or Y if the wave propagates along Z). V _q is the volume within which the field modes are assumed to exist, and its values reflect the fact that the field becomes less localized as the quantization volume increases. The volume sets the density of allowed wave vectors in the reciprocal space: for a finite volume, the wave vectors are discrete due to boundary conditions, but in an infinite volume, the modes become continuous (free-space quantization).

We refer to a laboratory-fixed coordinate system (X, Y, Z) for emission along the Z axis (Figure ). Using this frame, we can define two circularly polarized photon unit vectors as

$${ê}_R=\frac{1}{\sqrt{2}}({ê}_X-i{ê}_Y)$$ 16

$${ê}_L=\frac{1}{\sqrt{2}}({ê}_X+i{ê}_Y)$$ 17

4.

Laboratory (X, Y, Z) and molecular frame (x, y, z) with Euler’s angles (γ, δ, θ).

The transition probability per unit time for the spontaneous emission of each circularly polarized photon (right or left) can be calculated in a perturbative manner, and to first order a Fermi golden rule-type expression can be written assuming that all electric transition moments are purely real and that all magnetic ones are purely imaginary. In this frame, the components of the molecular transitions’ dipolar and quadrupolar moments are evaluated between the initial electronic excited state (n) and the final ground state (0). The differential spontaneous emission rate takes therefore the following expression ,

$$\Delta {\mathrm{W}}_e={\mathrm{W}}_L-{\mathrm{W}}_R=\left(\frac{{{\omega }_l}^3}{\pi c^3}\right)[-Im({\mu }_X^{n0}{m}_X^{0n}+{\mu }_Y^{n0}{m}_Y^{0n})+k_{l}Re(-{\mu }_X^{n0}{Q}_{YZ}^{0n}+{\mu }_Y^{n0}{Q}_{XZ}^{0n})]$$ 18

where μ_α and m _α (α = X, Y, Z) are the components of μ _n0 and m _0n . Equation is valid assuming a collection of molecules with identical orientations with respect to a fixed laboratory coordinate system (here assumed to be Z). In this frame, the components of the molecular transitions dipolar and quadrupolar moments are evaluated between the initial electronic excited state (n) and the final ground (0) state. We recall that $k_l=\left(\frac{{\omega }_l}{c}\right)$ is the norm of the wave vector k _l .

To relate this expression to observed emission intensities, one has to consider that, under the experimental conditions, molecules are not typically aligned in a single orientation but are distributed randomly in space. An extension of the initial formalism to include a random orientation of molecules in their ground state distribution prior to excitation is therefore necessary. When molecules are exposed to polarized excitation light, a subset of orientations is preferentially excited due to photoselection, resulting in a nonrandom distribution of molecules in the excited state immediately after absorption. This photoselected distribution subsequently evolves in time due to molecular reorientations driven by Brownian motion. These reorientations redistribute the molecular orientations in the excited state before the emission of light.

The observed differential (spontaneous) emission intensity for natural CPL is related to the differential transition rate

$$\Delta I(\omega ,\mathrm{t})=I_L-I_R={\omega }_l{N}_n(t)L_{CPL}(\omega )⟨\Delta {\mathrm{W}}_e(t)⟩$$ 19

where N _n (t) is the time-dependent population of the emitting state n, which may be different from the final state populated by the initial absorption process. The brackets around the emission rate in eq denote a spatial average over the orientational distribution of emitting molecules in the sample. L _CPL (ω) is a normalized line-shape function centered on ω _l , whose form will be extensively treated in Section .

The effects of photoselection and reorientation imply that the rate is transformed from the laboratory frame (X, Y, Z) to the molecular frame (x, y, z) defined by a set of Euler angles (γ, δ, θ, Figure ), also including an orientation-dependent factor (F) that gives excitation polarization and excitation–emission geometry information. If α is the angle between the direction of the incident beam and the detector and β is the one between the electric vector of the incident beam and the normal direction of the laboratory system, then F takes the expression

$$F=1-3{\sin }^2\alpha {\sin }^2\beta$$ 20

The orientational average of the rate is therefore

$$⟨\Delta {\mathrm{W}}_e(t)⟩=\left(\frac{{{\omega }_l}^3}{\pi c^3}\right)i[(\frac{2}{3}-\frac{1}{15}FC(t))({\mu }_x^{n0}{m}_x^{0n}+{\mu }_y^{n0}{m}_y^{0n})+(\frac{2}{3}+\frac{2}{15}FC(t)){\mu }_z^{n0}{m}_z^{0n}]+\frac{1}{5}FC(t)k_l(-{\mu }_x^{n0}{Q}_{yz}^{0n}+{\mu }_y^{n0}{Q}_{xz}^{0n})$$ 21

Here, the function C(t) describes the average random thermal reorientation of the molecules in solution and depends on the molecular shape. For fully asymmetric ellipsoids, it comprises five distinct exponential terms. This complexity reduces to three terms for symmetric ellipsoids and further simplifies to a single exponential for spherical molecules. In practice, many large molecules are treated as spherical rotors with diffusion constant D (Einstein sphere). Under such conditions, C(t) takes a straightforward exponential expression

$$C(t)=e^{-6Dt}$$ 22

Assuming a first-order decay of the excited state n, its population at time t is given by

$$N_n(t)=N_0{e}^{-t/\tau }$$ 23

where N ₀ is the population of state n at t = 0 after the excitation and τ is the total lifetime of the state that includes both radiative and nonradiative decay pathways. The final steady-state CPL intensity is therefore obtained by integrating over long time:

$$\Delta I(\omega )={\omega }_l{N}_0\tau L_{CPL}(\omega )\left(\frac{{{\omega }_l}^3}{\pi c^3}\right)i[(\frac{2}{3}-\frac{F}{(90D\tau +15)})({\mu }_x^{n0}{m}_x^{0n}+{\mu }_y^{n0}{m}_y^{0n})+(\frac{2}{3}+\frac{2F}{(90D\tau +15)}){\mu }_z^{n0}{m}_z^{0n}]+\frac{F{k}_l}{(30D\tau +5)}(-{\mu }_x^{n0}{Q}_{yz}^{0n}+{\mu }_y^{n0}{Q}_{xz}^{0n})$$ 24

This equation is general, with only two assumptions: the molecular transition dipoles during absorption are oriented along z and the molecule is approximately spherical. If the emitting molecules are small enough to reorient rapidly and/or have a long lifetime, such that Dτ ≫ 1, then eq reduces to an expression that is independent of the angle between the absorption and emission electric dipole transition moments (included in F), the diffusion coefficient, and the total emission lifetime:

$$\Delta I{(\omega )}_{relaxed}=N_0\tau L_{CPL}(\omega )\left(\frac{{{\omega }_l}^4}{\pi c^3}\right)\frac{2}{3}[Im({\mathit{\mu }}_{n0}·{\mathbf{m}}_{\mathbf{0}\mathit{n}})]=N_0\tau L_{CPL}(\omega )\left(\frac{{{\omega }_l}^4}{\pi c^3}\right)\frac{2}{3}{R}_{n0}$$ 25

In eq , R _n0 is the rotational strength defined in eq , as given by the Rosenfeld equation. Note that in the limit the molecules are completely “reorientationally relaxed” prior to emission and the electric dipole–electric quadrupole terms do not contribute to the differential rate and therefore to the differential intensity. However, an isotropic orientational distribution of emitting species must be considered to be a special case in CPL spectroscopy, even when one is dealing with fluid samples of homogeneous composition. In general, the sample is excited along just one direction, and this results in the selective excitation of molecules with specific orientational distributions.

2.2.1. Luminescence Anisotropy Factor (*)

The measurement of absolute emission intensities is not an easy task: the usual procedure in CPL experiments is to determine both the total and differential intensity (in arbitrary or relative units) and to determine their ratio, the so-called emission dissymmetry factor or emission anisotropy factor (Introduction), by analogy to ECD spectroscopy:

$$\mathrm{g}{(\omega )}_{lum}=\frac{[I_L(\omega )-I_R(\omega )]}{\frac{[I_L(\omega )+I_R(\omega )]}{2}}=\frac{2\Delta I(\omega )}{I(\omega )}$$ 26

The steady-state total luminescence intensity, I(ω), following an analogous derivation as done before to include photoselection and rotational relaxation, is given by

$$I(\omega )=\frac{1}{2}{N}_0\tau L_{Emi}(\omega )\left(\frac{{{\omega }_l}^4}{\pi c^3}\right)[(\frac{2}{3}-\frac{F}{(90D\tau +15)})({|{\mu }_x^{n0}|}^2+{|{\mu }_y^{n0}|}^2)+(\frac{2}{3}+\frac{2F}{(90D\tau +15)}){|{\mu }_z^{n0}|}^2]+O(highorders)$$ 27

One can therefore express g­(ω) _lum in terms of molecular parameters, total and circularly polarized band-shape functions, and an orientation-dependent factor appropriate to the specific photoselection/orientational relaxation model:

$$\mathrm{g}{(\omega )}_{lum}=\frac{4i{L}_{CPL}(\omega )}{L_{Emi}(\omega )}·\left[\begin{array}{c}\frac{[(\frac{2}{3}-\frac{F}{(90D\tau +15)})({\mu }_x^{n0}{m}_x^{0n}+{\mu }_y^{n0}{m}_y^{0n})+(\frac{2}{3}+\frac{2F}{(90D\tau +15)}){\mu }_z^{n0}{m}_z^{0n}]}{[(\frac{2}{3}-\frac{F}{(90D\tau +15)})({|{\mu }_x^{n0}|}^2+{|{\mu }_y^{n0}|}^2)+(\frac{2}{3}+\frac{2F}{(90D\tau +15)}){|{\mu }_z^{n0}|}^2]}\\ +\frac{\frac{iF{k}_l}{(30D\tau +5)}(-{\mu }_x^{n0}{Q}_{yz}^{0n}+{\mu }_y^{n0}{Q}_{xz}^{0n})}{[(\frac{2}{3}-\frac{F}{(90D\tau +15)})({|{\mu }_x^{n0}|}^2+{|{\mu }_y^{n0}|}^2)+(\frac{2}{3}+\frac{2F}{(90D\tau +15)}){|{\mu }_z^{n0}|}^2]}\end{array}\right]$$ 28

Obviously also here, in the limit that the molecules are completely “reorientationally relaxed” prior to emission (i.e., Dτ ≫ 1), the electric dipole–electric quadrupole terms do not contribute to the differential rate and the luminescence anisotropy factor reduces to

$$\mathrm{g}{(\omega )}_{lum}^{relax}=\frac{4L_{CPL}(\omega )R_{n0}}{L_{Emi}(\omega )D_{n0}}$$ 29

where D _n0 is the dipolar strength defined in eq . The same isotropic description results from selecting an experimental configuration such that F = 0, which can be obtained by playing with the polarization of the excitation beam and the relative excitation–emission geometry. ,

At the contrary, we can find cases where the emitting molecules are effectively “frozen” in their initial distribution, such as systems dissolved in viscous solvents or molecules with short total excited state lifetimes (i.e., Dτ ≪ 1):

$$\begin{gathered} \mathrm{g}{(\omega )}_{lum}^{frozen}=\frac{4{iL}_{CPL}(\omega )}{L_{Emi}(\omega )}\times \\ \left[\frac{[(10-F)({\mu }_x^{n0}{m}_x^{0n}+{\mu }_y^{n0}{m}_y^{0n})+(10+2F){\mu }_z^{n0}{m}_z^{0n}-3iF{k}_l(-{\mu }_x^{n0}{Q}_{yz}^{0n}+{\mu }_y^{n0}{Q}_{xz}^{0n})]}{[(10-F)({|{\mu }_x^{n0}|}^2+{|{\mu }_y^{n0}|}^2)+(10+2F){|{\mu }_z^{n0}|}^2]}\right] \end{gathered}$$ 30

2.3. Modeling of Transition Energies and Transition Moments (*)

Irrespective of the different geometrical arrangements, beam polarization, and even lifetime or molecular shape, eqs – clearly show that the modeling of g­(ω) _lum needs an accurate estimation of both dipolar and rotational strengths as well as excitation energies. It is therefore important to consider how transition electric and magnetic dipoles are computed by different electronic structure calculation approaches and the limits of their descriptions. The calculation of the dissymmetry factor needs information on the electric dipole and magnetic dipole transition moments in the ground- and excited-state geometries, respectively, for ECD and CPL. This implies that one needs to accurately describe not only transition properties but also geometries and even frequencies if it is necessary to include vibronic coupling (Section ).

In the context of chiroptical properties, the time-dependent perturbation theory is particularly significant because the rotational strength is related to the residues of the frequency-dependent optical activity tensor (via a Fourier transform), G′(ω):

$${\mathbf{G}}^{\prime }(\omega )=-2\omega \sum _{n\ne 0}\frac{Im[⟨0|\mathit{\mu ̂}|n⟩⟨n|\mathbf{m̂}|0⟩]}{{{\omega }_{n0}}^2-{\omega }^2}$$ 31

Here, |0⟩ and |n⟩ are shorthand notations for the wave function of the ground state and a generic n state. However, the sum-overstates definition given in eq is not usually computed in practice due to its slow rate of convergence. A more convenient approach is that of response theory, − where the Fourier transform of time-dependent expectation values are written as an expansion in the frequency-dependent perturbation operator and many electronic structure formalisms allows for the computation of molecular properties through extensions like response theory. One of the main advantages of response theory over a time-dependent perturbation theory approach, as demonstrated by Olsen and Jo̷rgensen, lies in the fact that the response of the wave function needs not be formulated as an expansion of excited states but may instead be computed for each perturbation operator in any complete yet convenient set of functions. This approach requires only the solution of systems of linear equations for each frequency-dependent perturbation and thus is far less expensive than the computation of the complete set of excited states. In the same spirit, the linear response (LR) theory is also developed in the time-dependent density functional theory (TD-DFT), but here it is the electronic density that is relaxed with respect to an external perturbation.

In the LR framework, the optical activity tensor in eq is related to the mixed electric–magnetic dipolar linear response function:

$${\mathbf{G}}^{\prime }(\omega )=Im⟨⟨\mathit{\mu ̂};\mathbf{m̂}⟩⟩$$ 32

$$R_{n0}=-Tr[\underset{\omega \to {\omega }_{n0}}{\lim }(\omega -{\omega }_{n0}){\mathbf{G}}^{\prime }(\omega )]$$ 33

Several electronic structure methods for the calculation of chiroptical properties of chiral molecules have been used over the years, such as time-dependent Hartree–Fock (TD-HF) and time-dependent density functional theory (TD-DFT), complete and restricted active space self-consistent field (CAS-/RAS-SCF) wave function methods with perturbative corrections (CASPT2), as well as coupled-cluster theory (CC) with singles and doubles (CCSD). These models are different in their respective implementations, and each has its drawbacks and merits: for instance, TD-DFT can be applied to molecular systems larger than those tractable by wave function methods such as compete active space or coupled-cluster theory but remains more limited in scalability compared to TD-HF.

In the next sections, we will briefly review how the response can be modeled in the different electronic structure frameworks. Atomic units will be used to present the theory, but it is easy to convert the dipole and rotational strengths from atomic units to cgs units employed in the experiments: the calculated quantities are multiplied by $\frac{1}{{(ec{a}_0)}^2}\times {10}^6$ and by $\frac{\hslash e^{2}c{a}_0}{m_e}\times {10}^4$ respectively, where e is the elementary charge, m _e is the mass of an electron, and a ₀ is the Bohr radius. ,

Finally, it is important to recall the well-known problem of gauge invariance of rotational strength: in the length gauge, the results obtained depend on the choice of the origin, due to the origin dependence of the magnetic dipole moment. For variational methods (including DFT), this problem can be overcome by using London atomic orbitals (gauge-invariant atomic orbitals, GIAOs). ,

Alternatively, one can express the transition electric dipole in the velocity gauge, using the commutator of the position operator with the Hamiltonian (hyper virial theorem):

$$⟨0|\mathrm{\nabla ̂}|n⟩=⟨0|[Ĥ,\mathit{r̂}]|n⟩$$ 34

$$R_{n0}^{Len}=-\frac{\mathbf{1}}{\mathbf{2}}\sum _i⟨n|{\mathbf{r̂}}_{\mathit{i}}|0⟩⟨0|{\mathbf{r̂}}_{\mathit{i}}\times {\mathrm{\nabla ̂}}_{\mathit{i}}|n⟩=-\frac{\mathit{i}}{\mathbf{2}}\sum _i⟨n|{\mathbf{r̂}}_{\mathit{i}}|0⟩⟨0|{\mathbf{m̂}}_{\mathit{i}}|n⟩$$ 35

$$R_{n0}^{Vel}=-\frac{\mathbf{1}}{\mathbf{2}{\mathit{\omega }}_{\mathbf{0}\mathit{n}}}\sum _i⟨n|{\mathrm{\nabla ̂}}_{\mathit{i}}|0⟩⟨0|{\mathbf{r̂}}_{\mathit{i}}\times {\mathrm{\nabla ̂}}_{\mathit{i}}|n⟩=\frac{\mathbf{1}}{\mathbf{2}{\mathit{\omega }}_{\mathbf{0}\mathit{n}}}\sum _i⟨n|{\mathbf{p̂}}_{\mathit{i}}|0⟩⟨0|{\mathbf{m̂}}_{\mathit{i}}|n⟩$$ 36

It is worth noting that the use of the hyper virial theorem in eq is valid for exact wave functions or for variational methods in the case of basis set completeness. Therefore, rotational strengths calculated in the two gauge representations are expected to converge to the same value only when approaching basis set completeness, , that is, when relatively large basis sets are employed.

2.3.1. Time-Dependent DFT (TD-DFT) and Tamm-Dancoff Approximation (TDA) (*)

One of the major issues in calculating excited-state energies and related properties is the computational cost affecting the chosen computational method: from one side, it is well known that post-Hartree–Fock (HF) methods, accurately describing the electronic correlation, are too expensive for medium and large molecules, despite the enormous progress made during the last years to improve their speeding-up. On the other hand, TD-DFT emerged as a promising approach to the study of excited states, and experience has shown that TD-DFT excitation energies are generally in good agreement with experiments. As a matter of fact, the optimal compromise between accuracy and computational cost makes TD-DFT the most widely used method of calculating excitation energies of chemically relevant molecules. This is also true in the context of CPL calculations, as will clearly emerge from our survey (Sections and ). However, the LR formulation of TD-DFT sums up the typical problems of ground-state DFT with those originating in the LR approximation and the adiabatic assumption. , These approximations do not significantly affect the calculation of electronic transitions involving valence electrons (valence excitations) which, as extensively shown in literature, , are accurately reproduced. On the contrary, numerous numerical benchmarks showed that the LR-TD-DFT approach, combined with approximate exchange-correlations functionals (XCFs), encounters difficulties in describing charge transfer (CT) excitations, multielectron excitations, and absorption spectra of systems with delocalized or nonpaired electrons. ,, A relevant example in the context of CPL-active compounds is that of f–f transitions in lanthanide complexes (Section ).

The approach commonly used in quantum chemistry is based on linear perturbation theory and takes advantage of the fact that poles of linear response functions correspond to excitation energies and can be calculated as eigenvalues from the response equation (often referred to as the Casida equation)

$$\left[\begin{array}{cc}\mathit{A}& \mathit{B}\\ \mathit{B}& \mathit{A}\end{array}\right]\left[\begin{array}{c}\mathit{X}\\ \mathit{Y}\end{array}\right]=\omega \left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}\mathit{X}\\ \mathit{Y}\end{array}\right]$$ 37

that links the electronic Hessian terms (i.e., the Fock matrix variation with respect to change in the density, contained in the A and B matrices) to the residues of the generalized eigenvalue problem (i.e., the transition density terms due to single electron excitations and de-excitations, respectively in X and Y).

Further, molecular properties such as dipolar or rotational strengths correspond to residues of response functions and can be obtained from the eigenvectors:

$${\mu }_{\alpha }^{0I}=\sum _{j,b}⟨j|{\mathrm{\mu ̂}}_{\alpha }|b⟩(X_{jb}^I+Y_{jb}^I)$$ 38

$${\mathrm{m}}_{\alpha }^{0I}=\sum _{j,b}⟨j|{\mathrm{m̂}}_{\alpha }|b⟩(X_{jb}^I-Y_{jb}^I)$$ 39

The popularity of this approach for a wide range of systems is also due to the efficient algorithms available for solving eqs . However, since eigenvalue calculations normally proceed from the lowest excitation energy and the computational cost increases with the number of eigenvalues, its applications in high-frequency spectral regions and regions with high density of states remain challenging and require the development of special techniques. − Moreover, due to its perturbative nature, the response eigenvalue equation requires the evaluation of the derivatives of DFT exchange–correlation potentials (the so-called kernels) that must be formulated carefully, particularly in relativistic multicomponent theories with spin–orbit coupling. , Hirata and Head-Gordon introduced a simplification of eq by the Tamm-Dancoff approximation (TDA), which decouples excitation and de-excitation processes, leading to a Hermitian eigenvalue problem (both the B and Y terms are set to zero). This decoupling enhances computational efficiency and stability and is particularly beneficial in systems where the full TD-DFT approach encounters challenges such as triplet instabilities or negative eigenvalues.

2.3.1.1. TD-DFT and TDA: Performance

Concerning the performance, many benchmarks of different XC functionals have been performed over the last 15 years, mainly focusing on the accuracy in reproducing excitation energies. However, as is clear from the theory, this is just a part of the story: the accurate simulation of CPL signals requires indeed the determination of excited state structures and transition state properties. The former require accurate calculation not only of energies but also of the PES shape (gradient and Hessian), whereas the latter are related to the ground-to-excited transition density matrix. The performance of different functionals is of course linked to the type of molecular system and the character of excitation involved. The impact of DFT functionals on the calculation of CPL quantities will be extensively discussed in Section . Some general recommendations can be anticipated: for valence transition involving small density reorganization after the excitation, global hybrids like B3LYP , and PBE0 are generally sufficient, offering a good balance between accuracy and computational efficiency, but functionals with a higher percentage of exact exchange, such as PBE0-1/3, should be considered to improve the accuracy for intermediate cases (such as n−π* or π–π* transitions involving large resonances). In general, the best performances are obtained using a range-separated functional, such as CAM-B3LYP or ωB97X-D, or a hybrid meta-GGA (generalized gradient approximation) such as M06-2X with high exact exchange percentage included: they are more adaptable when a large density reorganization is involved, as in charge transfer or large molecules with high resonances, as the range separation is suitable to accurately capture the long-range electron–hole interactions. A study by Loos and Jacquemin points out that CAM-B3LYP appears to be the most accurate XCFs for the oscillator strengths with an MAE (mean absolute error) of 8%, noticeably outperforming LR-CC2 (Section ). Systems involving strong dynamic electron correlation effects need to consider double hybrid functionals, such as B2PLYP, but keeping in mind the increased computational resources required and the fact that the configuration interaction correction used for double hybrids is only an energy correction, which means other quantities, such as transition dipole moments and their related oscillator and rotational strengths, have hybrid quality. Of course, since transition properties are involved, when possible, at least a triple-ζ zeta basis set should be preferred, such as def2-TZVP. The impact of the basis set on TD-DFT CPL calculations will also be discussed in Section .

TDA often yields excitation energies comparable to those obtained from full TD-DFT calculations. In certain cases, it provides better agreement with experimental data, especially when combined with functionals like B3LYP.

2.3.2. Coupled-Cluster Methods (EOM, CC2, and CCSD) (*)

Coupled cluster (CC) theory is one of the most accurate and versatile tools in quantum chemistry, essential for predicting molecular properties with a high degree of reliability. Originating in nuclear physics, the CC method was adapted for electronic wave functions by Čížek and Paldus. , Unlike linear expansions used in the configuration interaction (CI), CC theory employs an exponential ansatz for the wave function |Ψ _CC ⟩ = e ^T̂ |Φ₀⟩, where |Φ₀⟩ is the reference wave function, typically obtained from Hartree–Fock. The cluster operator, T̂ = T̂ ₁ + T̂ ₂ + T̂ ₃··· generates excitations in the form of single, double, and higher-order substitutions from the reference determinant. The exponential nature of the wave function ensures that even when the cluster operator is truncated at a finite level, such as in the commonly used CCSD (CC with single and double excitations) or CCSDT methods (also with triple excitations), higher-order excitations are implicitly included.

Using the exponential ansatz in the Schrödinger equation and then multiplying on the left by the inverse of the exponential, one obtains

$$e^{-\mathrm{T̂}}Ĥe^{\mathrm{T̂}}|{\mathrm{\Phi }}_0⟩=E_{cc}|{\mathrm{\Phi }}_0⟩$$ 40

which acts as an effective operator. Projecting the Schrödinger equation onto the reference determinant |Φ₀⟩ gives the CC energy

$$E_{cc}=⟨{\mathrm{\Phi }}_0|e^{-\mathrm{T̂}}Ĥe^{\mathrm{T̂}}|{\mathrm{\Phi }}_0⟩$$ 41

whereas the left projection by the determinants corresponding to the substitution level of the cluster operator yields sets of equations that may be solved for the amplitudes.

The equations are nonlinear in the cluster amplitudes and must be solved simultaneously. Unlike CI, CC energy is not variational, meaning that it is not guaranteed to be an upper bound to the exact energy. However, the CC method is size-extensive, ensuring that the energy scales correctly with the number of electrons, and size-consistent, meaning it accurately predicts the energies of noninteracting fragments. These properties are critical for studying large molecules and chemical reactions.

An alternative approach to deriving CC equations uses a Lagrangian framework (which is particularly advantageous for calculating analytic energy derivatives and molecular properties):

$${\mathcal{L}}_{CC}=⟨{\mathrm{\Phi }}_0|(1+\mathrm{\Lambda })e^{-\mathrm{T̂}}Ĥe^{\mathrm{T̂}}|{\mathrm{\Phi }}_0⟩$$ 42

The Λ operator generates substituted Slater determinants in the bra on the left in a manner analogous to T̂ in the right-hand ket wave function. Minimizing ${\mathcal{L}}_{CC}$ with respect to Λ produces the amplitude equations for T̂, while minimization with respect to T̂ gives equations for Λ. Moreover, the Lagrangian formulation puts out that the coupled cluster wave function can be viewed as having two distinct forms:

$$|{\mathrm{\Psi }}_{CC}⟩=e^{\mathrm{T̂}}|{\mathrm{\Phi }}_0⟩$$ 43

$$⟨{\mathrm{\Psi ̃}}_{CC}|=⟨{\mathrm{\Phi }}_0|(1+\mathrm{\Lambda })e^{-\mathrm{T̂}}$$ 44

However, the extension of response theory to coupled cluster is problematic because of this asymmetric nature of the wave functions in eqs and , where the distinct left- and right-hand forms complicate the formulation of an expectation value: there is no guarantee that the time-dependent left- and right-hand coupled cluster wave functions will even yield a real expectation value of Hermitian operators as required in the exact theory, leading to an often-used coupled cluster expression that naturally extracts only the real component. Despite these difficulties, Koch and Jo̷rgensen reported the first application of coupled-cluster response theory in 1990, and several reformulations have been reported since, ,, including extension to higher levels of electron correlation beyond CCSD.

The optical activity tensor, to be used in eq to obtain rotational strengths, takes the following final expression

$$\begin{array}{c}{\mathbf{G}}^{\prime }{(\omega )}_{\mathrm{LR}-\mathrm{CC}}=Im\{{Ĉ}^{\pm \omega }\mathrm{P̂}(\mathit{\mu }(-\omega ),\mathit{m}(\omega ))[⟨{\mathrm{\Phi }}_0|(1+\mathrm{\Lambda })[\mathit{\mu ̅},{\mathrm{T̂}}_m^{\omega }]|{\mathrm{\Phi }}_0⟩\\ +\frac{1}{2}⟨{\mathrm{\Phi }}_0|(1+\mathrm{\Lambda })[[H,{\mathrm{T̂}}_m^{\omega }],{\mathrm{T̂}}_m^{-\omega }]|{\mathrm{\Phi }}_0⟩]\}\end{array}$$ 45

where the overbar denotes the similarity transformation (i.e., e ^–T̂ Ôe ^T̂ ) of the given operator by analogy to the CC-Hamiltonian in eq using the unperturbed operator T̂. The permutation operator Ĉ ^±ω simultaneously changes the signs on the chosen field frequency and takes the complex conjugate of the expression, and P̂ symmetrizes the expression with respect to the perturbations μ and m . T̂ _m represents the perturbed wave function amplitudes (i.e., the Fourier transform of the time evolution of the cluster amplitudes that are obtained by projecting the time-dependent Schrödinger equation for the right-hand wave function onto the set of substituted determinants).

An alternative approach to calculating transition properties within the single reference CC theory is the equation of motion (EOM). The central difference between the CC-LR and EOM-CC is in their respective definitions of the perturbed wave functions: while the former uses an exponentiated perturbation, the latter employs a linear expansion; therefore, the LR approach rigorously considers the relaxation contribution of the ground-state cluster amplitudes induced by the perturbation that produces the electronic excitation, whereas the EOM approach considers “frozen” ground state amplitudes and includes the contribution of the reference determinant to the transition properties. In this case, the optical activity tensor takes a simpler expression to be used in eq

$${\mathbf{G}}^{\prime }{(\omega )}_{\mathrm{EOM}\text{−}\mathrm{CC}}=-Im\{{Ĉ}^{\pm \omega }\mathrm{P̂}(\mathit{\mu }(-\omega ),\mathit{m}(\omega ))[⟨{\mathrm{\Phi }}_0|(1+\mathrm{\Lambda })[\mathit{\mu ̅},{\mathrm{T̂}}_m^{\omega }]|{\mathrm{\Phi }}_0⟩$$ 46

which differs from eq because the terms that are quadratic in T̂ _m are eliminated.

LR and EOM formulations are equivalent for the exact wave function, which considers all the possible excitation clusters, both for excitation energies and transition properties. This is not the case when the wave operator is truncated at a certain order, as happens in production calculations. The two formalisms are still equivalent for the definition of the transition energies, but they differ for the transition properties, such as dipolar and rotational strength: the LR formalism provides size-intensive transition properties, whereas the EOM does not. The counterbalance is the larger computational cost of the LR approach compared to the EOM one, as the ground-state relaxation must be calculated for each excitation.

For small molecular systems, previous studies from Koch et al. have shown, however, small differences in transition dipole moments computed at the CC-LR and EOM-CC levels, and very recently Coriani and co-workers observed good agreement also for ECD spectra computed for both valence and core excitations of L-edge X-ray ECD of methyloxirane, norcamphor, 1,1′-bi-2-naphthol, l-alanine, and l-chloroethanol. A few examples of EOM-CCSD calculations of CPL properties will be provided in Section .

As stated in Section , we note that while the choice of length or velocity representation for the electric dipole operator is inconsequential for exact wave functions, the use of finite basis sets leads to differences between the computed transition strengths as well as arbitrary origin dependence for the rotational strengths calculated using the length representation for all quantum chemical methods. However, CC transition strengths exhibit differences between the two representations even in a complete basis due to the lack of variational optimization of the CC energy with respect to the molecular orbitals used to represent the correlated wave function, like any truncated configuration interaction method. The use of GIAOs, while effective for NMR, does not solve the origin problem for CC approaches, because the response of molecular orbitals is neglected to avoid unphysical poles in the response functions. , However, different solutions to overcome this problem have been proposed. − Yet, it is important to note that for small systems the choice of length versus velocity representation of the electric dipole operator made little qualitative or quantitative difference for rotational strengths for both ECD , and CPL.

2.3.2.1. CC Methods: Performance

Concerning performances on excitation energies, some considerations from a computational cost perspective of CC methods follow. Coupled cluster is iterative, and, depending on the model, each iteration scales with O­(N⁶) (for CCSD), O­(N⁸) (for CCSDT), and more. Alternatives are approximated coupled cluster models, like CC2 and CC3, , which are approximations to CCSD and CCSDT and scale with O­(N⁵) and O­(N⁷), respectively. In particular, the CC2 model has become the “gold-standard” single-reference method for excitation energies of large molecules with single reference character excitations. Concerning transition dipoles, previous benchmarks demonstrated that CC theory including triples (CC3 and CCSDT) provide oscillator strength values that are very accurate; for instance, EOM-CC3 shows a MAE of around 0.002 compared to CCSDTQ (coupled cluster with single, double, triple, and quadruple excitations) or CCSDTQP (P stands for pentuples) results with the aug-cc-pVTZ basis set. We note that the QUEST (QUantum Excited STate) database project allows one to compare multiple data sets and perform statistical analyses to evaluate the accuracy of a given method.

Finally, CC methods provide a robust framework for determining excited-state geometries, balancing computational cost and accuracy. A recent benchmark from Jacquemin and co-workers pointed out that among the commonly employed levels of theory, CC3 delivers geometries that are in excellent agreement with respect to a database of 35 compounds determined at the CCSDR(3)/def2-TZVPP level. CCSD-based methods, such as CCSDR(3), represent a cost-effective compromise, closely approximating CC3 geometries but occasionally underestimating bond lengths by around 0.005 Å. Some care should be taken for CC2 simulations: while computationally efficient, they tend to overestimate bond lengths in excited states, particularly for polar bonds like in carbonyl (CO) and imine (CN) groups, where the deviation can exceed 0.01 Å compared to higher-level methods like CC3 or CASPT2. Additionally, CC2 often exaggerates bond elongation upon excitation, leading to an unbalanced representation of structural changes. These tendencies are particularly pronounced for systems with strong polarization or significant charge transfer character. Statistical analyses and comparisons with experimental data reveal that CC3 outperforms lower-level methods in reproducing accurate bond lengths and geometrical features, aligning closely with experimental observations when single-reference character dominates. For systems with multireference character, CASPT2 (Section ) or multireference CC methods should be considered. Overall, CC3 is recommended for high-accuracy predictions, whereas CCSDR(3) is suitable for larger systems where computational resources are limited. It is finally important to mention that for very large systems an effective approximation for CC methods has been proposed, namely, the domain-based local pair natural orbital coupled cluster. , This approach is a linear-scaling variant of canonical CCSD that exploits the locality of electron correlation to drastically reduce the computational cost while retaining near-canonical accuracy. In DLPNO-CCSD­(T), the virtual orbital space is partitioned into domains centered on localized occupied orbitals, and correlation contributions are evaluated using pair natural orbitals (PNOs) constructed from MP2 pair densities. This method achieves cost scaling close to linear with system size, making it suitable for large molecules and noncovalent complexes. While explicit benchmarks for DLPNO-CCSD are not available, including perturbative triples corrections in DLPNO-CCSD­(T) has shown MAE in excitation energies below 0.01 eV for singlet excitations in organic molecules. Therefore, it is reasonable to infer that DLPNO-CCSD might exhibit slightly larger errors in excitation energies due to the absence of the triples correction.

2.3.3. Algebraic Diagrammatic Construction (ADC) Scheme (*)

The algebraic diagrammatic construction (ADC) method for electronically excited states is based on propagator theory from many-body Green’s function theory. Green’s functions are mathematical tools used to solve inhomogeneous differential equations, and although a single Green’s function cannot be defined for many-body systems like molecular electronic Hamiltonians, useful “propagators” can be identified to solve specific problems. For instance, the one-electron propagator tracks the probability of an electron’s movement over time, while the two-electron propagator deals with correlated electrons. Knowing these propagators allows for the calculation of exact expectation values for one- and two-particle operators as well as properties such as ionization potentials and electron affinities.

The polarization propagator encodes the linear response of the electronic system to a time-dependent perturbation. This can be viewed as time-dependent fluctuations in the ground-state electron density. In mathematical terms, the polarization propagator acts on the ground-state wave function, propagating density fluctuations. To describe these fluctuations, it requires the wave functions of the excited electronic states, making the polarization propagator inherently linked to the electronic excited states of the system, such as those of a molecule. In the context of ADC, the ground-state wave function serves as the starting point for calculating excited states. In many cases, the ground-state wave function in ADC theory can be calculated using Mo̷ller-Plesset perturbation theory (MPn). The ADC method focuses on the response of the system to perturbations starting from this reference wave function.

The ADC hierarchy is denoted as ADC­(n), where n indicates the order of perturbation theory used. For instance, ADC(1) corresponds to the random phase approximation (RPA) and is equivalent to CIS (configuration interaction with single excitations), and ADC(2) and ADC(3) include correlated single and double excitations up to second and third order, respectively. A central feature of ADC is the intermediate state representation (ISR), which replaces the traditional configuration basis with an orthonormal set of intermediate states |Ψ̃ _I ⟩. These are not eigenstates of the electronic Hamiltonian but states generated by applying a set of creation and annihilation operators to the MPn ground-state wave function and are then orthogonalized block-wise according to their excitation class; therefore, ADC­(n) is also reported as “MPn for excited states” in the literature.

The excitation energies and transition momentsrequired to evaluate rotational strengthsare obtained by finding the eigenvectors and eigenvalues of the Hermitian ADC matrix M, which is a representation of the electronic Hamiltonian in the ISR basis, shifted by the exact ground-state energy. The ADC matrix is connected to the diagonal representation of the shifted Hamiltonian in exact states,

$${\mathrm{\Omega }}_{nm}={\delta }_{nm}h{\omega }_n=⟨{\mathrm{\Psi }}_n|Ĥ-E_0|{\mathrm{\Psi }}_n⟩$$ 47

by unitary transformation

$$\mathbf{MX}=\mathbf{\Omega }\mathbf{X}$$ 48

with the eigenvector matrix X. Ground- to excited-state transition moments for a given operator can be evaluated by contracting an eigenvector with the vector of so-called modified transition moments which correspond to the transition moments between the ground-state and the intermediate states. Therefore, ISR allows for efficient evaluation of transition properties (such as electric and magnetic dipoles) by projecting them into this reduced and physically motivated basis. , Full-ADC is formally gauge-invariant. However, this is not true for truncated ADC schemes. The remaining difference between the length and velocity forms of the dipolar and rotational strengths occurs in the highest-order contributions of their perturbational expansion or, in other words, in O­(n) for ADC­(n).

2.3.3.1. ADC: Performance

ADC­(2) and ADC(3) are widely employed for computing electronic excitation energies and exploring excited-state properties. Their performance varies depending on the specific application and the nature of the excited states involved. Benchmark studies have shown that ADC(2) provides excitation energies with mean errors of around 0.22 eV for singlet states and 0.12 eV for triplet states, compared to high-level theoretical estimates. These results indicate that ADC(2) is generally reliable for a broad range of molecules, though it may exhibit larger deviations for certain types of excitations. ADC(2) generally provides valence transition energies as accurate as those obtained with CC2, at a smaller computational cost.

Based on extensive comparisons with both vertical energies determined with higher levels of theory (CCSDT, CCSDTQ, and full CI) and accurate 0–0 energies measured in the gas phase for small- and medium-size compounds, Loos and Jacquemin pointed out that the transition energies computed with ADC(3) in organic compounds are significantly less accurate than their CC3 counterparts. In comparison with CC methods, ADC has the advantage of not requiring the solution of both the left- and right-hand side equations. The computational effort required for ADC­(n) schemes scales as N⁴ for ADC(1), N⁵ for ADC(2), and N⁶ for ADC(2)-x and ADC­(3/2), with N being the number of basis functions. For comparison, the corresponding CC schemes, CC2, coupled-cluster singles and doubles (CCSD), and CC3, formally scale as N⁵, N⁶, and N⁷, respectively, and moreover require an iterative solution for the corresponding ground state.

2.3.4. Multireference Methodologies (CASSCF, CASPT2, and NEVPT2) (*)

The electronic structure approaches based on a single electronic configuration, so-called single-reference methods, often struggle to accurately describe systems with nearly degenerate electronic states or those with significant character involving multiple electronic configurations. This inadequacy stems from their inherent limitations in capturing complex static electron correlation effects. In such cases, multiple electronic configurations contribute substantially to the wave function, and the single-reference nature of traditional methods like Hartree–Fock, DFT, and even CC becomes inadequate (even if multireference methodologies have been developed for CC and DFT). ,

The complete active space self-consistent field (CASSCF) method is a prominent multireference approach that effectively addresses static correlation (the remaining electron correlation is referred to as dynamic correlation and can be accessed by means of perturbation theory, vide infra). It does so by optimizing both the coefficients of a selected set of electronic configurations and the molecular orbitals simultaneously. A configuration is a certain occupation of molecular orbitals, and a given electronic configuration can yield several space- and spin-adapted determinantal wave functions; such functions are referred to as configuration-state functions (CSFs) and are a symmetry- and spin-adapted linear combination of Slater determinants. CSFs are constructed to be eigenfunctions of certain symmetry operators, such as the total spin and its projection, ensuring that the wave function adheres to the required spatial and spin symmetries of the system. CSFs are particularly useful in methods like CI and CASSCF, where an accurate and efficient representation of the wave function is essential.

In CASSCF, the total wave function |Ψ _CASSCF ⟩, is constructed as a linear combination of CSFs within a predefined active space:

$$|{\mathrm{\Psi }}^{CASSCF}(\mathit{\kappa },\mathit{C})⟩=e^{-\mathrm{\kappa ̂}}\sum _I^{CSFs}{C}_I|{\mathrm{\Phi }}_I⟩$$ 49

Here, C _I are the expansion coefficients over the different configurations, |Φ _I ⟩ represent the CSFs, and the orbital variations are described through a unitary transformation, which is conveniently parametrized by using an exponential map e ^–κ̂ . In the CASSCF method, spatial orbitals are divided into three subsets, which are referred to as inactive, active, and virtual orbitals (Figure ). Active orbitals are selected to describe a particular multireference problem (e.g., excitation or bond breaking). Inactive orbitals are always doubly occupied, virtual orbitals are kept empty, and all possible distributions of the remaining electrons among the active orbitals are allowed in the active space to build the different CSFs. Both molecular orbitals (e ^–κ̂ ) and expansion coefficients (C _I ) are variationally optimized using the iterative CASSCF procedure that minimizes the energy:

$$E^{CASSCF}=\underset{\kappa ,C}{\min }\frac{⟨{\mathrm{\Psi }}^{CASSCF}(\mathit{\kappa },\mathit{C})|Ĥ|{\mathrm{\Psi }}^{CASSCF}(\mathit{\kappa },\mathit{C})⟩}{⟨{\mathrm{\Psi }}^{CASSCF}(\mathit{\kappa },\mathit{C})|{\mathrm{\Psi }}^{CASSCF}(\mathit{\kappa },\mathit{C})⟩}$$ 50

5.

Configuration-state functions built in the frame of a CAS­(4,4) active space of single-excited and multiple-excited determinants with respect to the reference wave function.

Starting orbitals can be taken as any set of orbitals (e.g., as canonical orbitals from the Hartree–Fock (HF) method). However, the choice of initial orbitals is obviously important for the convergence of the CASSCF method and its rate. A faster convergence can, for example, be achieved if natural orbitals from preceding iterative steps are used. Sometimes it is also important to improve the description of the unoccupied orbitals of the active space in an initial set of orbitals since the HF virtual orbitals are constructed as an orthogonal complement to the occupied set of orbitals and hence are not well-defined from a physical point of view. In such cases, CIS natural orbitals averaged over several excited states can be used. In other cases, when it is important to define starting orbitals from the chemical point of view for describing processes of bond breaking and forming, an initial set of localized orbitals (e.g., natural bond orbitals, NBOs) can be useful.

Choosing an appropriate active space is crucial for the success of a CASSCF calculation. The active space, often denoted as CAS­(n,m), consists of n active electrons in m active orbitals, and the selection process needs so-called chemical intuition to identify the chemically relevant those orbitals that have the largest influence on the molecular properties of interest (i.e., molecular orbitals that are directly involved in bonding or have a significant role in chemical reactivity and excitation processes). Unfortunately, this can also involve quite a large amount of trial and error, although during the last few years some automation of the active space selection has also been proposed.

The choice of basis set is another crucial factor in CASSCF calculations. A sufficiently flexible basis set is necessary to accurately represent the molecular orbitals and capture electron correlation effects. However, larger basis sets increase computational cost. Therefore, a balance must be struck between accuracy and feasibility. Incorporating polarization functions and diffuse functions can enhance the description of electron distribution, particularly in systems with lone pairs or anionic species. Between the others, atomic natural orbital (ANO) basis sets are specifically designed to provide an efficient and accurate representation of molecular orbitals by incorporating the most significant atomic orbitals derived from correlated calculations: their construction involves averaging density matrices over multiple electronic configurations, including various states of the atom, its cations, anions, and responses to external fields, making them particularly advantageous due to their ability to offer a balanced description of both ground and excited states as well as ionized species.

In the CASSCF method, as well as in its restricted active space (RAS) version, two primary approaches are employed to handle electronic states: state-averaged (SA-CASSCF) and state-specific (SS-CASSCF). In SA-CASSCF, a single set of molecular orbitals is optimized to represent an average of multiple electronic states, typically including both ground and excited states. The weights assigned to each state in the averaging process can be equal or different, depending on the specific requirements of the study. Equal weighting is commonly used to maintain uniformity across states, but in certain cases, different weights may be applied to emphasize particular states of interest. However, assigning different weights should be done cautiously and is generally recommended only for experts in special cases, as it can influence the balance and accuracy of the results. This approach ensures that the orbitals are balanced across the selected states, facilitating the study of excited states and avoiding issues like nonorthogonality and contamination that can arise in single-state calculations. For instance, when optimizing the geometry of an excited state, SA-CASSCF is often preferred to maintain orthogonality and prevent contamination between states. However, state-averaging can give discontinuous energy surfaces due to “root-flipping” when electronic states cross due to torsional motion, and large active spaces are required to capture all of the relevant states; using a common set of orbitals does not account for bespoke orbital relaxation in charge transfer and Rydberg excitations. On the other hand, state-specific approximations can accurately describe high-energy and charge-transfer excitations, beyond the reach of state-averaged calculations with small active spaces. However, the SS-CASSCF energy landscape can have a large number of stationary points, which complicates the selection and interpretation of physically relevant solutions. SS-CASSCF can face convergence difficulties, particularly when multiple electronic states are closely spaced or nearly degenerate, and optimizing orbitals for a single state can lead to symmetry breaking, resulting in unphysical solutions. This is especially problematic when the active space is not adequately chosen.

In summary, SA-CASSCF provides a balanced orbital optimization across multiple states, making it advantageous for studies involving several electronic states. In contrast, SS-CASSCF offers a focused optimization for a particular state, which can yield more precise insights into that state’s electronic structure but may face challenges like root flipping and unphysical solutions. For most spectroscopic applications, SA-CASSCF is generally preferred due to its balanced treatment of multiple electronic states and its ability to mitigate issues like root flipping and state contamination increasing the active space. However, if the focus is on obtaining a highly accurate description of a specific electronic state and the challenges associated with SS-CASSCF can be managed, then SS-CASSCF may be appropriate. The decision should be guided by the specific goals of the study and the nature of the electronic states involved.

While CASSCF provides a balanced account of static correlation, it lacks the capability to incorporate the effects of dynamic correlation, which is crucial for accurately describing excitation energies and transition properties. This limitation motivates the use of perturbative corrections to the CASSCF wave function, leading to the development of multireference second-order perturbation theory approaches such as complete active space second-order perturbation theory (CASPT2) and N-electron valence state perturbation theory (NEVPT2). CASPT2 extends the CASSCF wave function by treating the correlation energy perturbatively. , The zeroth-order Hamiltonian is chosen to be diagonal in the reference space, avoiding strong coupling between states. The total Hamiltonian is partitioned as

$$Ĥ={Ĥ}_0+\lambda \mathrm{V̂}$$ 51

where λ is the perturbation parameter, Ĥ ₀ is a chosen zeroth-order Hamiltonian, and V̂ represents the perturbation. The determination of the second-order correction to energy needs one to compute the first-order correction to the wave function (i.e., to identify and manipulate the first-order interacting space). In practice, one starts by dividing the configuration space into four convenient subspaces: 1) the one-dimensional space of the reference CASSCF wave function ${\mathcal{W}}_0$ ; 2) the configuration space of all possible CASSCF CI expansions consistent with the same the number of active electrons, number of active orbitals, and spin multiplicity as the reference CASSCF wave function but excluding this configuration, ${\mathcal{W}}_K$ ; 3) the space spanned by all possible single and double electron replacements generated from ${\mathcal{W}}_0$ called ${\mathcal{W}}_{SD}$ ; and 4) the complement to the first three subspaces, ${\mathcal{W}}_{TQ···}$ . It is important to point out that only wave functions belonging to ${\mathcal{W}}_{SD}$ interact perturbatively with the reference state: indeed, the subspace ${\mathcal{W}}_K$ consists of configurations that are part of the complete active space but are orthogonal to the reference CASSCF wave function, whereas Slater–Condon rules make null the coupling with wave functions belonging to ${\mathcal{W}}_{TQ···}$ . Concerning ${\mathcal{W}}_{SD}$ , for practical purposes as in MP2, the subspace is sometimes limited by removing virtual and/or core orbitals (the latter is the so-called frozen core approximation). The second-order correction to the energy is derived from Rayleigh–Schrödinger perturbation theory and is given by

$$E^{(2)}=\sum _{i\in {\mathcal{W}}_{SD}}\frac{⟨{\mathrm{\Psi }}_0^{(0)}|\mathrm{V̂}|{\mathrm{\Psi }}_j^{(0)}⟩}{E_0^{(0)}-E_j^{(0)}}$$ 52

where Ψ₀ is the CASSCF reference wave function, Ψ _j are the configurations in the first-order interacting space ${\mathcal{W}}_{SD}$ (i.e., consisting of single and double excitations from Ψ₀ ), and E ₀ and E _j are the zeroth-order (i.e., CASSCF) energies of the reference and excited configurations, respectively. In the context of CPL calculations, CASSCF has been employed in several situations calling for a multiconfigurational approach. The most noteworthy examples are represented by metallohelicenes and other transition-metal complexes, and especially lanthanide complexes (Section ).

2.3.4.1. CASPT2 Formulations and NEVPT2 (*)

CASPT2 has evolved into various formulations to address specific computational challenges and enhance accuracy. Below is an overview of notable CASPT2 variants.

• 1)Multistate CASPT2 (MS-CASPT2) uses a multidimensional reference space that is spanned by two or more state-averaged CASSCF states. An effective Hamiltonian is computed perturbatively and diagonalized within the reference space, permitting the SA-CASSCF states to interact via this effective operator. The MS-CASPT2 formalism is based on multipartitioning of the total Hamiltonian, where a separate reference is used for each reference space state. Use of multipartitioning permits a simple generalization of CASPT2. The MS-CASPT2 wave operator is a simple linear combination of wave operators obtained from separate SS-CASPT2 computations, and the diagonal elements of effective Hamiltonian are equal to the SS-CASPT2 energies. This approach is particularly beneficial for systems with nearly degenerate states or when studying processes involving several electronic states, such as photochemical reactions. By constructing an effective Hamiltonian that encompasses interactions among selected states, MS-CASPT2 provides a more accurate description of electronic structures where state interactions are significant, such as the nonphysical mixings between excited valence and Rydberg states.
• 2)Extended multistate CASPT2 (XMS-CASPT2) further refines MS-CASPT2 by ensuring invariance under unitary transformations of the reference states. This modification is crucial for accurately describing regions near conical intersections and avoided crossings, where electronic states become nearly degenerate and their interactions are complex. By averaging the Fock operator over all states, XMS-CASPT2 maintains consistency in these challenging scenarios.
• 3)CASPT2 with a modified zeroth-order Hamiltonian (CASPT2-K) introduces a modified zeroth-order Hamiltonian to mitigate the intruder state problem, a situation where nearly degenerate, nonreference states cause large perturbative corrections, leading to inaccuracies. By adjusting the zeroth-order Hamiltonian, CASPT2-K aims to stabilize the perturbative expansion and improve the reliability of the computed energies.
• 4)Extended dynamically weighted CASPT2 (XDW-CASPT2) combines the strengths of MS-CASPT2 and XMS-CASPT2 by employing a dynamic weighting scheme. This approach allows the method to adapt between state-specific and state-averaged regimes, providing accurate transition energies and a correct description of avoided crossings and conical intersections. The dynamic weighting ensures that the method performs optimally across different regions of the potential energy surface. ,
• 5)Rotated multistate CASPT2 (RMS-CASPT2) addresses the sensitivity of CASPT2 calculations to the choice of reference states. By rotating the reference states, RMS-CASPT2 reduces the dependence on the number of roots used in the calculation, leading to more stable and reliable results. This variant enhances the robustness of CASPT2 in systems where the selection of reference states is challenging.
• 6)Regularized CASPT2 (σp-CASPT2) introduces a regularization technique to eliminate the intruder state problem without relying on empirical level shifts. This method applies a regularization parameter to the perturbative correction, effectively smoothing out divergences caused by nearly degenerate states. As a result, σp-CASPT2 offers a more systematic and theoretically grounded solution to the intruder state issue.

Each of the CASPT2 variants listed above offers unique advantages tailored to specific computational challenges, such as handling near-degeneracies, ensuring method invariance, and mitigating intruder state problems. The choice of variant depends on the requirements of the system under study and the desired balance between computational cost and accuracy.

To tackle the intruder state problem, various shift techniques have been proposed to be added to the zeroth-order Hamiltonian to mitigate their impact on second-order energy contributions. Another shift technique is the ionization potential-electron affinity (IPEA) shift, introduced to correct systematic errors in open-shell electronic states. It is added to the Hamiltonian, and an optimal IPEA value has been determined through fitting against experimental data. However, recent studies suggest that the IPEA shift may not be necessary for excited-state calculations of organic molecules: while CASPT2 with the IPEA shift slightly better estimate excitation energies for small di- and triatomic systems, it overcorrects for larger molecules, leading to excessively high excitation energies. Therefore, its use is not justified for small- and medium-sized organic molecules. An example of CPL calculations run with CASPT2 combined with IPEA shift will be discussed below (Sections and ).

Finally, a closely related approach is NEVPT2, a multireference perturbation method that, like CASPT2, builds upon a CASSCF reference wave function to account for both static and dynamic electron correlation. In N-electron valence state perturbation theory (NEVPT2), the strongly contracted (SC) variant simplifies computations by using a single perturber function for each excitation class, whereas the partially contracted (PC) variant employs a more extensive set of perturber functions, leading to increased accuracy at the cost of higher computational demand. A notable distinction between NEVPT2 and CASPT2 lies in the choice of the zeroth-order Hamiltonian: NEVPT2 employs the Dyall Hamiltonian, which inherently circumvents the intruder state problem without necessitating empirical adjustments, like using the IPEA shift. This characteristic renders NEVPT2 parameter-free and generally more stable across diverse chemical systems. In the context of CPL simulations, we observed that while NEVPT2 has been effectively applied to determine excitation energies and transition properties, its direct application to rotational strength calculations is less straightforward due to the complexity of incorporating magnetic dipole transition moments within its framework. Consequently, NEVPT2 is often combined with response theories or state interaction approaches to evaluate properties related to optical activity. For instance, the restricted active space state interaction (RASSI) method (vide infra) can be employed post-NEVPT2 to compute transition moments and rotational strengths, facilitating the simulation of CPL spectra (examples in Section ). This combination leverages the robust correlation treatment of NEVPT2 and the capability of RASSI to handle transition properties, thereby providing a comprehensive approach to modeling chiroptical phenomena.

2.3.4.2. Multireference Methods: Transition Moments (*)

The calculation of transition moments between different electronic states is straightforward if in the CI expansion the same orbital basis is used for all states. This is not always the case, especially, but not only, considering the CASSCF method where it may be desirable to perform independent calculations for different states. To take such cases into account, a state interaction approach has been proposed, within the framework of the CAS and RAS approaches, , based on the use of biorthogonal orbitals. The RASSI method makes use of linear transformations of the two different orbital sets, without a large increase in the number of Slater determinants or CSFs that diagonalize the overlap matrix, built up from different orbital sets (for instance those obtained for different CAS states). This can be done for wave functions of the CASSCF type or in general all those wave function types which are “closed under de-excitation”. The result is two new orbital sets (A and B), which fulfill the biorthonormality condition above, and two new CI expansion vectors, which still express exactly the original wave functions, using single determinants defined by the new orbital sets. The transition moments can therefore be obtained by identically the same equations used in a conventional (i.e., orthonormal) basis case for a generic single electron operator (Ô):

$${\mathrm{\Psi }}_1^{CASSCF}=\sum _{\mu }{C}_{\mu }^A{\mathrm{\Phi }}_{\mu }^A;{\mathrm{\Psi }}_2^{CASSCF}=\sum _{\nu }{C}_{\nu }^B{\mathrm{\Phi }}_{\nu }^B$$ 53

$$⟨{\mathrm{\Psi }}_1^{CASSCF}|Ô|{\mathrm{\Psi }}_2^{CASSCF}⟩=\sum _{pq}{O}_{pq}^{AB}{\mathrm{\Gamma }}_{pq}^{AB}$$ 54

$${\mathrm{\Gamma }}_{pq}^{AB}=\sum _{\mu \nu }{C}_{\mu }^A{C}_{\nu }^B⟨{\mathrm{\Phi }}_{\mu }^A|a_p^{†}{a}_q|{\mathrm{\Phi }}_{\nu }^B⟩$$ 55

An important point deserves attention: in the context of second-order perturbative corrections, the RASSI module is still employed to compute transition dipole moments. However, for CASPT2 calculations the method uses the CASSCF wave functions to evaluate the transition dipoles while adopting the CASPT2 energies to determine the energy differences between states. This hybrid approach yields a robust approximation for transition properties, even if it is not a purely CASPT2 approach. In the multistate extension, MS-CASPT2, the transition moments are still computed from the underlying CASSCF wave functions, but these functions are mixed in the same manner as in the MS-CASPT2 procedure, with the corresponding MS-CASPT2 energies providing the correct energy separations.

Multiconfigurational linear response formalisms can also be applied to CASSCF reference wave functions to obtain excitation energies to avoid root flipping. LR-CASSCF can describe excitations that are “outside” the active space: indeed, rather than determining higher states directly from the time-independent Schrödinger equation as with SS- and SA-CASSCF, the time-dependent Schrödinger equation is solved in the framework of time-dependent perturbation theory, as already discussed for TD-DFT and CC methods. LR-CASSCF excitation energies and properties are independent of the number of requested excited states, as in TD-DFT, and often it is sufficient to choose an active space that accounts for degeneracies in the ground-state wave function only to avoid artificial or unphysical mixing of states. However, as any linear response formalism, this approach is still limited to one-electron excitations relative to the ground state and will struggle for problems with a quasi-degenerate ground-state wave function.

2.3.4.3. Multireference Methods: Performance

Benchmark studies evaluating the performance of CASPT2 and NEVPT2 methods in predicting excitation energies and transition properties have been conducted only recently. A comprehensive assessment involving 284 excited states across various molecular systems demonstrated that CASPT2 with IPEA shift and PC-NEVPT2 provide fairly reliable vertical transition energy estimates, with slight overestimations and mean absolute errors of 0.11 and 0.13 eV, respectively. These values are found to be rather uniform for the various subgroups of transitions. The development of the XDW-CASPT2 method has further enhanced the accuracy of excitation energy predictions: benchmarking against a set of 26 organic compounds revealed that XDW-CASPT2 achieves a mean absolute deviation of 0.01–0.02 eV, closely matching the performance of multi-state CASPT2 (MS-CASPT2) and significantly outperforming the XMS-CASPT2 variant, which exhibited a mean absolute deviation of 0.12 eV.

Benchmarks on excited-state geometries are limited: CASPT2 showed excellent agreement with high-level coupled-cluster methods like CC3, with bond length differences typically within 0.003 Å when using large basis sets such as aug-cc-pVTZ. The method accurately predicts CO and CN bond lengths, with deviations from experimental values typically below 0.01 Å. Puckering angles, such as those in formaldehyde derivatives, are well reproduced, with differences within 2–3° compared to CC3.

2.4. Electrostatic/Electrodynamical Environment Effects (From Solvent to a Generic Embedding)

The interaction between a molecular system and its surrounding environment can significantly impact the computed excitation energies, transition dipole moments, and rotational strengths, all of which are critical for the accurate simulation of CPL spectra. The environment influences molecular electronic states through a hierarchy of contributions, including electrostatic and polarization effects, specific phenomena such as hydrogen bonding, and terms of quantum origin like dispersion and repulsion effects. A multiscale approach to the description of molecular processes in the condensed phase must also account for all of these effects, or at least the most relevant ones depending on the specific case. This is quite a challenging task because the network of interactions that determine the behavior of the system and its response properties can be rather complex. Several strategies have been developed, which we list in order of increasing accuracy in the next paragraphs.

2.4.1. Electrostatic Contribution: QM/MM Approaches

At the lowest level of approximation, one can find the QM/MM (quantum mechanics/molecular mechanics) approach, in which the environment is treated just as a source of an electrostatic field that perturbs the system. In most cases, just the partial charges are considered, q _MM, but there are instances where multipoles up to the quadrupoles are included. The drawback of this simple formulation is the lack of any polarization interaction, as the environment is not polarizable. Probably one of the first models developed in the context of QM/MM methods is “our own N-layered integrated molecular orbital and molecular mechanics” (ONIOM) method. , ONIOM is a computational technique designed to model large molecular systems by applying different levels of theory to distinct regions within the system. ONIOM divides the molecular system into multiple layers, each treated with a different computational method. Typically, a high-level QM method is applied to the most critical region (the “model”), while less computationally demanding methods, such as MM or lower-level QM methods, are used for the surrounding environment. This stratification balances accuracy and computational efficiency. The total energy of the system is calculated by combining the energies of the individual layers, and contrary to usual QM/MM approaches, this is done in a subtractive scheme: for instance, in a two-layer ONIOM calculation, the energy of the high-level QM region is corrected by the difference in the energy of the entire system treated at the lower level and the energy of the model region treated at the lower level.

Traditional QM/MM approaches can be computationally demanding when extensive sampling of environmental configurations is required. An alternative to address these challenges by providing a computationally efficient framework that allows for the inclusion of a large number of atomic configurations is provided by the perturbed matrix method (PMM). PMM treats the quantum center (QC) as a quantum system embedded in an external perturbing field, with the surrounding environment modeled at a semiclassical level: the fluctuations of environmental effects with respect to the reference configuration are computed a posteriori by a perturbative approach. In detail, for each frame of the MD trajectory, the system Hamiltonian is written as the diagonal matrix of the eigenvalues of the reference configuration plus a perturbation matrix representing the difference in the electrostatic potential between the considered frame and the reference value (usually a gas-phase one). Diagonalization of this matrix provides a set of eigenvalues (electronic states) representing the instantaneous effects of the embedding environment as provided by the MD trajectories. This approach enables the calculation of perturbed eigenstates and related properties of the QC by expanding the perturbation operator in terms of the external field. Such an expansion allows for efficient computation and facilitates the inclusion of extensive environmental sampling, enhancing statistical accuracy. ,

2.4.2. Including Mutual Polarization: Continuum and Atomistic Polarizable Approaches (*)

The main limit of the standard QM/MM approaches is the missing of the mutual polarization between the QM subsystems and its embedding, but hybrid QM/classical polarizable approaches have been shown to be extremely versatile and easily adaptable to very different problems in chemistry, biology and physics. Here we will focus on the description of the embedding polarization response following an excitation and the proper coupling with the system’s electronic structure. By applying this simplified but effective picture, a quite accurate and complete interpretation is generally achieved for systems of increasing complexity.

Several computational approaches have been developed to incorporate environmental effects into quantum chemical calculations. These can be broadly categorized into continuum solvation models and explicit solvent models. Continuum solvation models, , such as the polarizable continuum model (PCM), treat the solvent as a polarizable dielectric medium, offering a computationally efficient way to capture bulk electrostatic effects. In contrast, explicit solvent models describe individual solvent molecules explicitly, typically using a hybrid QM/MM framework or fully quantum mechanical approaches for small solute–solvent clusters. The electronic structure of the system is determined by the effective Hamiltonian operator, Ĥ _eff , which includes the unperturbed Hamiltonian, Ĥ ₀, plus the potential arising from the presence of the classical environment, V̂ _env :

$${Ĥ}_{eff}|{\mathrm{\Psi }}_I(\mathit{r})⟩=[{Ĥ}_0+{\mathrm{V̂}}_{env}]|{\mathrm{\Psi }}_I(\mathit{r})⟩$$ 56

QM/continuum dielectric models are characterized by a representation of the environment as a structureless dielectric, mainly described by its macroscopic dielectric function, ε, which determines the environment polarization potential (historically called the reaction potential or reaction field) as a response to the presence of a solute. The most widespread formulations of these models, such as the PCM, of which different variants exist, adopt an apparent surface charge (ASC) formulation. The environment polarization is represented by a charge density, σ _S , spread over the surface of the molecular cavity containing the QM system and obtained by solving a classical Poisson problem. The electrostatic problem is discretized by partitioning the cavity surface into small elements (tesserae, τ) and representing the charge density as a collection of partial charges, q _PCM (ε, s _τ), placed on the tesserae positions ( s _τ). More recent implementations of PCM make use of a continuous description of the ASC and employ a geometrically smooth discretization of the solute–solvent interface (Figure ). Another largely used continuum dielectric approach, very similar to PCM, is the conductor-like screening model (COSMO).

6.

(a) PCM cavity (modeled as interlocked spheres) and (b) point charges representing the discretized solvent reaction field to the electronic density of 2-bromo[6]­helicene.

The environment reaction potential, in the case of a continuum model, is written as

$${\mathrm{V̂}}_{PCM}=\sum _{\tau }^{N_{Tess}}\mathrm{V̂}(\mathit{r},{\mathit{s}}_{\tau })q_{PCM}(\epsilon ,{\mathit{s}}_{\tau })$$ 57

The electrostatic potential operator between the quantum electron density at a given position r and the charge on a tessera at s _τ is

$$\mathrm{V̂}(\mathit{r},{\mathit{s}}_{\tau })=-\frac{1}{|\mathit{r}-{\mathit{s}}_{\tau }|}$$ 58

The polarization charge centered at the tesserae position s _τ on the cavity surface is

$$q_{PCM}(\epsilon ,{\mathit{s}}_{\tau })=\sum _{\lambda }^{N_{Tess}}{Q}_{PCM}(\epsilon ,{\mathit{s}}_{\lambda },{\mathit{s}}_{\tau })⟨{\mathrm{\Psi }}_I(\mathit{r})|\mathrm{V̂}(\mathit{r},{\mathit{s}}_{\lambda })|{\mathrm{\Psi }}_I(\mathit{r})⟩$$ 59

Here, N _Tess is the number of tesserae, and Q _PCM (ϵ, s _λ, s _τ) represents the response kernel of the continuum medium and depends on the cavity geometry, discretization, and dielectric constant.

The QM/discrete models are all characterized by retaining information on the environment atomic structure. The interactions within environment, as well as the nonelectrostatic interactions between the system and the environment, are commonly included by means of parametrized force fields (FFs), comprising bonded and nonbonded terms. The electrostatic interactions between the two parts are accounted for in the effective Hamiltonian through an operator which contributes to polarizing the chromophore’s wave function.

2.4.2.1. Polarizable Embedding Schemes: Equilibrium and Nonequilibrium Regimes (*)

Several polarizable embedding schemes have been devised, where the environment polarization is included either through induced dipoles (μ ^ind , ID), , polarizable embedding, , Drude oscillators, or effective fragment potential models.

In the so-called induced dipoles approach, a set of atomic polarizabilities are placed on the environment nuclei, generating electric dipoles in response to the system’s electric field. The problem of finding the response dipoles through an interaction tensor is formally and conceptually analogous to the PCM problem. The form of the classical environment, V̂ _env , is modified to

$${\mathrm{V̂}}_{MMpol}=\sum _i^{N_{at}}\mathrm{V̂}(\mathit{r},{\mathit{s}}_i)q_{MM}({\mathit{s}}_i)-\sum _j^{N_{cls}}\mathrm{F̂}(\mathit{r},{\mathit{s}}_j)·{\mathit{\mu }}^{\mathit{ind}}({\mathit{s}}_j)$$ 60

By analogy to the polarization charges of continuum dielectric models, the induced dipoles are determined as

$${\mathit{\mu }}^{\mathit{ind}}({\mathit{s}}_j)=\sum _k^{N_{cls}}{\mathit{Q}}_{MMpol}({\alpha }_j,{\mathit{s}}_k,{\mathit{s}}_j)⟨{\mathrm{\Psi }}_I(\mathit{r})|\mathrm{F̂}(\mathit{r},{\mathit{s}}_k)|{\mathrm{\Psi }}_I(\mathit{r})⟩$$ 61

In eq , F̂( r , s _k ) is the electric field operator (including also the part due to the fixed charges q _MM ) and Q _MMpol is the rank-2 tensor that describes the response of the environment and depends on the polarizabilities α _j of the N _cls classical atoms of the environment. Discrete approaches are generally better suited to capture anisotropic distributions, which is particularly relevant to account for the interaction with structured environments, such as the biological ones, especially, but not only, at short range. This implies, however, that an average over the possible system-environment configurations must be carried out to study the environment effects in real systems. Conversely, continuum approaches are designed to immediately capture the whole bulk effect of the environment and need no, or rarely very little, configuration sampling. It is worth mentioning that recently a new numerical paradigm for continuum solvation models has been proposed to extend the range of applicability toward very large molecular or supramolecular systems, based on Schwarz’s alternating domain decomposition (dd). Originally proposed in the context of the COSMO continuum approach, dd-COSMO, the domain decomposition paradigm has been extended to PCM.

Once the initial-state quantum mechanical problem is solved, system and embedding polarizations are mutually equilibrated: in the absorption process, the starting electronic wave function will be that of the ground state, whereas in an emissive process (like CPL) there will be an excited-state one. The equilibrium is broken if the system’s charge distribution changes suddenly, as in so-called vertical transition processes. The different characteristic response times of the various degrees of freedom of the embedding may lead to a polarization regime in which the slow components are no longer equilibrated with the final electronic state of the chromophore (i.e., the ground state in CPL), while the fast (electronic) ones are. The resulting regime is usually referred to as nonequilibrium, eventually slowly evolving into a new final equilibrium once all of the environment degrees of freedom, including the slow ones, have relaxed to the final electronic state of the QM system. The equilibrium and nonequilibrium polarization regimes , represent very different environment configurations (particularly for highly polar media), and the energy difference between the two is generally referred to as the environment reorganization energy, λ _env . We will see in Section that this quantity is linked to the line shape inhomogeneous broadening. The nonequilibrium polarization can be properly described within the continuum framework by the separation of a fast (dynamic) and a slow (inertial) polarization where the former is determined by the optical value (ε _∞) of the dielectric permittivity while the latter is calculated as the difference between the full polarization and the fast component, both calculated in equilibrium with the initial charge distribution of the solute. In QM/discrete frameworks, the distinction between dynamic and inertial polarization is even more straightforward, as the atomic multipoles are already naturally associated with the environment nuclei (and therefore with the slow degrees of freedom), while the polarization response, if included in the FF, reflects the fast electronic polarization.

When the specific problem of calculating excited-state energies (which can be thought of as the result of a dynamic process prompted by an external perturbation) is investigated, a further element has to be considered in polarizable models (i.e., the QM electronic scheme used to describe excited states (section )): the state-specific (SS) description requires the explicit calculation of the wave function of all the states of interest whereas the linear response (LR) approach solves for the entire spectrum of excited-state energies (or at least for a large subspace) and the transition densities corresponding to each electronic transition. For gas-phase systems, both approaches are equivalent (assuming that the corresponding equations are solved exactly), but in the presence of a polarizable environment, the two formalisms give rise to a complementary description: the SS-polarizable approach explicitly accounts for the rearrangement of the environment electronic degrees of freedom to adapt to the system excited-state wave function, while the LR-polarizable approach includes the term due to the dynamical response of the environment to the system charge density oscillating at the Bohr frequency. The reason for such discrepancy is mainly due to the presence of the nonlinear polarizable reaction field operator (V̂ _env ) in the effective Hamiltonian and therefore is common to all polarizable methods, both continuum and discrete.

The difference between the two approaches can be clarified by reformulating the deexcitation (or excitation) in solution as a two-step process: in the first step, the molecule in its initial state in equilibrium with the solvent polarization is deexcited (or excited in absorption) to the final state in the presence of a solvent polarization frozen in the initial electronic state; let us call ω _n0 the resulting deexcitation energy. In the second formal step, the dynamic component of the solvent polarization response (R _dyn ) rearranges to equilibrate with the final electronic state charge density of the chromophore. The deexcitation energy therefore changes, and one can adopt two different expressions corresponding to an LR or an SS polarization response description. A comparable expression for both frameworks can be written as

$$\Delta E_{n0}^{(LR)}={\omega }_{n0}^{(0)}+R_{dyn}^{LR}(P_{n0}^{trans})$$ 62

$$\Delta E_{n0}^{(SS)}={\omega }_{n0}^{(0)}+R_{dyn}^{SS}(\Delta P_{n0})$$ 63

The SS-polarization approach explicitly accounts for the environment’s polarization response to changes in the system’s charge density upon electronic transitionfully in the case of the equilibrium response and partially for the nonequilibrium onethereby encompassing both electrostatic and polarization effects. In contrast, the LR-polarization scheme models the dynamic response of the environment to the system’s transition density, hence partially approximating dispersion interactions between the solute and solvent. In principle, both effects (i.e., SS polarization and dispersion) should be considered to properly describe solute–solvent interactions. ,

The SS-polarization scheme, requiring that the change in electron density due to the excitation is known, is naturally coupled with state-specific wave function formulations such as CI or complete active space (CAS) approaches and accounts for the electrostatic equilibration of the solvent to the excited-state density. , A known limitation of state-specific solvation models is that the electronic states are not eigenfunctions of a common Hamiltonian, since each state is optimized in its own reaction field. Therefore, the resulting adiabatic states are not mutually orthogonal. This nonorthogonality poses conceptual and computational challenges for the evaluation of transition propertiesmost notably transition dipole moments and oscillator strengthssince such quantities rely on the assumption of orthonormal electronic wave functions. Some workarounds exist, such as adopting perturbative correction to the energy. − This problem is instead automatically solved when the solvent polarization is coupled to a CAS electronic structure description by the state-interaction procedure (Section ).

The LR-polarization formulation is the most widely used in combination with single reference methods that make use of the linear response electronic structure description, such as TD-DFT, TD-HF, LR-CC, or ADC(2). In terms of computational convenience, electronic structure LR formulations are particularly advantageous compared to state-specific wave function ones, mainly because a whole set of transition energies and properties can be obtained at once. For this reason, they often represent the optimal compromise between accuracy and computational cost and are the most widely used methods for the calculation of excitation energies of chemically relevant systems. To overcome the shortcomings discussed above when a polarizable environment is present, various models have been introduced to recover an SS-polarization of the environment response in the context of a linear response electronic structure formalism. In the framework of time-dependent self-consistent field (TD-SCF) methods, like TD-DFT and TD-HF, this correction can be achieved through the extension to analytical gradients and the ES relaxed density.

In the context of continuum models and single reference noncorrelated methods, three main approaches have been implemented to date: the corrected LR (cLR), the external iteration scheme (sometimes erroneously reported as SS-PCM in the literature), and the vertical excitation method (VEM). The cLR of Caricato et al. was the first to be introduced. The solvent polarization response, R _dyn (ΔP _n0), is computed using the excited-state relaxed density obtained from an LR-PCM calculation and then added as a perturbative correction to the solvent polarization frozen in the initial electronic state, ω _n0 , obtained solving the TD-SCF equations in a gas-phase-like calculation. Later, Improta et al. introduced an external iteration approach that iterates over the complete excited-state density to compute a new ground-state reaction field operator. This is added to the Fock operator as an external field and used to recompute the orbitals, the TD-SCF transition energy, and a new relaxed density. The process is repeated until self-consistency. This scheme is reminiscent of the perturbation-to-energy and density (PTED) procedure introduced in the context of correlated methods (see below). However, it is largely prone to numerical instabilities because the ground state is no longer variational. The VEM approach is a different iterative implementation where the reference ground state is kept unchanged, and only the response contribution, R _dyn (ΔP _n0), is updated using the relaxed density matrix until self-consistency. The first iteration of the VEM corresponds to a cLR scheme, but here the relaxed density is obtained from the frozen reaction field. The VEM model has been proven to be self-consistent and variational, and this opened the way to implement analytical gradients of the energies to compute not only state-specific polarization absorption and emission energies but also electronic properties (both for state and transition properties) and excited-state geometries. , To date, the VEM approach is the only SS-polarization approach in the context of TD-DFT that includes analytical gradients.

In the context of correlated wave function theory, the situation is even more complicated due to the influence of the treatment of electron correlation in the system–environment interaction on an equal footing. , Three different schemes to include the explicit correlation in the PCM contribution were proposed originally, which were coupled to the SS description. − In the perturbation-to-energy (PTE) scheme, the self-consistent field problem is solved in the presence of the environment reaction field. The resulting orbitals are used to compute the amplitudes without modifying in-vacuo-like equations. In the perturbation-to-density (PTD) scheme, a post-SCF one-body density matrix is computed in vacuo and then used to compute the solvation energy. The PTED (perturbation to energy and density) scheme can be thought of as the iterative combination of PTE and PTD, where the PTE relaxed density is used to update the reaction field operator, which is used to solve a new set of QM equations until self-consistency. The cLR approach has also been extended to CC methods in solution. An important difference between this formulation and that used in TD-DFT is that no orbital relaxation is considered, for the sake of consistency with the other SS formulation of CCSD-PCM. The coupling between the reaction field and the electronic correlation is neglected in both PTE and PTD schemes. Concerning the PTED implementations, self-consistency can also be achieved with approximations where the GS-SCF solution is recomputed with an updated operator or where only the post-SCF equations are solved again. , These latter cases, indicated as PTE­(S) and PTES, are analogous to a VEM scheme within correlated wave function methods. The PTES scheme has been implemented in both the SS and LR formalisms including their gradients for ES geometry optimization within the EOMCCSD formalism. ,

Very recently, these schemes have also been included to couple continuum models to the algebraic diagrammatic construction (ADC) electronic-structure methods. ,,

In the context of many-body perturbation theory, the coupling of the so-called GW approach and PCM was implemented within an integral equation formalism (IEF, including also the nonequilibrium effects). One of the main (GW + PCM) formalism features is that the polarization energies for all occupied and virtual energy levels are state-specific. Finally, the extension to the Bethe-Salpeter equation of PCM has been introduced recently. For an extensive discussion, interested readers are referred to a recent review and references therein.

Irrespective of which SS-polarization scheme is in use, it is very important to properly simulate emission energy including the nonequilibrium solvent polarization of the excited state at the ground one. In particular, the inertial part of the reaction field due to the excited state is determined with an equilibrium excited-state calculation and saved. It is then used in the second calculation to recompute the GS reaction field but with the inertial part due to the ES.

Concerning the atomistic models of the environment, the largest part of them were first implemented in an LR-polarization fashion, such as the QM/MMpol (that is based on induced dipoles, eq ) the QM/MM fluctuating charges and fluctuating dipoles model QM/FQFμ, and the polarizable embedding approach. In the QM/FQFμ force field, each MM atom is endowed with both a charge and an atomic dipole that can vary according to the external electric potential and electric field. Both charges and dipoles are described as s-type Gaussian distribution functions (to avoid any “polarization catastrophe”). It is an extension of the pristine QM/FQ approach based on just the inclusion of charges that can vary as a response to the differences both in electronegativity between MM atoms and in electric potential generated by the QM density. The PE model employs a fragment-based quantum-classical explicit embedding scheme: the environment is represented by a multicenter multipole expansion to model electrostatic interactions, with polarization effects accounted for by dipole–dipole polarizabilities placed at the expansion points. This setup allows for fully self-consistent mutual polarization between the quantum region and its environment. However, over the years, some state-specific polarization approaches have also been presented. For instance, the cLR scheme has been included in both QM/MMpol , and QM/FQFμ. State-specific implementations have also been proposed in the contest of the polarizable embedding approach. By analogy to the continuum methods, multireference approaches have been coupled to polarizable MM methods and therefore can be considered to be intrinsically state-specific approaches. In view of the application of explicit solvent models to emissive phenomena such as fluorescence and CPL, it is important to stress that, unlike continuum models that represent an averaged solvent response and thus allow for meaningful excited-state optimizations, atomistic models correspond to a single frozen configuration of the environment. As a result, optimizing the excited-state geometry of the solute in such a fixed configuration is not physically meaningful. A rigorous description would require excited-state molecular dynamics to sample solvent relaxation around the excited-state solute, but such simulations are computationally demanding and not always routinely feasible. Further discussion of this point is presented in Section , in the context of nuclear ensemble approaches.

A further point deserves a comment: in principle both state-specific medium polarization and dispersion interactions should be included to properly treat the complex and retarded response of a polarizable bath (the environment) to a perturbation of a quantum system (the solute). Depending on the expression used for dispersion, LR-PCM energy can recover part of the solute–solvent dispersion interaction, and recently the cLR² method was proposed in the context of TD-DFT both in continuum and atomistic models. A similar approach was also presented in the context of coupling ADC(2) and PCM. To the best of our knowledge, however, none of these corrections to the excited-state energies have been applied to CPL calculations yet.

2.5. CPL Band Shape Simulation

The complete spectroscopic characterization of a molecular system requires an accurate understanding of the structure of a spectrum (i.e., the line shape that modulates the intensity of a signal). Simulating the band shape of an absorption or an emissive phenomenon, like CPL, in molecular systems, requires accounting for various physical effects influencing electronic transitions. Key factors to consider include the following:

• 1)Vibronic Modulation: The interaction between electronic and nuclear vibrational degrees of freedom. The vibronic couplings that affect the intensity through the modulation of transition dipole moments are due to intramolecular vibrations and can allow for intensity borrowing in otherwise weak transitions: they do not inherently cause broadening but determine the vibronic peak structure.
• 2)Static and Dynamic Disorder of Embedding: The disorder within the systems, both static (structural differences in the local environment around the chromophore) and dynamic (thermal fluctuations of the local environment), affects the distribution of electronic energies and transition probabilities. The first kind is mainly related to the inhomogeneous broadening of the spectral bands, and the second, to the homogeneous one. The vibrational degrees of freedom (of both the chromophore and environment) can modulate the line shape function, L _CPL (ω), inducing homogeneous and inhomogeneous broadening. If a vibrational mode is fast, compared to emission lifetime, then it contributes to homogeneous broadening by causing a pure dephasing (i.e., the loss of phase coherence of the final state that in general is a coherent superposition of vibrational states in the electronic ground state) without energy relaxation. This process results in the broadening of spectral lines, providing insight into the dynamics of molecular systems and their surroundings. Vice versa, if a vibrational mode is slow or if different conformers have different vibrational characteristics, then it leads to inhomogeneous broadening.
• 3)Embedding Polarization Effects: We showed in Section that electrostatic/electrodynamical environment effects can influence the energies, ES structures, vibrations, and transition moments. However, the environment polarization response can also modulate the band shape by an inhomogeneous broadening due to the solvent reorganization energy.
• 4)Aggregation Effects: When molecules aggregate, their electronic states can interact, leading to the formation of excitonic states. This interaction results in spectral shifts of excitonic energy levels. The coupling of excitonic states with vibrational modes (both intramolecular and intermolecular) results in specific spectral features, including broadening and shifts due to discussed static and dynamic disorder. Additionally, aggregation can enhance electron–vibration coupling, further contributing to line broadening. (These phenomena are particularly relevant in systems exhibiting aggregation-induced emission, AIE, where nonemissive monomers become emissive upon aggregation). We will discuss these latter effects in Section .

2.5.1. Modeling of Vibronic Effects (*)

The vibronic structure in emission spectra, and therefore also CPL spectra, arises from the coupling between electronic transitions and nuclear vibrations: indeed, the values of transition moments, and therefore the rotational strength, are shaped by the underlying nuclear dynamics. In a fully quantum mechanical treatment, the total wave function of a molecule encompasses both electronic and nuclear degrees of freedom, meaning that optical transitions must be considered in terms of vibronic states rather than purely electronic ones. Therefore, in principle, one must describe vibronic modulation in a fully nonadiabatic framework, treating the molecular wave function as an entangled electronic–nuclear state. From a computational point of view, there are two broad categories for the treatment of nonadiabatic dynamics: wave function-based approaches, which solve the time-dependent Schrödinger equation (TDSE) for the nuclear degrees of freedom beyond the adiabatic approach, and trajectory-based approaches, which evolve classical or semiclassical nuclear trajectories while allowing for quantum electronic transitions.

If nonadiabatic effects can be safely ignored (i.e., avoiding nearly degenerate electronic states, rapid nuclear dynamics, and strong nonadiabatic coupling), then a system can be treated under the Born–Oppenheimer approximation and the molecular wave function is factorized as the product of electronic and nuclear wave functions:

$${\mathrm{\Psi }}_I^{BO}={\psi }_I^{el}{\psi }^{nuc}\approx {\psi }_I^{el}{\chi }^{Tr}{\chi }_l^{Rot}{\chi }_{\nu }^{vib}$$ 64

For what follows, we will consider explicitly only the vibrational contribution (χ_ν ) to the nuclear part, integrating out the translational (χ ^Tr ) and rotational (χ _l ) degrees of freedom. The transition moments (here generically indicated by the operator Ô) that are included in the definition of the rotational strength take the following general expression:

$$⟨{\psi }_n^{el}{\chi }_{{\upsilon }^n}^{vib}|\mathit{Ô}|{\psi }_0^{el}{\chi }_{{\upsilon }^0}^{vib}⟩=⟨n{\upsilon }_n|\mathit{Ô}|0{\upsilon }_0⟩=⟨{\upsilon }^n|{\mathit{O}}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})|{\upsilon }^0⟩$$ 65

The shorthand notation introduced in eq for electronic and vibrational states should be self-evident. We integrated over the electronic coordinates, obtaining the so-called electronic transition moment O _{n 0} ( Q _n ). This can be expanded in a Taylor series with respect to the normal coordinates ( Q _n ) of the n electronic states around the equilibrium geometry ( Q _n ):

$${\mathit{O}}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})={\mathit{O}}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})})+\sum _a{\frac{\partial {\mathit{O}}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})}{\partial {\mathit{Q}}_{\mathit{n},a}}|}_{{\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}}{\mathit{Q}}_{\mathit{n},a}+···$$ 66

The sum is here extended over all of the normal modes, a, and the expansion is truncated to the second term. From eqs and , one finally obtains the vibronic expansion of electric and magnetic dipole transition moments:

$$⟨{\upsilon }^n|{\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})|{\upsilon }^0⟩\approx {\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})})⟨{\upsilon }^n|{\upsilon }^0⟩+\sum _a{\frac{\partial {\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})}{\partial {\mathit{Q}}_{\mathit{n},a}}|}_{{\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}}⟨{\upsilon }^n|{\mathit{Q}}_{\mathit{n},\mathit{a}})|{\upsilon }^0⟩$$ 67

$$⟨{\upsilon }^n|{\mathit{m}}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})|{\upsilon }^0⟩\approx {\mathit{m}}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})})⟨{\upsilon }^n|{\upsilon }^0⟩+\sum _a{\frac{\partial {\mathit{m}}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})}{\partial {\mathit{Q}}_{\mathit{n},a}}|}_{{\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}}⟨{\upsilon }^n|{\mathit{Q}}_{\mathit{n},\mathit{a}})|{\upsilon }^0⟩$$ 68

The first term on the right side of eqs and represents the Franck–Condon (FC) term, whereas the second is the Herzberg–Teller (HT) one. FC, HT, and FCHT simulations of the spectra therefore refer to including respectively only the first, only the second, or both terms to compute the total rotational strength R _n0 in eq . The HT term recovers the fact that even a small displacement along normal coordinates can mix electronic states involved in the transition with close-lying states, due to the vibronic coupling. This effect is captured within the framework of first-order perturbation theory and provides an approximate treatment of nonadiabatic effects.

Within the Born–Oppenheimer approximation, the simulation at room temperature of the vibrational line shape associated with electronic spectral bands is nowadays at hand for large semirigid molecules thanks to the advances in both time-independent (TI) and time-dependent (TD) techniques.

2.5.1.1. Time-Independent (Sum-Overstates) Approach (*)

The time-independent approach (often referred to as the sum-overstates method) − calculates vibronic spectra by summing over all possible transitions from the thermally populated vibrational levels |υ _i ⟩ in the initial electronic state |n⟩ whose Boltzmann population is ρ_{υ i} , to those |υ _f ⟩ in the final ground electronic state |0⟩, for an emission phenomenon

$$\begin{array}{c}\Delta I{(\omega )}_{relaxed}=\frac{2}{3}{\omega }_l{N}_0\tau L_{CPL}(\omega )\left(\frac{{{\omega }_l}^3}{\pi c^3}\right)[Im({\mathit{\mu }}^{\mathit{n}\mathbf{0}}·{\mathit{m}}^{\mathbf{0}\mathit{n}})]\\ =\frac{2\tau {\omega }_l{N}_0}{3\pi c^3}{{\omega }_l}^3{L}_{CPL}(\omega )[Im({\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}){\mathit{m}}_{\mathbf{0}\mathit{n}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})})]\sum _{{\upsilon }^n,{\upsilon }^0}^{N\mathrm{mod}}{\rho }_{{\upsilon }_i^n}{|⟨{\upsilon }_i^n|{\upsilon }_f^0⟩|}^2\\ +\sum _{a}Im\left({\frac{\partial {\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})}{\partial {\mathit{Q}}_{\mathit{n},a}}\frac{\partial {\mathit{m}}_{\mathbf{0}\mathit{n}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})}{\partial {\mathit{Q}}_{\mathit{n},a}}|}_{{\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}}\right)\sum _{{\upsilon }^n,{\upsilon }^0}^{N\mathrm{mod}}{\rho }_{{\upsilon }_i^n}{|⟨{\upsilon }^n|{\mathit{Q}}_{\mathit{n},\mathit{a}})|{\upsilon }^0⟩|}^2\\ +\sum _{a}Im({\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}){\frac{\partial {\mathit{m}}_{\mathbf{0}\mathit{n}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})}{\partial {\mathit{Q}}_{\mathit{n},a}}|}_{{\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}}+{\mathit{m}}_{\mathbf{0}\mathit{n}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}){\frac{\partial {\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})}{\partial {\mathit{Q}}_{\mathit{n},a}}|}_{{\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}})\sum _{{\upsilon }^n,{\upsilon }^0}^{N\mathrm{mod}}{\rho }_{{\upsilon }_i^n}⟨{\upsilon }_i^n|{\upsilon }_f^0⟩⟨{\upsilon }^n|{\mathit{Q}}_{\mathit{n},\mathit{a}})|{\upsilon }^0⟩]\\ =C({\omega }_l){{\omega }_l}^3{L}_{CPL}(\omega )[w_{CPL}^{FC}+w_{CPL}^{HT}+w_{CPL}^{FC/HT}]\end{array}$$ 69

where the three latter symbols refer to the FC, HT, and mixed FC/HT components of vibronic rates with respect to the electric and magnetic dipole transition definition.

The calculation of FC and HT vibrational integrals is computationally demanding. Early analytical formulations were proposed by Sharp and Rosenstock for FC integrals involving up to four simultaneously excited modes and later extended to the HT case by Baranov et al., though their approach was impractical for general application. The need for truncating infinite summations remains a general challenge in vibronic spectroscopy, requiring prescreening methods to identify relevant transitions: while energy-based prescreening is commonly used, it becomes inefficient for broad spectral windows or temperature-dependent simulations. Alternative approaches have been suggested, each with varying efficiency, generality, implementation ease, and computational cost. − Most vibronic CPL calculations found in the literature rely on the methods implemented in the FCclasses code by Santoro and co-workers.

2.5.1.2. Time-Dependent Vibronic Approach (*)

Despite the developments of efficient prescreening techniques, time-independent methods still suffer from the fundamental problems related to the huge number of transitions to be considered for large-size systems and of the intrinsic arbitrariness of any prescreening algorithm. The alternative time-dependent approach offers a viable route for avoiding these shortcomings since it exploits the properties of the Fourier transform, leading to fully converged spectra that include temperature effects, without additional computational cost. ,−

The expression in eq can be reformulated in the time domain (TD approach) as the Fourier transform of the finite-temperature time correlation function, χ­(t, T): ,

$$\Delta I{(\omega )}_{relaxed}=\frac{C({\omega }_l){{\omega }_l}^3{L}_{CPL}(\omega )}{2\pi }{\int }_{-\infty }^{+\infty }dt\chi (t,T)e^{-i({\omega }_{n0}-\omega )t}$$ 70

The time correlation function is

$$\chi (t,T)=Tr[⟨{\upsilon }^n|{\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})|{\upsilon }^0⟩e^{-i{H}_{n}t}⟨{\upsilon }^n|{\mathit{m}}_{\mathbf{0}\mathit{n}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})|{\upsilon }^0⟩e^{-i(\beta -it)H_0}]=\int d\mathit{Q}[⟨\mathit{Q}|⟨{\upsilon }^n|{\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})|{\upsilon }^0⟩e^{-i{H}_{n}t}⟨{\upsilon }^n|{\mathit{m}}_{\mathbf{0}\mathit{n}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})|{\upsilon }^0⟩e^{-i(\beta -it)H_0}|\mathit{Q}⟩]$$ 71

where β is the inverse thermal energy (1/k _B T) and H ₀ and H _n are the Hamiltonians of states 0 and n, respectively. The use of a Taylor expansion of both transition dipole moments in eqs and , by analogy to the TI approach, gives rise to the FC, FC/HT, and HT/HT terms:

$${\chi }^{FC}(t,T)=Im({\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}){\mathit{m}}_{\mathbf{0}\mathit{n}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}))\int d\mathit{Q}[⟨\mathit{Q}|e^{-i{H}_{n}t}{e}^{-i(\beta -it)H_0}|\mathit{Q}⟩]$$ 72

$$\begin{array}{c}{\chi }^{FC/HT}(t,T)=\sum _{a}Im\left({\frac{\partial {\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})}{\partial {\mathit{Q}}_{\mathit{n},a}}|}_{{\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}}{\mathit{m}}_{\mathbf{0}\mathit{n}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})})\right)\int d\mathit{Q}[⟨\mathit{Q}|{\mathit{Q}}_{\mathit{n},\mathit{a}}{e}^{-i{H}_{n}t}{e}^{-i(\beta -it)H_0}|\mathit{Q}⟩]\\ +\sum _{a}Im\left({\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}){\frac{\partial {\mathit{m}}_{\mathbf{0}\mathit{n}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})}{\partial {\mathit{Q}}_{\mathit{n},a}}|}_{{\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}}\right)\int d\mathit{Q}[⟨\mathit{Q}|e^{-i{H}_{n}t}{\mathit{Q}}_{\mathit{n},\mathit{a}}{e}^{-i(\beta -it)H_0}|\mathit{Q}⟩]\end{array}$$ 73

$${\chi }^{HT/HT}(t,T)=\sum _a\sum _{b}Im\left({\frac{\partial {\mathit{\mu }}_{\mathit{n}\mathbf{0}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})}{\partial {\mathit{Q}}_{\mathit{n},a}}\frac{\partial {\mathit{m}}_{\mathbf{0}\mathit{n}}^{\mathit{el}}({\mathit{Q}}_{\mathit{n}})}{\partial {\mathit{Q}}_{\mathit{n},\mathit{b}}}|}_{{\mathit{Q}}_{\mathit{n}}^{(\mathbf{0})}}\right)\int d\mathit{Q}[⟨\mathit{Q}|{\mathit{Q}}_{\mathit{n},\mathit{a}}{e}^{-i{H}_{n}t}{\mathit{Q}}_{\mathit{n},\mathit{b}}{e}^{-i(\beta -it)H_0}|\mathit{Q}⟩]$$ 74

For harmonic PES, analytical expressions have been derived for FC and FC + HT (FC/HT and HT/HT) terms using path integral techniques and Gaussian integration. The interested reader can find details and extended references in the original publication by Barone and co-workers.

2.5.1.3. Harmonic Approximation of PES: Adiabatic vs Vertical Models and the Dushinsky Effect (*)

The harmonic approximation of potential energy surfaces is an essential simplification used in the simulation of vibronic spectra, including CPL. Since electronic transitions occur between two PESseach associated with a different electronic stateone must define an appropriate reference geometry where the expansion of the PES is performed. The choice of this reference leads to different harmonic models, each with advantages and limitations. The most widely used models are the adiabatic Hessian (AH) and vertical Hessian (VH) approaches, along with the vertical gradient (VG) model, which is an even simpler approximation of VH. ,

In the AH model, the final-state PES is expanded around its own equilibrium geometry (x ₀ ), ensuring that normal-mode frequencies and vibrational displacements are properly defined within each electronic state. Mathematically, the PES expansion is given by

$$V(\mathit{Q})=E_{ad}+\frac{1}{2}{\mathit{Q}}^{\mathit{T}}\mathit{HQ}$$ 75

where Q represents the normal coordinates, H is the Hessian, and E _ad is the adiabatic (i.e., at the GS and excited minima) excitation energy difference. The relationship between the normal coordinates in the initial and final electronic states is governed by the Duschinsky transformation

$${\mathit{Q}}_{\mathbf{0}}=\mathit{JQ}+\mathit{K}$$ 76

where J is the Duschinsky rotation matrix, describing the mixing of vibrational modes, and K is the displacement vector between equilibrium geometries:

$$V({\mathit{Q}}_{\mathbf{0}})=E_{ad}+\frac{1}{2}{\mathit{K}}^{\mathit{T}}\mathit{JH}{\mathit{J}}^{\mathit{T}}\mathit{K}-{\mathit{K}}^{\mathit{T}}\mathit{JH}{\mathit{J}}^{\mathit{T}}{\mathit{Q}}_{\mathbf{0}}+\frac{1}{2}{{\mathit{Q}}_{\mathbf{0}}}^{\mathit{T}}\mathit{JH}{\mathit{J}}^{\mathit{T}}{\mathit{Q}}_{\mathbf{0}}$$ 77

The VH approach is an alternative quadratic model of the final-state PES that can be built simply on the grounds of vertical data: data computed at the initial state geometry Q ₀ , namely, the vertical excitation energy (E _V ), the gradient ( g ₀ ), and the Hessian ( H ₀ ) of the excited PES:

$$V({\mathit{Q}}_{\mathbf{0}})=E_V+{\mathit{g}}_{\mathbf{0}}^{\mathit{T}}{\mathit{Q}}_{\mathbf{0}}+\frac{1}{2}{{\mathit{Q}}_{\mathbf{0}}}^{\mathit{T}}{\mathit{H}}_{\mathbf{0}}{\mathit{Q}}_{\mathbf{0}}$$ 78

This approach is more suitable for describing spectral line shapes near the band maximum, as it better represents the actual FC region of the PES. However, the VH model has practical limitations, as the Hessian is computed at a nonstationary point, leading to deficiencies in the description of low-frequency modes and possible spurious imaginary frequencies. These issues can be mitigated by using curvilinear internal coordinates, rather than Cartesian ones, for the vibrational analysis. It is possible to settle a simpler model than VH, called VG and also known in the literature as the linear coupling model (LCM) or the independent-mode displaced harmonic oscillator (IMDHO): it is assumed that in the initial and final states the normal modes are simply displaced. A simplified representation of adiabatic and vertical models is shown in Figure .

7.

Illustration of (a) vertical and (b) adiabatic emission processes between an excited state S₁ and a ground state S₀. Legend: VH, vertical Hessian process; VG, vertical gradient; AH, adiabatic Hessian; Q ^j _eq equilibrium geometry for the jth state; E ⁱ (Q ^j _eq), energy for the ith state evaluated at the equilibrium geometry for the jth state; ΔE ^j _reorg, reorganization energy for the jth state; ΔQ, displacement; ∂E/∂Q, energy gradient.

Finally, it has been suggested that marked differences between AH and VH predictions suggest the existence of relevant anharmonic effects. Anharmonicity arises from the deviations of a PES from the idealized harmonic approximation, where molecular vibrations are assumed to behave as independent harmonic oscillators. In real systems, the PES is often anharmonic, meaning that vibrational energy levels are not equally spaced, and mode couplings introduce complex spectral features beyond the predictions of harmonic models. While the AH and VH models rely on the harmonic approximation, anharmonic corrections become necessary for flexible molecules and low-frequency modes such as torsions.

2.5.2. Modeling the Broadening Due to Static and Dynamic Disorder (*)

Beyond the possible vibronic modulation of intensities, the bands of experimental spectra have a non-negligible width due to various broadening effects, whose origins have been summarized at the beginning of this section (Section ) and usually are included in a line-shape function L _CPL (ω), adopting different strategies.

Homogeneous broadening refers to the uniform widening of spectral lines in molecular spectra, arising from mechanisms that affect all molecules in a sample equally. The primary contributors to homogeneous broadening are the finite lifetimes of excited states, the pure dephasing due to the vibration of both chromophore and the environment (usually referred generically as phonons), and the collisional broadening (observed particularly in dense phases or high-pressure gases), where frequent collisions with surrounding molecules perturb the energy levels, causing fluctuations in transition energies.

The homogeneous broadening can be effectively modeled with a Lorentzian line shape function:

$$L_{CPL}^{HB}(\omega -{\omega }_{0n})=\frac{1}{\pi }\frac{\mathrm{\Gamma }}{{(\omega -{\omega }_{0n})}^2-{\mathrm{\Gamma }}^2}$$ 79

The finite excited-state lifetime introduces an intrinsic spectral width, related to a total decay rate (including radiative and nonradiative channels), the collision frequency, and the pure dephasing rate (i.e., $\mathrm{\Gamma }=\frac{1}{\tau }+{\mathrm{\Gamma }}_{coll}+{\mathrm{\Gamma }}_{deph}$ ). This broadening is therefore mapped into a dynamic disorder (i.e., when the fluctuations of energy transitions occur on a time scale comparable to or even faster than the excited-state lifetime). Often one can extract Γ as an experimental parameter or simulate it ab initio (vide infra).

Inhomogeneous broadening arises from the static distribution of different molecular conformations and environment configurations affecting the energy levels of a chromophore. The relationship between solvent reorganization energy and spectral broadening can be understood through the energy fluctuations induced by solvent dynamics. Variations in solvent polarization lead to a distribution of solvation energies, which translates into a spread of electronic transition energies of the solute molecules. This spread manifests as inhomogeneous broadening in the absorption or emission spectra. The methods based on implicit solvent models, such as those of the PCM family, allow one to account for “mean” (in the statistical sense) solvent effects on the solute PES: not only the vibrational progressions but also the inhomogeneous broadening in molecular spectra will change.

Marcus’s theory, originally formulated to describe electron-transfer reactions, provides a foundational framework for understanding how solvent dynamics contribute to this broadening, due to polar interactions between the solute and the medium. It is based on a treatment of solvent nuclear degrees according to classical statistics. − The spectral shift due to the solvent is obtained by averaging over all possible solvent configurations while broadening arises from the fluctuations in the energy difference in the initial and final electronic states of the transition. The solvent inhomogeneous broadening takes a Gaussian line shape with a half-width at half-maximum (HWHM)

$$L_{CPL}^{IB}(\omega -{\omega }_{0n})=\frac{1}{\sqrt{2\pi \sigma }}{e}^{-{(\omega -{\omega }_{0n})}^2/2{\sigma }^2}$$ 80

The variance of the electronic transition energy is

$${\sigma }^2=2\lambda k_{\mathrm{B}}T$$ 81

where λ is the polar reorganization energy (i.e., the polar contribution to the difference between the nonequilibrium and equilibrium Helmholtz free energy in the final electronic state at the FC solute geometry).

In the context of emissive phenomena simulated with polarizable dielectric models, the computation of reorganization energy involves the state-specific energy difference at the ES minima of emissive states between the equilibrium (i.e., using the static dielectric constant) and the nonequilibrium (i.e., setting the optical dielectric constant) values.

While continuum models provide a rapid estimation of solvent-induced shifts, atomistic descriptions are necessary to capture local solvation and disorder effects beyond the electrostatic/electrodynamical ones. The simplest model considers the solvent as a bath of harmonic modes, and an analogous expression to that of Marcus can be obtained in a fully quantum approach describing the solvent as a set of normal oscillators displaced by the electronic transition. Therefore, one can extract σ from the distribution of the vertical transition energies, computed over the snapshots extracted from MD simulations, and then include it in the computation of eq . To enforce the separation of solute and solvent motions, during the MD simulation the solute is kept frozen at the GS equilibrium geometry. Therefore, at the simplest level, one can compute the total CPL spectrum including vibronic effects using a TI or a TD description and the band shape broadening due to the product of eqs and using a continuum dielectric or a frozen chromophore-MD simulation to estimate static disorder. ,

In the nuclear ensemble approach, , the electronic spectra are simulated using molecular configurations that are sampled from different snapshots during a molecular dynamics (MD) trajectory at a specified temperature. For emission, excited-state MD is performed, with the energy gap calculations on these snapshots representing the vertical deexcitation energy. To obtain the final optical spectrum, often a Gaussian function dresses the bare vertical energies. Alternatively, one can obtain the nuclear configurations from a quantum quasi-probability distribution function, and the Wigner distribution is one of the most used.

A further development with respect to this approach includes the ensemble Franck–Condon approaches: for each independent snapshot sampled from a classical MD, one assumes a separation of time scales for the nuclear redistribution of the environment upon electronic excitation of the chromophore to justify freezing the atomistic environment during a geometry optimization of the chromophore in its ground and excited states. At these optimized geometries, the normal modes are computed within a harmonic approximation to the potential energy surfaces and the frequencies, displacements, and relative rotations between ground- and excited-state normal modes are used to compute Franck–Condon spectra. The final spectrum is an average over the different snapshot of the different FC spectra. Different approximations to this procedure exist, and interested readers can find an extensive review.

Along the same line of approaches is the adiabatic molecular dynamics generalized vertical Hessian (Ad-MD|gVH) method: it is a mixed quantum-classical approach rooted in a partition of stiff/flexible modes that makes use of a vertical Hessian approach that avoids optimizing the geometry of the excited state and therefore decreases the computational cost of computing the FC spectra. Ad-MD|gVH has been shown to accurately reproduce absorption and emission spectra of flexible organic dyes in different solvents (including hydrogen bond interactions) , but also flexible dyes in environments of increasing complexity. On the other hand, one can microscopically model the homogeneous broadening and then use MD sampling to include the inhomogeneous one. Indeed, in the Condon approximation, the absorption/emission line shape depends on the evolution of the nuclear wave packet on the final PES, which can be expressed perturbatively through a cumulant expansion of the so-called spectral density (SD). This function describes the interaction of the system with vibrational modes in a time-averaged manner and is derived from time-correlation functions of energy gaps, since the electric dipole correlation function, which defines the spectra in the TD formalism, can be expressed in terms of energy gap fluctuations as a consequence of the perturbation theory in the interaction picture.

The spectral density is used in the response function formalism to compute the line shape function directly. This approach naturally incorporates homogeneous broadening due to vibrational dephasing, which includes intramolecular vibrations and system–bath interactions. Given the equilibrium wave packet trajectory of the nuclei on the excited-state PES, the energy gap fluctuations are defined as

$$U(t)=\mathrm{\Delta }E(t)-⟨\mathrm{\Delta }E⟩$$ 82

where ΔE is the de-excitation energy gap and its autocorrelation function is

$$C(t)=⟨U(t)U(0)⟩$$ 83

The real part of C(t) is even, whereas the imaginary part is odd and its Fourier transform C̃(ω) is a real and positive function. The even C̃′(ω) and odd C̃′′(ω) parts of C̃(ω) are related: ,

$${\mathrm{C̃}}^{\prime }(\omega )=\coth \frac{(\beta \hslash \omega )}{2}{\mathrm{C̃}}^{″}(\omega )$$ 84

Equation expresses a detailed balance relation for positive frequency ω. C̃″(ω) is usually called SD in the framework of theoretical spectroscopy and is indicated as J(ω). Since C(t) represent a statistical distribution, it can be expanded in terms of its moments, which are statistical measures describing the shape and characteristics of the distribution of U(t). The cumulant expansion offers an alternative framework that focuses on the cumulants of the distribution. The first cumulant is the mean, the second cumulant is the variance, and the third cumulant is the same as the third central moment, whereas the fourth and higher-order cumulants are not equal to central moments.

By truncating the cumulant expansion to the second order, one defines the line shape function g(t):

$$g(t)={\int }_0^{t}d{t}_1{\int }_0^{t_1}⟨U(t_2)U(0)⟩d{t}_2={\int }_0^{+\infty }\mathrm{d\omega }\frac{J(\omega )}{{\omega }^2}[\coth \frac{(\beta \mathrm{\hslash }\omega )}{2}·(\cos (\omega t)-1)-i(\sin (\omega t)-\omega t)]$$ 85

From the line shape function, the homogeneous absorption line shape can be obtained as the real part ( $\mathfrak{R}$ ) of the Fourier transform of the line shape function: ,

$$L_{CPL}^{HB}(\omega -{\omega }_{0n})=\mathfrak{R}{\int }_{-\infty }^{+\infty }{dte}^{i(\omega -{\omega }_{0n})t}{e}^{-g(t)}$$ 86

Valleau et al. derived the expressions for J(ω) for a collection of harmonic oscillators whose initial conditions follow the Boltzmann distribution, suggesting a relationship between J(ω) and a classical autocorrelation function C _Cl (t):

$$J(\omega )=\frac{(\beta \omega )}{\pi }{\int }_{-\infty }^{+\infty }{dte}^{i\omega t}{C}_{Cl}(t)$$ 87

Therefore, one can use classical simulations to get an estimate of line shape broadening, since the classical autocorrelation function of the de-excitation energy gap can be obtained from the de-excitation energies ΔE(t _i ) at given times t _i along the MD trajectory:

$$C_{Cl}(t_j)=\frac{1}{N-j}\sum _{i=1}^{N-j}\mathrm{\Delta }E(t_i+t_j)\mathrm{\Delta }E(t_i)$$ 88

Accurate computation of the classical autocorrelation function necessitates careful consideration of simulation parameters to effectively capture vibrational dynamics across different frequency ranges: low-frequency vibrations (≤100 cm^–1), corresponding to periods of around 300 fs, require simulation trajectories extending over several ps to ensure adequate sampling. In contrast, high-frequency modes demand shorter time steps, typically less than 5 fs, to accurately resolve their rapid oscillations. A common challenge in these calculations is the artificial persistence of C _Cl (t) at longer times due to the finite length of the simulation window, leading to spurious fluctuations. To mitigate this, the original time window can be divided into smaller, overlapping segments, and the resulting autocorrelation functions can be averaged. Additionally, applying damping functions, such as Gaussian or exponential decays, ensures that C _Cl (t) approaches zero at the window’s end, effectively simulating vibrational dephasing and resulting in Gaussian or Lorentzian broadening of spectral density peaks. Interested readers can find extended reviews and tutorials in the recent literature. , The inhomogeneous distribution is finally obtained from a set of N_runs independent trajectories: from each dynamics run, an average vertical emission energy and rotational strength are obtained and finally combined with the homogeneous line shape (as combining eqs –) to obtain a homogeneous broadened spectrum for each run. In this framework, the final inhomogeneous CPL spectrum is just the average over different runs, by analogy to what is done for absorption spectra. Interestingly, one could possibly also include vibronic effects in each run: indeed, even when the vibronic structure is hidden by broadening, the coupling of electronic transitions to nuclear degrees of freedom can result in an asymmetrical line shape.

A final point needs a comment concerning the molecular dynamics approach used to sample the molecular structures in the excited state: there are substantially two types of possible approaches: performing a completely classical MD based on an excited-state parametrization of the force field − or Born–Oppenheimer molecular dynamics (BOMD) in which the classical nuclei evolve on top of the excited-state PES, directly coupling the electronic structure of the excited state into the nuclear motion provided the analytical gradients of the selected method are available. Each approach has its advantages and limitations: classical MD with parametrized force fields offers computational efficiency but may lack accuracy in capturing complex excited-state phenomena whereas BOMD provides a more precise depiction of excited-state dynamics but at a higher computational cost. The choice between these methods depends on the dimension of the system and balancing accuracy and computational resources.

2.6. Modeling of Molecular Aggregates

Chiroptical spectroscopies are intrinsically sensitive to supramolecular organization, where the molecular emitters can form dimers, oligomers, or extended aggregates. The interactions modify the photophysical response significantly compared to single-molecule behavior, often inducing new spectral features or amplifying chiroptical signals: the emission energy and rotational strength are no longer properties of an isolated chromophore but emerge from delocalized excitations (so-called excitons) influenced by geometry, intermolecular couplings, vibrational structure, and environmental disorder. Below we describe different strategies for the modeling of CPL in aggregates, ranging from atomistic models to exciton-based approaches, with a focus on the Frenkel model extended to include vibronic effects. A few literature examples of CPL calculations based on such approaches will be described in Section . Finally, an overview of periodic boundary conditions will also be given.

2.6.1. Atomistic Models: Simulation of Oligomers

The simplest strategy to account for aggregate effects is the explicit construction of small oligomers (dimers, trimers, tetramers, and so on) whose initial geometry is sampled from molecular dynamics (MD) simulations or extracted from crystal structures. Quantum chemical calculations on these supramolecular assemblies can yield absorption and emission energies as well as transition electric and magnetic dipole moments. The CPL spectrum is then computed using the same formalism as for isolated molecules but applied to a larger quantum system: this allows for direct access to the chiroptical response of the aggregate and includes, in principle, all relevant interactions (π–π stacking, hydrogen bonding, and charge transfer). However, such calculations are computationally demanding; therefore, the electronic structure description is almost invariably limited to TD-DFT. In this case, during optimization, it becomes important to include dispersion correction effects like Grimme’s dispersion empirical corrections series DFT-D. ,

Vibronic structure in aggregate spectra is of course harder to analyze and often not included explicitly. The sampling of small but flexible oligomers often needs an exhaustive description of the first excited-state PES: in this case, some ps-long TD-DFT BOMD can give access to a representative set of configurations , to account for the thermal and structural fluctuations within the aggregate. The calculated electronic properties (excitation energies, dipolar strengths, and rotational strengths) from each snapshot can then be statistically averaged to obtain the overall optical response of the aggregate, as already discussed in Section .

When aggregates grow beyond a few monomer units or form extended structures like crystals or polymers, periodic boundary conditions (PBC) and embedding techniques become necessary. A common scheme is a hybrid QM/MM partitioning, where the chromophore and nearby molecules are treated quantum mechanically while the rest of the matrix is described classically, either as static charges or via polarizable embedding. An effective way to include the presence of the other systems composing the matrix can be the use of a multilayer ONIOM approach (Section ), such as QM:QM′ or QM:QM′:MM ONIOM. These kinds of approaches can include mechanical, electrostatic, and even quantum mechanical (at the QM′ level) interactions of the oligomer with the rest of the system.

Systems characterized by long-range ordering effects, layered structures, or flexible chiral molecules, where high-energy barriers between conformations can hinder exhaustive sampling, could in principle benefit from enhanced sampling techniques that improve the exploration of relevant conformational landscapes. These methods facilitate a more thorough understanding of the conformational characteristics of chiral systems, thereby enabling more accurate predictions of their optical responses. Within these methods, a bias potential is added to the Hamiltonian of the systems to overcome the free-energy barrier, thereby reaching the transition states, like in umbrella sampling, metadynamics, the adaptive biasing force method, and steered molecular dynamics, among many others. To date, to the best of our knowledge, such an approach has been suggested but never applied to CPL simulations.

2.6.2. Large Aggregates: The Frenkel Exciton Model Including Vibrations and Disorder (*)

In physics, an exciton is a quasiparticle formed due to an electronic transition to a higher energy state, leaving behind a hole in its original state. The electron and hole are bound together by their mutual Coulomb attraction, forming a neutral entity that can transport energy without transporting a net electric charge. In molecular aggregates, due to the interactions between neighboring molecules in the aggregate, the electron and hole can “delocalize” and spread across multiple molecules. This delocalization leads to collective behavior, where the excitation is no longer confined to a single molecul, but is spread across a region of the aggregate called the coherence length.

A powerful approach for treating extended molecular aggregates is the Frenkel exciton model, , where the electronic excitation is delocalized over multiple molecules and the aggregate is treated as a set of coupled systems through interactions due to their electronic transitions. The electronic excitonic Hamiltonian that describes the eigenstates of the multichromophoric system is approximated in terms of the excitations of the single chromophores (with site energies ε _i and wave functions on site |Φ _i ⟩) and the corresponding electronic couplings among them (V _ij or off-diagonal elements):

$${Ĥ}_{exc}=\sum _i^{N_{sites}}{\epsilon }_i|{\mathrm{\Phi }}_i^{0n}⟩⟨{\mathrm{\Phi }}_i^{0n}|+\sum _{i\ne j}^{N_{sites}}{V}_{ij}|{\mathrm{\Phi }}_i^{0n}⟩⟨{\mathrm{\Phi }}_j^{0n}|$$ 89

In the most general form, V _ij includes Coulomb, exchange, and overlap contributions: these terms can be computed ab initio (e.g., through the efficient grid-based integration methods used in DFT or with transition density cube or fragment-based methods) or approximated using transition multipolar moments models (point charges, dipoles, quadrupoles, and so on). The excitonic ground state (Ψ₀ ) can be written as a Hartree product of the localized unperturbed ground states of each single chromophore (φ _i )­

$${\mathrm{\Psi }}_0^{Exc}=\prod _i^{Nsites}|{\phi }_i^0⟩$$ 90

whereas the excitonic wave functions (Ψ _K ) can be written by considering a linear combination of wave functions (Φ _i ) expressed as a product of a single electronic transition (0 ← n) per chromophoric unit (φ _i ) at a certain time:

$${\mathrm{\Psi }}_K^{Exc}=\sum _i^{N_{sites}}{C}_{i,0n}^K|{\mathrm{\Phi }}_i^{0n}⟩$$ 91

$$|{\mathrm{\Phi }}_i^{0n}⟩={\phi }_i^{0n}\prod _{j\ne i}^{Nsites}|{\phi }_j^0⟩$$ 92

The expansion coefficients (C _i,0n ) are obtained by diagonalizing the excitonic Hamiltonian matrix, eq , and the transition moments of the aggregate are then obtained as the linear combination of the corresponding ones relative to the transitions on each chromophore. A detailed derivation is provided in the Supporting Information of ref .

The rotational strength of an excitonic state K, originating from the coupling of couples of chromophores, i with transition (0 ← a) and j with transition (0 ← b), amounts to

$$R^{\mathit{K}\mathbf{0}}=\mathrm{Im}\left\{\sum _i^{Nsites}\sum _{a,b}^{exc}{C}_{i,a0}^K{C}_{i,0b}^K[{\mathit{\mu }}_{i,a0}·{\mathit{m}}_{j,0b}+i\frac{{\omega }_{j,0b}}{2}{\mathit{\mu }}_{i,a0}·{\mathit{R}}_{ij}\times {\mathit{\mu }}_{j,0b}]\right\}$$ 93

where μ and m are, respectively, the transition electric dipole moment and the intrinsic transition magnetic moment (i.e., with respect to the center of mass of the aggregate). R _ij is the relative position of the two chromophores and ω _j,0b is the transition frequency of the jth chromophore.

The very popular exciton chirality method , instead provides a simplified expression derived from a classical multipolar expansion, where only the second term of the sum in the brackets is considered. This expression relies solely on electric dipole–dipole coupling and assumes that the magnetic dipole contribution is negligible or that chirality arises purely from the spatial arrangement of the transition dipoles. While useful for quick qualitative insights, this approximation breaks down when the monomer transitions have non-negligible magnetic dipole moments or electric and magnetic dipoles are not colinear and their interference is significant or when there are multiple nearly degenerate transitions with strong excitonic coupling, where magnetic terms mix nontrivially. As shown by some of us, , systems like bis-phenanthrenes and bis-BODIPY exhibit substantial magnetic–electric contributions, and the sign of the rotational strength can be inverted compared to the chirality rule prediction. Hence, a fully quantum mechanical evaluation of both electric and magnetic transition moments is essential to capturing the true nature of the CPL and ECD signals in multichromophoric chiral systems. This feature is currently implemented in the excitonic analysis tool (EXAT) code , that includes the gauge-independent formulation of the excitonic rotational strength both in length and velocity formalism. Finally, it is also possible to define generalized Frenkel Hamiltonians, including more than one transition per monomer or defining oligomeric units to include charge-transfer states. ,

Disorder can have a strong influence on the photophysical behavior of molecular aggregates and may be considered by assuming a statistical distribution for the values of the site energies (diagonal disorder) and interchromophoric couplings (off-diagonal disorder), using the approaches discussed in Section . Diagonal disorder originates from local electrostatic variations, solvent effects, or structural heterogeneity in the aggregate, while off-diagonal disorder typically arises from fluctuations in intermolecular distances and orientations. These effects lead to localization of the excitonic wave function over a finite number of chromophores, thereby reducing the coherence length and altering the interference patterns among transition dipole and magnetic dipole moments that determine the CPL response.

2.6.3. Very Large Aggregates: Nearest-Neighbor Approximation (*)

Often, in the case of very large aggregates, a simplified version of the Frenkel exciton model can be employed by restricting the coupling to nearest neighbors only. This approximation is particularly appropriate for extended π-stacked systems, such as helically ordered oligomers or polymeric assemblies where identical chromophores repeat periodically. In these cases, restricting the coupling to adjacent chromophores provides an analytically tractable framework that captures the essential physics of the system. This simplification is justified by the rapid spatial decay of dipole–dipole (Förster-type) interactions and leads to a tractable form of the aggregate Hamiltonian. The excitonic states that result from this coupling are delocalized over the aggregate and can be described as collective excitations whose energies are spread into a band. The nature of this band and thus the optical properties of the aggregate are determined by the sign of the excitonic coupling: positive values lead to so-called H-aggregates, with blue-shifted absorption and optically forbidden lowest-energy states, while negative values give rise to J-aggregates, where the lowest excitonic state is optically allowed and the spectrum is red-shifted with enhanced emission. , Unlike inorganic materials, organic aggregates are structurally soft, and electronic excitations are accompanied by nuclear rearrangements. These systems therefore require a treatment that includes both electronic and vibrational degrees of freedom. This is typically accomplished using the so-called Frenkel–Holstein model, where each chromophore interacts with its own local intramolecular vibrational mode. The strength of this coupling is quantified by the Huang–Rhys factor, a dimensionless parameter that measures the displacement between the ground and excited states PES along a specific vibrational coordinate. Physically, it reflects how many vibrational quanta are typically excited during an electronic transition. A small Huang–Rhys factor indicates weak vibronic coupling and sharp spectral features, while larger values lead to pronounced vibronic progressions. In H-aggregates, vibronic coupling results in suppression of the purely electronic (0–0) transition due to destructive interference, enhancing higher-order vibronic bands (e.g., 0–1), while in J-aggregates, constructive interference enhances the 0–0 transition, making it the dominant feature in the spectrum. Finally, the disorder can be included by modeling site energies with a Gaussian distribution, effectively capturing the impact of energetic inhomogeneities on exciton dynamics. Such disorder leads to a reduction in the coherence length of excitons and induces localization. This directly impacts the chiroptical response and particularly the CPL dissymmetry factor, which becomes a sensitive probe of the spatial delocalization of the excitonic wave function: in the strong disorder regime, where excitons are confined to individual chromophores, the CPL signal vanishes. As disorder decreases and excitonic delocalization increases, the dissymmetry factor grows, reflecting enhanced coherence across the aggregate. Remarkably, in certain regimes, g _lum (ω) can even undergo sign inversion depending on the helical pitch and number of chromophores per turn, emphasizing the complex interplay among structural chirality, exciton delocalization, and vibronic coupling in determining the chiroptical behavior of the system.

2.6.4. Simulation of Solid-State CPL Spectra: Periodic Boundary Conditions and Methodological Challenges (*)

While simulations of ECD and CPL spectra in isolated molecules are now standard practice, their application to molecular crystals and thin films still pose some challenges. The inclusion of PBC ensures that the system’s bulk properties are accurately represented by replicating the unit cell infinitely in all directions. This treatment is essential for simulating materials like organic semiconductors and OLEDs, where the excitons are delocalized across the crystal lattice. Recently, Caricato presented a methodological perspective on the simulation of chiroptical signals in periodic systems, identifying both promising directions and critical gaps. The theoretical foundation for CPL (and ECD) in solids involves computing the full rotational strength tensor, and for periodic systems, a key complication arises from the lack of translational invariance of the standard electric and magnetic dipole operators or even the electric quadrupolar moment for nonisotropic systems (eq in Section ). This issue has been addressed through an approximate form of transition moments in reciprocal space, using k-space derivatives of crystalline orbitals to define transition moments in a gauge-consistent manner. − While these formulations have primarily been developed for optical rotation, they should, in principle, be extendable to ECD and CPL simulations, although as of now no direct applications to CPL have been reported.

A bridge between excitonic models and full periodic quantum chemical simulations is offered by tight-binding approaches, where the Hamiltonian includes only local interactions such as nearest-neighbor couplings. These models, sometimes coupled with excitonic model descriptions by Woody and Tinoco, , even if often efficient, are system-specific and require reference data for the parametrization of the Hamiltonian. Furthermore, longer-range interactions may be important when excited-state properties are involved, especially with sensitive properties such as optical activity. ,

From a methodological standpoint, the most rigorous and generalizable strategy would be to adapt molecular linear-response methods to periodic systems via PBCs. Given the size of the typical chromophores involved, probably the only accessible approach would most likely be obtained with TD-DFT due to the cost and accuracy ratio. Even if the first implementation for monodimensional periodic systems was proposed by Hirata and co-workers, the developed standalone program was just focused on excitation energies, and the method has not been generalized to tridimensional periodicity. More recent developments in TD-DFT for solids have targeted UV/vis absorption spectra , often employing the Tamm–Dancoff approximation (TDA) which improves efficiency but may compromise accuracy. Additionally, these implementations are frequently restricted to calculations at the Γ-point, which simplifies the equations approximating the coefficients of crystalline orbitals as real, thus reducing the accuracy. Choosing an appropriate exchange-correlation functional presents further complications. While range-separated hybrids are generally preferred for describing optical activity and charge-transfer excitations, their integration into PBC frameworks remains difficult due to the computational challenges associated with evaluating long-range exact exchange. Basis set limitations also persist: while molecular calculations benefit from large Gaussian-type basis set for accurate rotational strengths, such basis functions are problematic in PBC simulations due to the linear dependence between images of the unit cell. Similarly, plane-wave basis sets (which inherently respect translational symmetry) require a high cutoff and large basis expansions, making them computationally demanding. Hybrid schemes involving both Gaussian and plane-wave functions have been proposed, but they still suffer from similar issues related to basis set overlap and redundancy. In conclusion, despite meaningful advancements, several methodological developments are still necessary to make solid-state CPL simulations broadly accessible and predictive. Key areas for future progress include (i) TD-DFT-PBC implementations that correctly incorporate gauge-invariant transition moments and long-range exchange; (ii) efficient algorithms for excited-state geometry optimization under periodic conditions; and (iii) rigorous treatment of vibronic and exciton–phonon couplings within a periodic framework. Additionally, machine leaning (ML) approaches, if trained on proper reference data for aggregated states, can offer an alternative and computationally efficient way toward the prediction of CPL spectra in the solid state. Finally, we note that fragment-based methods, like the molecules-in-molecules (MIM) approach, are employed to simulate chiroptical properties of large systems like polypeptides, particularly those with vibrational optical activity. One possible development of these methods could be the extension to periodic systems, allowing for efficient calculation of ECD or CPL signals that are otherwise intractable with traditional methods.

2.7. Key CPL Parameters and Features for Experiment–Calculation Comparisons

In many cases, a CPL spectrum is much simpler with respect to its ECD counterpart. Nevertheless, it is characterized by a number of experimental features. The first apparent feature is the number of bands observed, related to the possible presence of multiple emitting states. Most of the time, in the case of organic compounds only a single emission band is observed, as emission stems almost exclusively from the lowest excited state, according to Kasha’s rule. More involved situations may occur in the presence of different emitting states, as in the case of excimers and especially for lanthanide and other metal complexes. Another straightforward feature to consider is the (main) emission wavelength of the compound (λ_em). Most of the organic CPL-active compounds emit in the blue region (400–500 nm), but there is significant interest in more red-emitting compounds, which could find applications in devices or as CPL-reporters. In this latter case, low-energy emission allows for better penetration of tissues and minimizes photodamage. Also relevant is the energy or wavelength difference between the emission and red-most absorption maxima, namely the Stokes shift, which offers a glimpse of several different phenomena, such as solvent relaxation in the excited state, excited-state reactions, and excimer or exciplex formation. Parameters such as g _lum and its ratio with g _abs, λ_em, and the Stokes shift are both experimentally and computationally accessible, and represent the main means to evaluate the quality of calculations, on one side, and to use them for molecular design, on the other side. Additionally, QM calculations offer several other pieces of information which have no immediate experimental counterpart but also help in the interpretation of CPL data and in the rational design of efficient CPL emitters. Both kinds of parameters and quantities are summarized in this section.

2.7.1. Emission Dissymmetry Factor (g _lum)

As in emission spectroscopy, CPL intensity is always given in arbitrary units, which are instrument- and experiment-specific; therefore, it is useful to discuss the results in terms of the g factor. For most organic compounds, g _lum is constant throughout the emission bands, and for this reason, most authors report the g _lum estimated only on the CPL/fluorescence intensity maximum (I _max). On the other hand, in many cases it is still worth considering the actual dependence of g _lum versus wavelength. To this end, as the denominator of the g _lum expression goes to zero at the spectral edges, it may be convenient to slightly correct it to avoid fluctuation of the plots for I ≈ 0, as ,

$${\mathrm{g̃}}_{\mathrm{lum}}=\frac{\Delta \mathit{I}}{\mathit{I}+\delta }$$ 94

with δ being a numerical constant chosen in a way such that δ ≪ I _max, so that g̃ _lum = g _lum away from the edges of the spectrum. In general, even in the presence of a single band, rotational strength may be scattered over several vibronic contributions. In this case, the g _lum calculated for I _max is not representative of the emission optical activity. If a single g value is needed, as for comparison with the results of nonvibronic calculations, then an average g _lum should be calculated as

$$g_{\mathrm{lum}}=\frac{\int \Delta \mathrm{I}(\mathrm{\nu ̅})·{\mathrm{\nu ̅}}^{-4}\mathrm{d}\mathrm{\nu ̅}}{\int \mathrm{I}(\mathrm{\nu ̅})·{\mathrm{\nu ̅}}^{-4}\mathrm{d}\mathrm{\nu ̅}}=\frac{\int \Delta \mathrm{I}(\lambda )·{\lambda }^2\mathrm{d\lambda }}{\int \mathrm{I}(\lambda )·{\lambda }^2\mathrm{d\lambda }}$$ 95

where the integrals in the numerator and denominator are proportional to R _n0 and D _n0 respectively. From a computational perspective, a purely electronic calculation affords R ₁₀ and D ₁₀ and then g _lum at the S₁–S₀ transition energy (wavelength), for example, at the poles of LR functions for TD-DFT (Section ). Simulation of g _lum at different wavelengths, namely, the g _lum(λ) dispersion, is fully available through CPL band shape simulation (Section ), although this is seldom done in the literature.

In order to correctly estimate g _lum from experimental spectra, the possible presence of self-absorption and circular extinction due to the sample itself must be considered. This is especially important if the sample shows a small Stokes shift and relatively strong ECD bands (partially) overlapping with the CPL spectrum and the measurement is carried out on relatively concentrated conditions. For the sake of example, considering a positive ECD band (A _L > A _R) overlapping with a same sign CPL signal (I _L > I _R), the emitted light will be partially self-absorbed and in particular left-polarized light will be more absorbed than right-polarized light. This leads to a decrease in the CPL signal, and in some extreme cases, it can also cause an apparent reversal of the CPL sign. A simplified way to account for this issue is to correct the observed CPL (CPL_obs) following the method proposed by Abbate, Castiglioni, Longhi, and co-workers. , The corrected CPL (CPL_corr) can be recovered to first order by taking into account the observed fluorescence (F_obs) and the transmittance (T), as

$${\mathrm{CPL}}_{\mathrm{corr}}=\frac{1}{\mathrm{T}}·{\mathrm{CPL}}_{\mathrm{obs}}+\frac{1}{\mathrm{T}}{\mathrm{F}}_{\mathrm{obs}}·\alpha ·\ln (10)·\mathrm{\Delta A}$$ 96

with α being a constant depending on the excitation-collection geometry. Equivalently, the corrected g _lum (g _corr) is obtained from the observed one (g _obs) as

$$g_{\mathrm{corr}}=g_{\mathrm{obs}}+\alpha ·\ln (10)·\mathrm{\Delta A}$$ 97

Figure shows a simulation of the g _lum profile in the presence of partially overlapping ECD bands of different intensity, with an ECD/CPL maximum shift of 30 nm. Interestingly, an ECD signal of ∼6 mdeg overlapping with the CPL spectrum in a certain spectral region will offset the true g _lum by ∼1 × 10^–3, which may be of the same order of magnitude as the true g _lum itself, and a larger ECD intensity will even change the CPL sign. In most cases, these issues are avoidable by working with diluted solutions.

8.

Simulation of g _lum profile (dashed lines) in the presence of corresponding ECD bands (continuous lines). Solid and dashed black lines represent CPL and g _lum, respectively, in the absence of any self-extinction. g _lum curves are simulated according to eq for a 90° excitation geometry (α = 2.5); a corrected g _lum = 1 × 10^–3 is assumed with Gaussian-shaped ECD and CPL bands centered at λ = 470 and 500 nm, respectively (bandwidth was 30 nm in all cases).

2.7.2. Ratio of Emission and Absorption Dissymmetry Factors (g _lum/g _abs)

The fundamental equations gauging the emission and absorption dissymmetry factor (eqs and ) are functionally identical, as they have the same mathematical dependency on μ, m, and θ _μm . If the ground (|0⟩) and excited (|j⟩) electronic states involved in the emission (|j⟩ → |0⟩) are the same as those involved in the |0⟩ → |j⟩ absorption transitions, then one may expect that the emission and absorption dissymmetry factors are similar in magnitude and have the same sign. This should be especially true if vibronic coupling does not play a significant role, and |0⟩ and |j⟩ states have a similar geometry. Note that in most organic compounds |j⟩ corresponds to the S₁ state due to Kasha’s rule; therefore, the relevant g _abs to consider in this context is the one associated with the most red-shifted ECD band.

In 2018, Mori et al. analyzed the g _lum -vs-g _abs correlation for several compounds belonging to different classes of organic compounds (namely, chiral cyclic ketones, planar chiral cyclophanes, axially chiral biaryls, chiral BODIPY derivatives, and helicenes and helicenoids), finding significant, although scattered, correlations. More recently, a good g _lum -vs-g _abs correlation was found by focusing on 170 helicenes and helicenoids showing a great extent of structural variety. The analysis results indicate that the g _lum/g _abs ratio is narrowly distributed around 0.77, with 95% of the ratio values being positive, which means that the same sign is expected for CPL and the lowest-energy ECD band in virtually all cases for this class of compounds. On the other hand, this correlation is expected to fail when very different excited states are involved in absorption and emission, such as in the case of excimer emission and lanthanide coordination compounds. Moreover, the presence of Herzberg–Teller effects, dominating pure Franck–Condon contributions, may lead to different rotational strengths in CPL and ECD. Finally, even when a correlation holds, it may be difficult to extract the correct g _abs value from an experimental ECD spectrum where several bands overlap. Despite all of these limitations, a standard ECD calculation may highlight the presence of multiple nearly overlapping bands and give insight into the expected CPL behavior. Indeed, a comparison between CPL and ECD computational results constitutes a good sanity check for the reliability of the calculations themselves. If large discrepancies are found, then additional care must be taken, which in turn may lead to unraveling nontrivial photophysics in excited states. For such a reason, during our survey we put a large amount of effort into extracting both g _abs and g _lum data available from the literature on CPL calculations, and whenever statistically relevant, the calculated g _lum/g _abs ratio will be scrutinized in Section .

As an example of the heuristic value of this correlation, recently an order of magnitude discrepancy between experimental g _lum and g _abs in some [7]­helicenes bearing phenanthrene-like moieties reported in the literature prompted us to investigate the case computationally (Section ). All of the computational results suggested an error in the reported experimental CPL data, which was later experimentally confirmed by an independent group.

2.7.3. CPL Brightness (B _CPL)

As anticipated in the Introduction, a more comprehensive metric to quantify the CPL efficiency of an emitter is the CPL brightness B _CPL (eq ). Despite being an empirical tool easy to assess from experimental data, its computational estimation is not straightforward, as it requires an ab initio calculation of the quantum yield (QY), which in turn would require computational access to radiative and nonradiative lifetimes. Although possible, this task is far from trivial, and at present, no computation of B _CPL seems to be present in the literature. Other figures of merit (FM) have been proposed in the literature to be used when the extinction coefficient is not easily experimentally recovered, as may be the case with solid samples. , The simplest one considers only QY (Φ_f) and g _lum as

$$\mathrm{FM}={\mathrm{\Phi }}_{\mathrm{f}}·|g_{\mathrm{lum}}|$$ 98

2.7.4. Emission Maximum and Stokes Shift

The measurement of Stokes shifts (ΔE_S = E_em – E_abs in energy units, commonly eV or cm^–1) is relatively straightforward. Exceptions may be encountered when the S₀–S₁ transition is dark and its absorption band is submerged by flanking bands or when the absorption or emission envelopes display a pronounced vibrational structure, making the 0–0 transition not immediately identified. At the same time, the calculation of energy differences between the S₀–S₁ and S₁–S₀ transitions is implicit in any combination of Abs/ECD and Emi/CPL calculations (Section ). Therefore, the comparison between experimental and calculated Stokes shifts is immediate. From a computational viewpoint, moreover, it should in principle cancel or reduce the systematic error made in transition energies by several levels of calculations (“performance” subsections in Section ). However, the Stokes shift depends crucially on the solvent relaxation in the excited state; therefore, a careful estimate would require a state-specific solvent treatment (Section ), which is seldom done in the literature. In Section , we will run further considerations on Stokes shifts based on the largest available data set for a homogeneous family, namely, helicenes.

2.7.5. Excited-State Geometry

The excited-state (S₁ or T₁) geometry is perhaps the most valuable piece of information confered by QM calculations with no immediately accessible experimental counterpart. Especially useful is its comparison with the ground state (S₀) geometry calculated at the same level, which may instead be checked against several kinds of experimental data (X-ray diffractometry, NMR, etc.). As explained in the Introduction and in Section , any discrepancy between the sign of the CPL band and the red-most ECD band and between g _lum and g _abs values must first be traced back to a substantial difference between S₀ and S₁ geometries due to a large structural rearrangement or solvent relaxation in the S₁ state possibly due to excited-state reactions, excimer or exciplex formation, and so on. Another relevant phenomenon, especially for ketone derivatives (Section ), is the existence of multiple minima on the S₁ PES, which translates to multiple or altered CPL bands. In the quest for large g _lum values, moreover, it is worth anticipating that a highly symmetrical S₀ structure does not necessarily lead to a similarly symmetrical S₁ geometry, due again to excited-state relaxation (Section ).

2.7.6. Transition Density and Transition Dipole Moments

The impact of intensity and the reciprocal arrangement of the electric dipole transition moment μ ₁₀ (EDTM) and the magnetic dipole transition moment m ₀₁ (MDTM) on the rotational strengths and on g _lum is obvious from eq . This piece of information, which has no immediate experimental counterpart, is easily extracted from any QM calculation. Therefore, plotting μ ₁₀ and m ₀₁ vectors (see the example in Figure ) is a very common practice in the literature of CPL calculations, mainly with the aim of justifying or rationalizing observed g _lum values; several examples of these plots will be provided in Section . In order to maximize g _lum, one can act both on EDTM and MDTM intensities and their angle θ _μm (Figure ). Although several strategies have been suggested for such a purpose, it is apparent that any structural modification aimed at maximizing |m ₀₁ | will unavoidably impact the other two parameters as well. With that in mind, the main strategies employed to maximize |m ₀₁ | for organic compounds include increasing the degree of helicity (Section ), increasing the size of a macrocycle (Section ), and manipulating the orbital manifold favoring short-range charge transfer (SRCT) transitions (Section ). For transition-metal and lanthanide complexes, |m ₀₁ | is intrinsically higher but also more difficult to control by rational design. Minimizing |μ ₁₀ | is usually a less successful strategy because, especially for purely organic molecules, it has as a consequence a decrease in quantum yields and hence a decrease in B _CPL. An example is offered by ketones (Section ), whose electric-dipole-forbidden n−π* transition is associated with large MDTM, small EDTM, and large g _lum and g _lum; here, low quantum yields and small extinction coefficients impart B _CPL < 10^–2 M^–1 cm^–1. Finally, the most successful way of regulating angle θ _μm is by acting on the molecular symmetry, that is, building molecular structures belonging to high-symmetry chiral point groups (D ₂, D ₄) where the orientation between EDTM and MDTM is restricted by symmetry rules (Sections , , and ).

9.

Plot of EDTM and MDTM vectors (μ ₁₀ and m ₀₁ ) shown for a carbo[7]­helicenoid. The vector intensity is not to scale.

2.7.7. Molecular Orbitals (MO) and Natural Transition Orbitals (NTO)

Frontier molecular orbitals (FMOs), especially HOMO (highest occupied MO) and LUMO (lowest occupied MO) or their equivalent in Kohn–Sham theory, are almost invariably shown in any literature paper involving excited-state calculations, including CPL calculations. In general, different electronic configurations (i.e., Slater determinants specifying an electron occupation of molecular orbitals) contribute to the wave function description of excited states, so the significance of FMOs should not be overemphasized: indeed, electronic transitions are hardly described, even for very simple chromophores, in terms of HOMO-to-LUMO single-electron excitations. Still, chemical reasoning related to the substituent effect is often based on HOMO and LUMO plots as a starting point for their manipulation (Section ).

The use of natural transition orbitals (NTO) offers a compact orbital representation for the electronic transition density matrix to characterize electron excitation from single-reference methods like CIS and TD-DFT or even CC. The method is based on a singular value decomposition (SVD) of the one-electron transition density matrix T, whose elements vary depending on the electronic structure method. For instance, for a TD-DFT approach they can be written as

$$T_{ia}=\sum _{\sigma }⟨{\mathrm{\Psi }}_e|{â}_{i\sigma }^{†}{â}_{a\sigma }|{\mathrm{\Psi }}_0⟩=X_{ia}^I+Y_{ia}^I$$ 99

where i and a label occupied and virtual orbitals, respectively, â _iσ and â _aσ are the creation and annihilation operators, and σ is the spin index. As a note for nonexpert readers, it can be helpful to point out that often, in the electronic structure literature, one-particle transition density matrix components are also indicated as γ _ia . The SVD approach is derived by solving the eigenvalue problems for the two associated square matrices T _occ = TT ^T (in the occupied space) and T _virt = T ^T T (in the virtual space), yielding two sets of orbitalsthe hole and particle NTOsand their associated eigenvalues, which represent the weights of the corresponding transitions. Therefore, each NTO pair consists of an occupied NTO and a virtual NTO sharing the same (sorted) eigenvalue, with the latter representing the contribution to the electron excitation. NTOs offer a simplified and intuitive picture of the orbitals involved in any electronic transition (Figure ). Regrettably, their use in the literature concerning CPL calculations is much less widespread than canonical FMOs.

10.

(a) MOs involved in the second vertical excitation of 1-fluoropyrene. (b) NTOs for the same transition.

An intuitive picture of how locally the electron density changes during the electronic transitions can be obtained by applying the “effective electron displacement” metric Γ_NTO that defines an effective hole–particle distance covered during the excitations. , The Γ_NTO index is the sum of two contributions: the orbital centroid distance (centered on the NTO barycenter) and the variation of the electron delocalization around each centroid of the involved orbitals. This procedure allows one to identify the principal components of the transition electron density (by using NTO) and provide a measure of the interaction distance of the hole–particle pairs. An example is shown in Figure .

22.

NTOs of S₁–S₀ transitions for carbo[7]­helicenoids H21, H22b, and H23b. On each arrow, the charge-transfer index Γ_NTO is reported. Adapted from ref .

2.7.8. Electron–Hole Densities and Transition Moment Densities

A complementary approach to the analysis of the transition density matrix T is the investigation of the difference density matrix, which is simply given as the difference between the single electron density matrices of the excited state and the electronic ground state. , A similar approach to describing electronic transitions resulting from multiple singly excited configurations is the so-called electron–hole density difference analysis, which consists of plotting the charge density difference (ΔP) between the “electron” (P ^elec ) and the “hole” (P ^hole ) derived from the transition density matrix from eq : ,

$$\Delta {\mathrm{P}}^{unrel}={\mathrm{P}}^{elec}-{\mathrm{P}}^{hole}$$ 100

$${\mathrm{P}}_{ab}^{elec}(\mathbf{r})=\sum _i{({\mathrm{X}}^{†}+{\mathrm{Y}}^{†})}_{ai}{(\mathrm{X}+\mathrm{Y})}_{ib}$$ 101

$${\mathrm{P}}_{ij}^{hole}(\mathbf{r})=\sum _a{(\mathrm{X}+\mathrm{Y})}_{ia}{({\mathrm{X}}^{†}+{\mathrm{Y}}^{†})}_{aj}$$ 102

Using these definitions, however, does not afford the relaxed difference density matrix of the ES and GS, ,, but instead the unrelaxed one, that is, the contribution due to the orbital coefficient relaxation (usually computed by solving the coupled perturbed equations or the Z-vector approach) is not included.

Some groups employ a different approach named transition moment density analysis, where the transition density matrix is fragmented in electric and magnetic contributions (ρ _tot , ρ _tot ) both expressed as linear combination of axis components (e.g., the x-components are ρ _x , ρ _x ). , Integrating the electric dipole transition moment density over the whole space yields the dipole transition moment. For instance, in a TD-DFT formalism, the x component of the electric dipole transition moment density reads

$${\rho }_x^{\mu }=-Tr[\mathrm{x̂}·\mathbf{T}]=\sum _{\mathit{i},\mathit{a}}({\mathit{X}}_{\mathit{ia}}+{\mathit{Y}}_{\mathit{ia}})⟨{\phi }_i|\mathrm{x̂}|{\phi }_a⟩$$ 103

Similar relations hold for the magnetic dipole moment. In this way, one can conveniently study the contributions to dipole transition moments μ ₁₀ and m ₀₁ of various molecular regions by plotting the electric dipole transition moment density. Transition density plots and/or transition moment density analysis were displayed in a few papers covered by our survey, and some examples will be shown in the figures in Section .

3. Literature Overview

In this section, we will comprehensively survey all systems, ranging from small organic molecules to metal complexes to supramolecular species, whose calculations of CPL spectra, and most often of the quantities mentioned in Section , have been reported in the literature. As a minimum requirement to be included in the survey, we will consider the calculation of the g _lum value or of the quantities it may be evaluated from, that is, rotational and dipolar strengths for the S₁–S₀ transition or the corresponding electric and magnetic transition moments μ ₁₀ and m ₀₁ . In practically all cases covered, the calculations will be compared with experimental values. A second useful comparison, which will be made whenever possible, is with the calculated (and experimental) quantities relative to the S₀–S₁ transition, in particular, g _abs values for the red-most ECD band. A final parameter that we will take into account, whenever relevant, is the Stokes shift, which combines both absorption and emission processes.

This section is organized into three main subsections devoted to small organic systems, larger organic systems and supramolecular assemblies, and metal complexes. Each of them is further divided into subsections, each devoted to a specific molecular system. Organizing the systematic survey according to the nature of the compounds will make it easier to follow with respect to other criteria like a chronological one or based on the calculation level. Nonetheless, each family of compounds has its specific properties, and most often, their simulation requires specific strategies. During the survey, the reader will also encounter several distinctive phenomena, which will be illustrated through the relevant literature examples. They include ECD vs CPL sign inversion, the appearance of multiple CPL bands, excimer and exciplex formation, excited-state tautomerism (e.g., ESIPT), intramolecular charge transfer (ICT), thermally activated delayed fluorescence (TADF), circularly polarized phosphorescence (CPP) from both metal complexes and organic emitters, triplet emitters and other spin-forbidden transitions, and so on.

In all cases where the number of examples for each class of compounds is statistically relevant and/or several compounds are structurally homogeneous (for instance, ketones, helicenes and helicenoids, and 1,1′-binaphthyls), a statistical analysis of selected parameters will be run, focused on the quantitative comparison of simulated and experimental values of dissymmetry factors g _abs, g _lum, and g _lum/g _abs. To quantify the accuracy of calculated g _abs and g _lum values, we will employ the signed relative error (RE) and the absolute relative error (ARE)

$$\mathrm{RE}=\frac{|g_{lum}^{\exp }|-|g_{lum}^{calc}|}{|g_{lum}^{\exp }|}$$ 104

$$\mathrm{ARE}=|\mathrm{RE}|=\left|\frac{|g_{lum}^{\exp }|-|g_{lum}^{calc}|}{g_{lum}^{\exp }}\right|$$ 105

with similar formulas for g _abs. The absolute value taken for each g factor allows it to be independent of the specific enantiomer considered; for the same reason, except where explicitly signed, for the sake of readiness, all listed g factors must be intended as absolute values.

This analysis will offer the starting point for Section where all surveyed data will be critically reevaluated with a twofold aim: 1) identify the best practices to calculate CPL data of families of compounds in an accurate way and 2) recognize the chief structural and electronic properties which can be modulated for the design of more efficient CPL emitters.

3.1. Small Organic Molecules

Chiral small organic molecules (SOMs) represent a very large majority of CPL-active compounds analyzed via QM calculations of chiroptical properties, especially CPL. Despite the tremendous efforts made in the last few years, ,,− SOMs are still outperformed in terms of CPL performance by both lanthanide complexes and some transition-metal complexes, on one side, and by polymeric or aggregated systems, on the other side. , Yet, the research on CPL-active SOMs is far from discontinued, for several reasons discussed in the Introduction (low cost, ease of functionalization, facile modulation and control of optical and chiroptical properties, etc.). From our viewpoint, the prediction of chiroptical properties of SOMs is by far simpler and less computationally demanding than complexes containing heavy metals, especially lanthanide ions, and supramolecular systems. Therefore, SOMs also represent the field of research where QM calculations may provide the most significant contribution to the design of more efficient compounds and materials.

3.1.1. Ketones and Diketones

Chiral ketones and diketones are one of the most investigated families of CPL-active organic compounds, since the seminal works of Pritchard and Autschbach, Pecul and Ruud, and Longhi et al. The attention paid to this family of compounds as CPL emitters is due to the same reasons that chiral ketones, and especially cyclic ketones, have been for decades among the most studied ones by ECD. Carbonyl compounds feature an S₀–S₁ n−π* transition around 280–300 nm which is magnetic-dipole-allowed and electric-dipole-forbidden and well separated in energy from the S₀–S₂ π–π* transition at around 190–200 nm. Consequently, the typical g _abs measured for cyclic ketones is between 10^–1 and 10^–2, among the strongest observed for SOMs. The associated g _lum values are similarly strong, usually between 10^–2 and 5 × 10^–2, corresponding to g _lum/g _abs ratios normally around 0.1–0.5. However, the electric-dipole-forbidden character implies low quantum yields, which, associated with small extinction coefficients (∼10² M^–1 cm^–1), cause very small B _CPL values (below 10^–2 M^–1 cm^–1). Finally, the relatively high-energy emission at around 400 nm thwarts any practical technological application of these compounds as CPL emitters. On the other hand, the simplicity of the carbonyl chromophore and the peculiarity of the S₁–S₀ transition has made chiral carbonyl compounds the ideal scaffolds to benchmark different CPL calculation schemes and test high-level approaches such as coupled-cluster methods (Section ) and vibronic calculations. In fact, a relatively small set of chiral compounds based on a few skeletons (camphor, norcamphor, hydrindane, adamantane, norbornene) has been the subject of intensive investigations by several research groups (Chart and Table ).

1.

1. Collection of Experimental and Computational Data for Ketones and Diketones .

	Experiment						Calculation						Number or abbreviation
Type	λmax	g abs	λem	g lum	ΔES	g lum/g abs	g abs	g lum	ΔES	g lum/g abs	Method	Basis set	This review	Original publication	ref.
β,γ-enone	302	17.8	397	6.3	0.98	0.35	31.2	16.1/–11.9	1.53/1.33		CAM-B3LYP	aug-cc-pVDZ	K3	1
β,γ-enone	302	5	400	–12	1.0	2.40	6.5	–16.5	1.4	2.5	CAM-B3LYP	aug-cc-pVDZ	K4	1a
β,γ-enone	302	56.8	403	29.4	1.0	0.52	69.8	45.9	1.32	0.66	CAM-B3LYP	aug-cc-pVDZ	K5	2
β,γ-enone	302	45.8	403	15.7	1.0	0.34	57.4	47.8/–2.1	1.28		CAM-B3LYP	aug-cc-pVDZ	K6	3
β,γ-enone	302	–32.5	427	3	1.2	0.09	–52.6	–26.2/4.3	1.92/1.71		CAM-B3LYP	aug-cc-pVDZ	K7	4
β,γ-enone	302	17.8	397	6.3	0.98	0.35	22.5	25.9/–10.8			B3LYP	aug-cc-pVDZ	K3	1
β,γ-enone	302	5	400	–12	1.0	2.40	3.5	–13.0		3.7	B3LYP	aug-cc-pVDZ	K4	1a
β,γ-enone	302	56.8	403	29.4	1.0	0.52	53.2	34.1		0.64	B3LYP	aug-cc-pVDZ	K5	2
β,γ-enone	302	45.8	403	15.7	1.0	0.34	41.3	19.0/–0.59			B3LYP	aug-cc-pVDZ	K6	3
β,γ-enone	302	–32.5	427	3	1.2	0.09	–34.4	–13.0/2.62			B3LYP	aug-cc-pVDZ	K7	4
ketone	287	50	419	4	1.4	0.07	120	3	1.2	0.02	CAM-B3LYP	aug-cc-pVDZ	K8	1
β,γ-enone	303	60	430	5	1.2	0.1	70	45	1.2	0.6	CAM-B3LYP	aug-cc-pVDZ	K9	2
1,4-diketone	290	40	455	5	1.5	0.1	150	3.5	1.2	0.02	CAM-B3LYP	aug-cc-pVDZ	K10	3
1,2-diketone	467	10	514	10	0.24	0.9	35	35	0.26	1.0	CAM-B3LYP	aug-cc-pVDZ	K11	4
1,4-diketone	294	12	438	2	1.4	0.17	50	24	1.2	0.04	CAM-B3LYP	aug-cc-pVDZ	K12	5
1,4-diketone	297	80	438	30	1.34	0.4	90	80	1.17	0.9	CAM-B3LYP	aug-cc-pVDZ	K13	6
β,γ-enone	295	60	456	5	1.48	0.1	95	50	1.19	0.5	CAM-B3LYP	aug-cc-pVDZ	K14	7
β,γ-enone	302	56.8		29.4		0.52	79.4	37.4	1.41	0.47	EOM-CCSD	aug-cc-pVDZ	K5	a
β,γ-enone	302	56.8		29.4		0.52	79.1	35.9	1.35	0.45	B3LYP	aug-cc-pVDZ	K5	a
ketone	323	153	408	27	0.81	0.33	391	34.9	1.07	0.09	EOM-CCSD	aug-cc-pVDZ	K2	f
ketone	323	153	408	27	0.81	0.33	342	52.4	0.97	0.15	B3LYP	aug-cc-pVDZ	K2	f
ketone	323	153	408	27	0.81	0.33	415	46.8	0.97	0.11	CAM-B3LYP	aug-cc-pVDZ	K2	f
β,γ-enone	302	45.8	403	15.7	1.0	0.34	65.2	28.5/–1.2	1.33		EOM-CCSD	aug-cc-pVDZ	K6	g
β,γ-enone	302	45.8	403	15.7	1.0	0.34	41.5	60.2/–0.26	1.24		B3LYP	aug-cc-pVDZ	K6	g
β,γ-enone	302	45.8	403	15.7	1.0	0.34	57.3	45.1/–1.9	1.28		CAM-B3LYP	aug-cc-pVDZ	K6	g
β,γ-enone	302	–32.5	427	3	1.2	0.09	–64.5	–15.2/4.6	1.92/1.64		EOM-CCSD	aug-cc-pVDZ	K7	h
β,γ-enone	302	–32.5	427	3	1.2	0.09	–34.6	–44.5/2.6	1.95/1.67		B3LYP	aug-cc-pVDZ	K7	h
β,γ-enone	302	–32.5	427	3	1.2	0.09	–52.7	–26.9/4.4	1.93/1.71		CAM-B3LYP	aug-cc-pVDZ	K7	h
ketone	323	153	408	27	0.81	0.33	293	55.2	1.02	0.19	B3LYP/COSMO	QZ4P	K2	Mol-1
ketone	323	153	408	27	0.81	0.33	398	47.4	1.02	0.12	CAM-B3LYP/COSMO	QZ4P	K2	Mol-1
ketone	323	153	408	27	0.81	0.33	709	29.8	1.29	0.12	CASSCF/C-PCM	TZP	K2	Mol-1
ketone	323	153	408	27	0.81	0.33	698	28.7	1.05	0.12	CASPT2/C-PCM	TZP	K2	Mol-1
β,γ-enone	302	56.8	403	29.4	1.0	0.52	51.5	31.9	1.29	0.62	B3LYP/COSMO	QZ4P	K5	Mol-2
β,γ-enone	302	56.8	403	29.4	1.0	0.52	67.3	40.2	1.32	0.60	CAM-B3LYP/COSMO	QZ4P	K5	Mol-2
β,γ-enone	302	56.8	403	29.4	1.0	0.52	48.8	15.8	1.85	0.32	CASSCF/C-PCM	TZP	K5	Mol-2
β,γ-enone	302	56.8	403	29.4	1.0	0.52	57.3	22.1	1.49	0.39	CASPT2/C-PCM	TZP	K5	Mol-2
β,γ-enone	302	45.8	403	15.7	1.0	0.34	56.8	61.4/0.4	1.26/1.25		B3LYP/COSMO	QZ4P	K6	Mol-3
β,γ-enone	302	45.8	403	15.7	1.0	0.34	41.2	46.3/–1.7	1.29/1.28		CAM-B3LYP/COSMO	QZ4P	K6	Mol-3
β,γ-enone	302	45.8	403	15.7	1.0	0.34	57.4	37.1/–1.5	1.73/1.70		CASSCF/C-PCM	TZP	K6	Mol-3
β,γ-enone	302	45.8	403	15.7	1.0	0.34	41.2	35.5/0.7	1.16		CASPT2/C-PCM	TZP	K6	Mol-3
β,γ-enone	302	–32.5	427	3	1.2	0.09	–33.5	–46.4/2.4	1.96/1.67		B3LYP/COSMO	QZ4P	K7	Mol-4
β,γ-enone	302	–32.5	427	3	1.2	0.09	–51.6	–27.6/4.2	1.95/1.73		CAM-B3LYP/COSMO	QZ4P	K7	Mol-4
β,γ-enone	302	–32.5	427	3	1.2	0.09	–66.4	–13.3/2.1	2.37/2.17		CASSCF/C-PCM	TZP	K7	Mol-4
β,γ-enone	302	–32.5	427	3	1.2	0.09	–78.0	–17.6/1.2	1.96/1.83		CASPT2/C-PCM	TZP	K7	Mol-4

^aData for electronic calculations only (no vibronic calculations). See Chart for structure numbering.

^b g _abs and g _lum multiplied by 10³. The sign indicates the presence of multiple bands. ΔE_S in eV.

^cExperimental data from ref .

^dWeighted average over two or more excited-state geometries.

^eExperimental data from ref .

In 2010, Pritchard and Autschbach reported the first CPL calculation of a chiral diketone (d-camphorquinone, K1, Chart ) and a chiral ketone (trans-β-hydrindanone, K2) using vibronic calculations based on DFT. This paper will be summarized later in this section when we deal with vibronic CPL spectra of chiral ketones and diketones. This work also represents the very first CPL calculation with ab initio methods reported in the literature. The only previous report we could find dates to 1974. It concerns a biaryl system (1,1′-dianthracene-2,2′-carboxylic acid) to be discussed in Section , and the calculation employed a semiempirical method based on the Pariser–Parr–Pople (PPP) approximation. Some publications quote a 2008 paper by Coughlin et al. as the first example of a CPL calculation. Although the ECD calculation of the chiral octahedral Ir^III complexes discussed in the paper was run by TD-DFT, the CPL from the triplet state was extrapolated but not explicitly calculated from the triplet geometry. Therefore, we dispute the attribution of the first reported ab initio CPL calculation to Coughlin et al. and rather assign it to Pritchard and Autschbach.

The first electronic-only CPL calculation of chiral cyclic ketones, specifically β,γ-enones (K3-K7, Chart ), was reported by Pecul and Ruud in 2011. Interestingly enough, the paper was submitted in April 2010, before the first online publication of Pritchard and Autschbach’s contribution (July 2010). Therefore, Pecul and Ruud claimed appropriately that their study was “the first computational study of CPL”. The authors presented the theoretical framework of the CPL calculation within density functional linear response theory (Section ), whereby the CPL intensity is obtained by the rotational strength R ₁₀ of the excited-state geometry and R ₁₀ is in turn evaluated as a residue of the linear response function in either the velocity or the length gauge. As stressed in Section , the rotational strengths R ₀₁ and R ₁₀ calculated in the length gauge with a finite orbital basis are origin-dependent, although for variational methods (such DFT) this problem can be overcome by using GIAOs. The dipolar strengths D ₀₁ and D ₁₀ were similarly calculated in the velocity and length gauges, hence the dissymmetry values g _abs and g _lum were obtained. Pecul and Ruud employed the B3LYP hybrid functional in combination with the aug-cc-pVDZ basis set for geometry optimizations of the S₀ and S₁ states. Due to the software limitations at that time, CAM-B3LYP was not available for excited-state geometry optimizations but was employed to evaluate excitation energies and R ₀₁ and R ₁₀ rotational strengths. Larger basis sets (daug-cc-pVDZ and aug-cc-pVTZ) were also tested; all basis sets were based on London atomic orbitals of the GIAO type. For the smallest β,γ-enone norbornenone (K5), the coupled-cluster method CC2 (Section ) was also used, including the resolution-of-the-identity (RI) approximation for geometry optimizations. While all compounds K3–K7 feature a single ground-state geometry with a planar carbonyl moiety, in the n−π* excited state S₁ the carbonyl moiety undergoes pyramidalization. Two energy minima, S_1(E1) and S_1(E2), can be isolated along the excited-state potential energy surface for compounds K3, K6, and K7, with the CO groups pointing in opposite directions (Figure ). The energy difference between the two minima was small in each case (0.02–0.06 eV or 0.46–1.38 kcal mol^–1); for K3 and K7, the ordering of the excited-state structures was reversed at the B3LYP and CAM-B3LYP levels, which has consequences for CPL predictions. Another interesting fact is that CC2 tends to overestimate the C–O bond length in the S₁ geometry (1.359 Å for CC2 and 1.267 Å with B3LYP, with 1.295 Å being a reasonable estimate). The experimental |g _abs| values range between 1.8 and 5.7 × 10^–2 for K3 and K5–K7 and 5 × 10^–3 for K4 (Table ). All of the calculated g _abs values at the CAM-B3LYP/aug-cc-pVDZ level had the correct sign, while the intensity was overestimated up to a factor of 1.75 as a consequence of the fact that the calculated rotational strengths were overestimated while the dipolar strengths were underestimated. The agreement was improved using the B3LYP functional; however, this is due to a cancellation of errors in the rotational and dipolar strengths. The calculations run with CC2 yielded results similar to CAM-B3LYP though they were much more dependent on the gauge employed, which was instead almost uninfluential for DFT calculations. Similarly, passing from London to non-London orbitals as well as increasing the basis set from aug-cc-pVDZ to daug-cc-pVDZ (with one extra diffuse function) and aug-cc-pVTZ had little influence on calculated rotational and dipolar strengths.

11.

Schematic representation of S₀ and S₁ PES for (1R)-7-methylenebicyclo­[2.2.1]­heptan-2-one (K6, Chart ) showing the two excited-state minima E1 and E2 and the corresponding CPL signs.

The simulation of CPL spectra proved to be more challenging for a series of reasons. For enones K4 and K5, the sign of the observed CPL band is the same as that of the n−π* ECD band around 300 nm. These compounds have a single excited-state energy minimum; the calculated g _lum and especially the g _lum/g _abs ratios were in good agreement with the experiment (Table ). On the contrary, for K3, K6, and K7, the sign of the observed CPL band is opposite to that of the n−π* ECD band. These three enones have two excited-state structures S_1(E1) and S_1(E2) (Figure for K6) with, in each case, rotational strengths of opposite sign. The CAM-B3LYP functional provided the correct S₁ energy ordering which reconciled the observed and calculated CPL signs. This is a nice demonstration of the power of DFT calculations in explaining observed phenomena, provided that the correct functional is employed. On more theoretical grounds, the authors additionally demonstrated the crucial role of the arrangement between the CC and CO moieties. The S₁–S₀ rotational strength depends subtly on the value of the C–C–CO dihedral, which may explain, at least in part, the discrepancy between the calculated and experimental CPL intensities of some enones. A distinctive feature of many chiral carbonyl compounds is the large difference between |g _abs| and |g _lum| values due to the pronounced rearrangement between S₀ and S₁ geometry, as already noticed by Schippers et al. in an experimental study. As a consequence, the |g _lum/g _abs| ratio may span from <0.1 to >2, and the correlation between the two values is very scattered, as evidenced later by Mori and co-workers.

A second in-depth analysis of the CPL spectra of camphor (K8, Chart ), a related β,γ-enone (K9) and two diketones (K1 and K10), plus other three [2.2.2] bicyclic ketones and diketones (K11–K13), was reported in 2012 by Longhi, Castiglioni, Abbate, and co-workers. Following the same calculation scheme as that of Pecul and Ruud, the authors ran TD-DFT calculations at the CAM-B3LYP/aug-cc-pVDZ level. This publication shows calculated ECD and CPL spectra (obtained by applying a Gaussian band shape with a heuristically assumed bandwidth). Such a procedure, which is anything but common in the literature concerning CPL calculations, has the merit of offering an immediate comparison with experimental spectra beyond numerical data. As an example, the experimental and calculated ECD and CPL spectra of camphor (K8) are reproduced in Figure . The striking feature in the CPL spectrum of camphor is its bisignate appearance, which is not due to a violation of Kasha’s rule (S₂–S₀ emission) but to the presence of two excited-state minima, as discussed above. Similar to the case of K3, K6, and K7, the two excited-state structures of K8 have opposite orientations of the CO moiety and rotational strengths opposite in sign (the inset and vertical bars in Figure ). The combination between the predicted signs, the energy ordering of the two structures (S_1(E1) being more stable than S_1(E2) by 2 kcal mol^–1), and the vertical energies (S_1(E1) giving red-shifted emission with respect to S_1(E2) by 0.05 eV) offered a plain interpretation of the bisignate CPL spectra of K8 and of the sign reversal between the major CPL band and the n−π* ECD band. The case of enone K9 is similar to K4 and K5, namely, a monosignate CPL spectrum allied with a single excited-state structure with the same sign as the corresponding ECD band. In diketones K1 and K10–K12, two n−π* excitations combine to define the low-energy ECD and the CPL band. So, for example, for K10 four distorted excited states were predicted, corresponding to the two distortions in opposite directions for each carbonyl group. From the perspective of calculation performance, it is advisible to concentrate on compounds featuring a single excited-state minimum, thus bypassing the possible error associated with the relative energies of the two minima. We notice that while |g _abs| values are nicely predicted (ARE < 0.2 for K9 and K13), some of the |g _lum| values are overestimated by 1 order of magnitude (e.g., K9 and K14, Table ).

12.

Experimental (top) and calculated (bottom) ECD, CPL, absorption and emission spectra of (+)-(R)-camphor (K8, Chart ). Bars represent calculated rotational and dipolar strengths, to which a Gaussian band shape was associated. Here and in all other figures taken from the literature, the level of calculation is not stated in the caption; the reader is referred to the text, tables, and/or the original publication. Adapted with permission from ref . Copyright 2013, John Wiley and Sons.

In 2015, McAlexander and Crawford reported the first application to CPL calculations of the equation-of-motion coupled cluster singles and doubles (EOM-CCSD, see Section ) level of theory. They chose eight compounds among ketones, diketones, and β,γ-enones (K2, K5–K7, K14, and K15, Chart ) plus two achiral compounds with the aim of demonstrating the importance of higher-order electron correlation effects on simulated CPL spectra. For comparison, TD-DFT calculations with B3LYP and CAM-B3LYP functionals were also run. In all cases, the aug-cc-pVDZ basis set was chosen. The first noteworthy observation is that the S₁ C–O bond length estimated for norbornenone (K5) by EOM-CCSD was 1.296 Å, in nearly perfect agreement with the “best” estimate by Pecul and Ruud (1.295 Å). The good agreement between excited-state geometries, g _abs and g _lum, calculated with EOM-CCSD and B3LYP for the two test molecules K2 and K14 led the authors to employ B3LYP/aug-cc-pVDZ geometry optimizations for both S₀ and S₁ states for the remaining compounds; transition energies and dipolar (D ₀₁ and D ₁₀) and rotational strengths (R ₀₁ and R ₁₀) were evaluated with the EOM-CCSD, B3LYP, and CAM-B3LYP methods. An interesting trend emerges in |g _abs| and |g _lum| values where, in nearly all cases, the values calculated with CAM-B3LYP lie between those calculated with EOM-CCSD and B3LYP. A direct comparison with the experiment was provided only for β,γ-enones K6 and K7. Consistent with the results reported by Pecul and Ruud, CAM-B3LYP performed best in terms of calculated D ₀₁ and R ₀₁ while B3LYP yielded the best agreement in terms of |g _abs| due to a mutual cancellation of errors (Table ). On the contrary, EOM-CCSD predicted a satisfactory R ₀₁ but underestimated D ₀₁, yielding a dissymmetry factor |g _abs| that was too large. For the emission counterpart, the comparison with the experiment is complicated by the presence of multiple S₁ minima, as discussed above. For both compounds K6 and K7, the energy order between the two minima was consistent between CAM-B3LYP and EOM-CCSD and in agreement with the experiment. Instead, B3LYP predicted the incorrect CPL sign for K7, again in agreement with Pecul and Ruud. , A comparison with the experiment is also possible for trans-β-hydrindanone whose ECD and CPL spectra were reported by Emeis et al.; the agreement is satisfactory for both |g _abs| and |g _lum| (Table ).

A second coupled-cluster method, based on the more computationally efficient CC2 scheme (Section ), was tested on the already encountered β,γ-enones K3 and K5–K7 as well as on the α,β-enone K16 and the two 1,2-diketones K1 and K17. CC2 results were compared with TD-DFT (B3LYP functionals), employing the aug-cc-pVDZ basis set. The paper reports only computed rotational strengths, thus an assessment of dissymmetry factors is not possible. In general, R ₀₁ and R ₁₀ computed with CC2 agree qualitatively with TD-DFT results; in a few cases, the rotational strengths differ in sign between the two methods because of a switch of close-lying excited states between TD-DFT and CC2.

An insightful re-evaluation of ECD and CPL calculations on four bicyclic ketones (K2 and K5–K7) at different levels of theory was reported by Gendron et al. in 2019. They ran CASSCF and CASPT2 calculations (Section ) compared with TD-DFT calculations (B3LYP and CAM-B3LYP functionals with a quadruple-ζ basis set) as well as with TD-DFT, CCSD, and CC2 results from the aforementioned studies. ,, A continuum solvent model was also added to both density functional and wave function calculations, respectively, the conductor-like screening model (COSMO) and the conductor polarizable continuum model (C-PCM; see Section ). Qualitatively, all calculation methods provided similar results, generally agreeing with the experiment (Table ). Going into detail, for K2, CASSCF and CASPT2 provided a 2- or 3-fold increase in |g _abs| with respect to TD-DFT, thus overestimating the experimental value; on the other hand, |g _lum| was better predicted by the former two methods. The authors stressed that the calculations reproduced the large magnitude decrease between the ECD and CPL dissymmetry factors, which we quantify through the g _lum/g _abs ratio. This effect is due to the pyramidal geometry of S₁, which causes an increase in the electric transition dipole so that D ₁₀ > D ₀₁. Similar results are noticed for K5. For K6 and K7, two energy minima were again obtained for S₁ with substantially different g _lum values, generally positive and large for S_1(E1) of K6, negative and large for S_1(E1) of K7, and almost negligible for S_1(E2) of both compounds. Interestingly enough, the energy ordering for the two excited-state geometries of K7 heavily depends on the calculation method, reinforcing the same observation made by Pecul and Ruud. Because compounds K2 and K5 allow for a direct comparison of several calculation methods without any complication related to multiple excited-state geometries, we summarize all available results from Table in Figure .

13.

Collection of calculated |g _lum| data from refs , and for ketones K2 and K5 with different methods. The same basis set was employed in all cases (aug-cc-pVDZ).

As anticipated at the beginning of this subsection, the first calculation ever of CPL spectra of chiral ketones appeared in 2010 by Pritchard and Autschbach. They simulated vibrationally resolved CPL spectra of d-camphorquinone (K1, Chart ) and trans-β-hydrindanone (K2), employing B3LYP, CAM-B3LYP, and BHLYP functionals in combination with TZVP and aug-cc-pVDZ basis sets. Franck–Condon factors were calculated at the same levels by means of the widely exploited FCclasses code by Santoro and co-workers, currently available in its 3.0.3 version. The intensities were evaluated by including Herzberg–Teller terms (HT, Section ). After the application of a red shift to the calculated transition wavelength, the agreement between experimental and calculated line shapes is satisfactory (Figure ). The vibronic calculation highlighted that while for K1 the intensity of the 0–0 transition is large (vertical bar in Figure ) it almost disappears for K2, whose emission and CPL spectra are dominated by a multitude of weak vibronic transitions. For the valence n−π* transition of interest, neither the basis set augmentation from TZVP to aug-cc-pVDZ nor the change in the HF exchange fraction (from 33% B3LYP to 50% BHLYP) had a sizable effect on the calculated spectra.

14.

Experimental and calculated emission and CPL spectra of d-camphorquinone (left, K1, Chart ) and trans-β-hydrindanone (right, K2). The vertical lines indicate 0–0 transitions. Adapted with permission from ref . Copyright 2013, John Wiley and Sons.

A series of vibronic CPL calculations on chiral ketones were reported by Barone and co-workers in the line of the development of their all-purpose computational procedure for spectral simulations including both vibrational models and environmental effects. The CPL spectrum of d-camphorquinone (K1) was calculated at the TD-DFT level, including a polarizable continuum model (IEF-PCM) for cyclohexane assuming a nonequilibrium regime for the excited state (Section ) and employing the adiabiatic Hessian–Franck–Condon and Franck–Condon/Herzberg–Teller (AH|FCHT and AH|FC, Section ) models for vibronic intensities within the time-independent approach (TI, Section ). Inclusion of HT effects improved the agreement with the experimental band shape while the effect of solvent was minor. More recently, the same group tackled the simulation of the CPL spectrum of camphor (K8) with a QM/MD approach to model the solvent effects, in combination with vibronic calculations (Section ). The characteristic bisignate appearance of the CPL spectrum of camphor, discussed above, could be nicely reproduced.

The description of circularly polarized phosphorescence (CPP) by ketones and diketones, as well as by other compounds, is postponed to Section .

3.1.2. Helicenes and Related Compounds

Helicenes and related compounds are the most extensively studied family of chiral SOMs. , Helicenes are polycyclic aromatic compounds formed by ortho-fused aromatic rings. For steric reasons, from a certain size on (four benzene rings for carbohelicenes), the skeleton deviates from planarity and is forced to assume a three-dimensional helical shape. In carbohelicenes, the ring system contains only carbon atoms; introduction of one or more heteroatoms leads to heterohelicenes. When one or more rings contain atoms devoid of p _π orbitals, the ortho-fused ring system is not aromatic as a whole, and it is referred to as a helicenoid. Moreover, two or more helicene systems can be fused together yielding additional chiral structures, named double helicenes or multiple helicenes. The geometry of the ring fusion may vary, providing different shapes named crossed or X-shaped, serial or S-shaped, and so on (Scheme ). Finally, if one or more rings embrace a metal atom or ion, then one obtains a metallohelicene system (to be discussed in Section ).

1. Schematic Representation of the Possible Arrangement of Double Helicenes Starting from a [6]­Helicene Core.

The reasons for the tremendous interest in helicenes and helicenoids are manifold. First of all, the helical shape is naturally fascinating as it embodies the prototypical chiral structure. Second, the synthesis of enantiopure helicene-like systems is nowadays a mature field of research and is able to afford a multiplicity of compounds with variable complexity, shape, type of aromatic rings, and substituents. − Third, the possible applications of helicene-like systems are countless. , The most relevant systems for the scope of the present review are those connected with the interaction with (circularly polarized) light, which include CP-OLEDs, ,, OFETs (organic field-effect transistors), , organic photovoltaic devices, solar cells, , chiroptical switches, − and so on. As already stressed in the Introduction, any application based on CPL emission necessitates maximizing key quantities such as g _lum and B _CPL. CPL data for hundreds of helicenes and helicenoids are available in the literature, ,, and they were collected in various surveys. ,, Typical g _lum values of helicenes are between 10^–4 and 10^–3, with a relatively small dependence on the specific skeleton. The CPL brightness median value is 5 M^–1 cm^–1. These figures are strikingly small if one considers the importance of these systems and justifies the tremendous effort expended in the design, synthesis, and characterization of new compounds. In that respect, helicene-like molecules exemplify the key message of the present review: since the chemical space for helicene synthesis is endless, a screening based on computational simulations aimed at the optimization of a few key quantities may be crucial for a rational molecular design. This concept was beautifully demonstrated in some cases described in the following sections.

From a computational viewpoint, there is little difference among carbohelicenes, heterohelicenes, and helicenoids; therefore, Section surveys the three classes of compounds at the same time; all structures are collected in Charts –. This will be followed by a presentation of a statistical analysis on all collected data concerning calculated g _abs, g _lum, g _lum/g _abs, and Stokes shift (ΔE_S) values, in comparison with corresponding experimental quantities. The data are presented in Table and will be discussed in Section . Section will be devoted to double helicenes (Charts and ).

2.

5.

2. Collection of Experimental and Computational Data for Helicenes and Helicenoids .

	Experiment						Calculation						Number or abbreviation
Type	λmax	g abs	λem	g lum	ΔES	g lum/g abs	g abs	g lum	ΔES	g lum/g abs	Method	Basis set	This review	Original publication	ref.
carbo [6]helicene	412	0.9	421	1.1	0.06	1.2	2.7	7.1	0.33	2.6	CAM-B3LYP	TZVP	H1	2
carbo [6]helicene	327	10	421	1	0.85	9.4	8.5	1.3	0.45	0.15	CAM-B3LYP	Def2-SV(P)	H1	H6
carbo [6]helicene	420	0.6	422	0.08	0.01	0.14	0.55	4.25	0.33	7.7	CAM-B3LYP	TZVP	H2	3
hetero [6]helicene	411	4.7	427	4.2	0.11	0.90	4.5	7.0	0.36	1.6	CAM-B3LYP	TZVP	H4	1
hetero [6]helicene	396	2.0	484	9.0	0.57	4.5	11	13	0.84	1.2	CAM-B3LYP	TZVP	H6	2
hetero [5]helicene	396	5.6	488	4.7	0.59	0.84	6.6	5.3	0.65	0.8	CAM-B3LYP	6-31G(d)	H7a	1
hetero [5]helicene	461	2.1	580	1.4	0.55	0.67	2.7	2.7	0.54	1.0	CAM-B3LYP	6-31G(d)	H7b	2
hetero [5]helicene	552	0.9	685	0.9	0.44	1.0	1.2	1.3	0.35	1.1	CAM-B3LYP	6-31G(d)	H7c	3
carbo [6]helicene	427	4.5	527	1.8	0.55	0.4	2.7	1.8	0.27	0.67	BHLYP	SV(P)	H8a	1 ,
carbo [6]helicene	427	6.5	520	3.1	0.52	0.5	2.7	3.1	0.26	1.1	BHLYP	SV(P)	H8b	2 ,
carbo [6]helicene	382	32	426	28	0.34	0.87	18	71	0.51	3.9	CAM-B3LYP	Def2-SV(P)	H9b	H6(CN)2
carbo [6]helicene	428	22	500	54	0.42	2.4	20	34	0.83	1.7	CAM-B3LYP	Def2-SV(P)	H9e	H6(NMe2)2
carbo [6]helicene	341	20	422	9	0.70	4.0	19	21	0.39	1.1	CAM-B3LYP	Def2-SV(P)	H10	H6(H)2
carbo [6]helicene	417	2.3	485	1.3	0.42	0.6	5.1	2.25	0.54	0.4	PBE0	SV(P)	H11a	H2
carbo [6]helicene	433	3.4	493	2.5	0.35	0.75	10.5	4.45	0.42	0.4	PBE0	SV(P)	H11b	H3
carbo [5]helicene	370	1.2	457	2.1	0.64	1.74	2.7	1.9		0.7	M08HX	6-31G(d)	H12a	1
carbo [5]helicene	368	1.7	459	1.6	0.67	0.92	2.6	2.4		0.9	M08HX	6-31G(d)	H12b	2
carbo [5]helicene	343	3.3	452	6.5	0.87	1.95	47	3.2		0.07	M08HX	6-31G(d)	H13a	3
carbo [5]helicene	337	6.2	508	12	1.24	2.01	21	20.5		1.0	M08HX	6-31G(d)	H13b	4
carbo [5]helicene	330	7.0	421	3.8	0.81	0.55	7.2	6.6		0.9	M08HX	6-31G(d)	H13c	5
carbo [5]helicene	381	1.5	453	1.9	0.52	1.3	1.4	1.3	0.85	1.1	MN15	6-311G(d,p)	H14b	2
carbo [7]helicene	391	30	450	6	0.42	0.2	50	–8	0.27	–0.16	B3LYP-D3BJ	6-311G(2d,p)	H15a	1
carbo [7]helicene	418	16	500	3	0.49	0.2	24	5	0.31	0.21	B3LYP-D3BJ	6-311G(2d,p)	H15b	2
carbo [7]helicene	421	16	550	13	0.69	0.8	18	18	0.59	1	B3LYP-D3BJ	6-311G(2d,p)	H15c	3
carbo[7] helicene	391	30	450	6	0.42	0.2		7.8	0.27		B3LYP	6-311G(2d,p)	H15a	[7]H
carbo[7] helicene				3				6			B3LYP-D3BJ	6-311G(2d,p)	H15b	2a
carbo[7] helicene				13				18			B3LYP-D3BJ	6-311G(2d,p)	H15c	2b
carbo [7]helicene	569	2.3	652	2.7	0.28	1.2	2.7	4.0	0.49	1.5	B3LYP	6-311G(2d,p)	H16	1
hetero[7] helicene				8				10			B3LYP-D3BJ	6-311G(2d,p)	H17	3
hetero[7] helicene	369	1.9	436	0.76	0.52	0.4		2.6			CC2	Def2-TZVP	H18	7
hetero[7] helicene	360	11	429	4.4	0.55	0.4		1.1			CC2	Def2-TZVP	H19	8
hetero [7]helicene	425	0.45	447	0.66	0.14	1.5	0.67	0.66	0.30	0.99	B3LYP-D3BJ	6-311+G(2d,2p)	H20	C-29
hetero [7]helicene	475	0.31	585	0.78	0.49	2.5	0.99	0.78	0.37	0.37	B3LYP-D3BJ	6-311+G (2d,2p)	H20·H +	C-29H+
carbo[7] helicenoid	400	1.2	417	3.0	0.13	2.5		5.1	0.54		B3LYP	6-311G(2d,p)	H21	[7]H–C
carbo [7]helicenoid	400	1.2	417	3	0.13	2.5	0.78	3.1	0.70	3.9	CAM-B3LYP LR	6-311+G(d,p)	H21	3
carbo [7]helicenoid	400	1.2	417	3	0.13	2.5	0.87	3.4	0.64	3.7	PBE0–1/3 LR	6-311+G(d,p)	H21	3
carbo [7]helicenoid	404	2.9	449	3.4	0.31	1.4	3.1	3.6	0.91	1.1	CAM-B3LYP LR	6-311+G(d,p)	H23a	1
carbo [7]helicenoid	404	2.9	449	3.4	0.31	1.4	3.8	4.0	0.84	1.4	PBE0–1/3 LR	6-311+G(d,p)	H23a	1
carbo[7] helicene	408	8.6	532	4.2	0.71	0.49	6.7	3.2		0.48	ωB97X-D	6-31G(p,d)	H25b	2
hetero [9]helicene	460	12	548	27	0.43	2.2	66	52	0.52	0.79	PBE0-D3	Def2-SVP	H26	9Ha
carbo [4]helicene	413	0.28	424	1.2	0.08	4.3	3.5	2	0.46	0.57	B3LYP-D3BJ	6-311+G (2d,2p)	H29a	2a
carbo [4]helicene	392	3.2	430	3.1	0.28	1.0	4.6	3.8	0.44	0.83	B3LYP-D3BJ	6-311+G (2d,2p)	H29b	2c
carbo [4]helicene	416	1.0	426	1.8	0.07	1.8	3.9	2.3	0.47	0.59	B3LYP-D3BJ	6-311+G (2d,2p)	H29c	2d
carbo [4]helicene	395	4.0	429	3.6	0.25	0.9	4.9	4.6	0.38	0.94	B3LYP-D3BJ	6-311+G (2d,2p)	H29d	2e
carbo [5]helicene	435	3.0	475	3.0	0.24	1.0	2.9	3	0.39	1.0	B3LYP-D3BJ	6-311+G (2d,2p)	H30	2p
carbo [6]helicene	409	10.8	430	3.4	0.15	1.9	1.0	4.1	0.06	4.1	B3LYP-D3BJ	6-311+G (2d,2p)	H31	2x
hetero [7]helicene	420		433	5	0.09		3.6	2.4		0.03	CAM-B3LYP	def2-SV(P)	H32	10
hetero[7] helicene				5				11			B3LYP-D3BJ	6-311G(2d,p)	H32	exo-4
hetero [9]helicene	392	31	466	4.2	0.50	0.13	382	19		0.05	CAM-B3LYP	def2-SV(P)	H33	1
hetero[9] helicene				4				28			B3LYP-D3BJ	6-311G(2d,p)	H33	exo-5
hetero [9]helicene	395	18	470	12.5	0.50	0.7	61	24		0.4	CAM-B3LYP	def2-SV(P)	H34	2
hetero[9] helicene				13				19			B3LYP-D3BJ	6-311G(2d,p)	H34	endo-6
hetero[7] helicene	431	15	450	10	0.12	0.67	13	60	0.59	4.6	B3LYP-D3BJ	6-311G(2d,p)	H35	TD[7]H
hetero[9] helicene	445	36	483	40	0.22	1.3	49	66	0.56	1.3	B3LYP-D3BJ	6-311G(2d,p)	H37	TD[9]H
carbo[5] helicene				2.3				3.7			B3LYP-D3BJ	6-311G(2d,p)	H38	1
hetero [7]helicene	388		398	0.81	0.08		4.5	0.83	0.68	0.18	MN15	6-311G(d,p)	H44	OH7
hetero [9]helicene	404		411	2.20	0.05		13.9	2.28	0.51	0.16	MN15	6-311G(d,p)	H45	OH9
hetero [5]helicene	430	0.7	480	0.25	0.49	0.36	0.50	0.30	0.66	0.6	CAM-B3LYP	Def2-SVP	H46a	10
hetero [5]helicene	452	0.56	500	0.95	0.48	1.7	0.48	0.77	0.67	1.6	CAM-B3LYP	Def2-SVP	H46b	11
hetero [5]helicene	458	2.0	590	3.5	0.74	1.75	2.19	4.56	0.59	2.1	CAM-B3LYP	Def2-SVP	H46c	12
hetero[5] helicene	383	9.9	407	4.2		0.42	13	10.3		0.79	B3LYP	cc-pVDZ	H48a	BN[5]
hetero[5] helicene	413	8.0	424	5.1	0.08	0.64	10.7	8.1		0.76	B3LYP	cc-pVDZ	H48b	BN[5](CN)2
hetero[6] helicene	424	10.8	434	13	0.06	1.2	22.4	22.9		1.0	B3LYP	cc-pVDZ	H49	BN[6]
carbo[5] helicene	375	6.0	409	4.4		0.73	9.55	8.5		0.89	B3LYP	cc-pVDZ	H50	CC[5]
hetero[6] helicene	424	10.8	434	13	0.06	1.2	21.3	15.6		0.73	CAM-B3LYP-D3BJ	6-311+G(d,p)	H49	BN[6]
hetero [6]helicene	435	1.6		1.6		1.0	1.34	1.39		1.0	M06-2X	Def2-SVP	H52	EH2
hetero [7]helicene	439	2.2		2.2		1.0	1.79	1.73		1.0	M06-2X	Def2-SVP	H53	EH3
carbo[4] helicene	629	0.45	655	0.5	0.08	1.1	0.42	0.43		1.0	ωB97X-D	Def2-SVP	H57	1
carbo[4] helicene	588	2.1	613	2.3	0.09	1.1	2.7	1.8		0.67	ωB97X-D	Def2-SVP	H58	2
carbo[4] helicene	470	4.0	532	0.9	0.30	0.22	4.0	3.2	0.41	0.8	CAM-B3LYP	6-31G(d,p)	H59a	7b
carbo[4] helicene	479	3.8	525	1.5	0.23	0.39	3.2	3.6	0.39	1.1	CAM-B3LYP	6-31G(d,p)	H59b	7c
carbo[4] helicene	462	14	468	14	0.03	1.0	23	19	0.23	0.83	B3LYP	6-31G(p,d)	H60a	[3]HA(Tip)
carbo[4] helicene	465	15	471	12	0.03	0.80	15	13	0.23	0.87	B3LYP	6-31G(p,d)	H60b	[3]HA(Ph)
carbo[4] helicene	464	13	469	12	0.03	0.92	14	13	0.22	0.93	B3LYP	6-31G(p,d)	H60c	[3]HA(C6F5)
hetero [9]helicene	546	5.5	613	5.8	0.25	1.1	12.8	17.9		1.4	B3LYP	Def2-SVP	H64	BN[9]H
hetero[7] helicene	504	2	550	1.4	0.21	0.7		1.24			M06-2X	6-311G(p,d)	H65	DCzAO1
hetero[7] helicene	516	0.7	555	0.8	0.17	1.1		0.16			M06-2X	6-311G(p,d)	H66	DCzAO2

^aData for electronic calculations only (no vibronic or nuclear ensemble calculations). Data for multiple helicenes, metallohelicenes, helicene hoops, and nanographenes not included. See Charts – for structure numbering.

^b g _abs and g _lum multiplied by 10³. ΔE_S in eV.

^cLarge overlap with neighboring bands of opposite sign (g _abs value underestimated).

^dExperimental data in DCM.

^eMultiple conformers present.

^fOpposite sign to the experiment; not included in the statistics.

^gExperimental data from ref .

6.

7.

3.1.2.1. Helicenes and Helicenoids

The first computational study of CPL properties of hexahelicene derivatives was reported by Abbate, Longhi, and co-workers in 2013. The set of compounds included carbo[6]­helicene (H1, Chart ), its 2-methyl and 2-bromo derivatives (H2 and H3), and 5-aza[6]­helicene (H4). In all cases, TD-DFT was employed using the long-range-separated CAM-B3LYP functional (Section ) in conjunction with the triple-ζ TZVP basis set. Figure reports the experimental and calculated absorption, emission, ECD, and CPL spectra of carbo[6]­helicene (H1), which can be considered to be the parent molecule of all hexahelicenes and lends itself to a preliminary discussion of the main spectroscopic features of the whole helicene family. The absorption and ECD spectra display several peaks of moderate intensity in the UV range. The bands centered around 250 and 320 nm are assigned as ¹B_a and ¹B_b transitions, respectively, according to Platt’s nomenclature. , It must be noted that at the excitation wavelength of 350 nm the extinction coefficient is ∼10⁴ M^–1 cm^–1. The following bands around 400 nm, due to ¹L_a and ¹L_b transitions, are much weaker in intensity in both the absorption and ECD spectra. In particular, the lowest-energy ¹L_b transition has substantial electric-dipole-forbidden character and gives rise to a faintly visible ECD band at 412 nm, negative for the P configuration, with g _abs = −0.9 × 10^–3. This latter band is associated with the CPL signal detected above 400 nm, which is also negative for the P configuration and has g _lum = −1.1 × 10^–3. The GS and ES geometries of H1 are, accordingly, very consistent with each other. Similar agreements were obtained for compounds H2–H4 (Table ). It must be stressed that all experimental spectra clearly display the presence of vibrational fine structure, which was taken into account by vibronic calculations in a study to be discussed below. The calculations reproduced all of the main spectroscopic features, in particular, the sign of the CPL band, although the calculated |g _lum| = 7.1 × 10^–3 overestimated the experimental value (Table ). Similarly overestimated is the Stokes shift (experimental ΔE_S = 0.11 eV; calculated ΔE_S = 0.36 eV). In a systematic investigation on ECD calculations on carbo­[n]­helicenes, Inoue and co-workers had highlighted that a careful choice of the DFT functional, and in particular the amount of exact exchange, is crucial for reproducing the chiroptical properties of this class of compounds. For instance, B3LYP and PBE0 functionals yield an incorrect sign for the ¹L_b band. The choice of the CAM-B3LYP functional by Abbate and co-workers was justified by the necessity of an accurate simulation of the excited-state geometry; although at that time no case history was available, the choice appears farsighted in light of the future results, presented in this review. The authors further evaluated the intensity and relative orientation of calculated transition moments, again inspired by the work by Inoue and co-workers who had emphasized the role played by angle θ _μm defined by m ₁₀ and μ ₀₁ for the carbo­[n]­helicene series. Angle θ _μm is 90° for the ¹B_b transition of planar anthracene (carbo[3]­helicene) and decreases progressively upon increasing n; however, for the most common values of n = 5–7, θ _μm is between 45 and 75° (i.e., still far from the ideal values of 0 and 180°). For the ¹L_b transition of H1, Abbate and co-workers calculated θ _μm = 96° in absorption (S₀–S₁ transition, arising from the dot product μ ₀₁ ·m ₁₀ ) and 130° in emission (S₁–S₀ transition, arising from the product μ ₁₀ ·m ₀₁ ; notice the exchange in the indices). A more favorable θ _μm = 161° for the S₁–S₀ transition was obtained for the hetero[6]­helicene H4, while H2 and H3 led to θ _μm closer to 90° and almost vanishing rotational strengths.

15.

Experimental (first and third row) and calculated (second and fourth row) ECD, CPL, absorption, and emission spectra of carbo[6]­helicene (H1, Chart ). Blue traces for (P)-enantiomer, red traces for (M)-enantiomer. Reprinted with permission from ref . Copyright 2014 American Chemical Society.

As already mentioned, 2 years later the same authors, in collaboration with Santoro and co-workers, re-examined [6]­helicenes H1–H4, adding a further dimension to their analysis, namely, estimating vibronic contributions to electronic spectra with special attention to HT and Duschinsky effects (Section ). The study could reproduce the peculiar appearance of absorption, emission, and ECD spectra of H1–H4 in the long-wavelength region with respect to their vibrational fine structure. Very importantly, it helped interpret the sign inversion seen in the ECD of H1 at around 400 nm (Figure ) as being due to a vibronic pattern within the S₀–S₁ transition and not to the superposition of S₀–S₁ and S₀–S₂ transitions. Such sign inversions in ECD profiles may be induced by HT contributions and can be confirmed by comparing vibronic calculations at FC and FCHT levels (Figure ). Turning to CPL simulations, vibronic calculations replicated well the appearance of the CPL spectrum of H4 (the strongest in the set) while they predicted an overall wrong sign for the CPL envelopes of H1 (Figure ) and H2, a failure attributed to a possible overestimation of HT effects.

16.

Vibronically resolved experimental and calculated ECD and CPL of carbo[6]­helicene (H1) with various methods. HT₀ and HT₁ mean that the transition dipoles were expanded around the GS or ES equilibrium geometries, respectively. Notice that the horizontal axis is expressed in energy rather than in wavelength. Reprinted with permission from ref . Copyright 2016 American Chemical Society.

Compounds H1 and H4 were further reconsidered by the same authors in comparison with two thia-bridged triarylamine heterohelicenes (H5 and H6, Chart ), formally a hetero[4]­helicene and a hetero[6]­helicene. TD-DFT calculations at the CAM-B3LYP/TZVP level reproduced well the key parameters (Table ), especially for H6. Notably, the g _lum value for this compound reaches almost 10^–2. The different behavior with respect to H1 was traced back to both geometric and electronic reasons. In fact, the S₁ geometries of H5 and H6 were found to be quite different from S₀ ones because, in order to attain a more compact structure, they lose the C ₂ symmetry. This, in turn, impacts the FMO involved in the S₁–S₀ transitions, which are not symmetric. Conversely, the S₁–S₀ transition densities for H1 were fully symmetric. Carbo[6]­helicene H2 and hetero[6]­helicene H6 were also revaluated by Chen et al. in a systematic analysis which will be treated in detail in a following section (Section ). These authors obtained S₁ geometries by TD-DFT calculations using two different approaches. What they refer to as the excited-state stationary point corresponds to the energy minimum (or minima) found on the S₁ PES by geometry optimization, which is the standard approach in CPL calculations outlined in Section . Alternatively, the nuclear ensemble approach was employed to sample different molecular configurations in the S₁ state (Section ), corresponding to all possible molecular vibrations. For H6, the calculations run at the CAM-B3LYP/def2-SVP level led to similar results for the two approaches. For H2, on the contrary, the use of the nuclear ensemble approach led to strongly reduced g _lum with respect to stationary-point calculations (5.6 × 10^–5 vs 2.4 × 10^–3), in much closer agreement with the experiment. In some conformations generated by the nuclear ensemble approach, the terminal rings of the [6]­helicene system lie so close to allow for complete delocalization of transition density over the entire molecule during vibrations.

A family with a skeleton similar to that of H5–H6 but with oxygen atoms instead of sulfur atoms was described by Wakamiya, Murata, and co-workers (H7a–c, Chart ). Upon substitution at position 4 of the naphthyl ring with electron-withdrawing groups (formyl, H7b, and 2,2-dicyanovinyl, H7c), the compounds undergo a strong bathochromic shift in both absorption and emission, which is captured well by calculations run at the CAM-B3LYP/6-31G­(d) level; the use of a relatively small double-ζ basis set is noticed here (Table ). Molecular orbitals’ plots nicely illustrated the substituent effects, with the LUMO being progressively lowered in energy and being dislocated on the naphthalene moiety upon substitution while the HOMO is localized on the phenoxazine ring. The S₁ excited state is an intramolecular charge transfer (ICT) state which, expectedly, showed a marked dependence on the solvent: the emission color of H7b was modulated from green to red when passing from cyclohexane to DMSO. The solvent-dependent emission wavelength was, however, not theoretically investigated.

A different family of carbo[6]­helicene derivatives purposely designed to undergo ICT transitions was reported by Faveraeu, Autschbach, Crassous, and co-workers (H8a–b and H9a–e, Chart ). , The ICT behavior is bestowed by molecular skeletons of the type A-D or A-D-A (A = π-acceptor, D = π-donor), where the carbo[6]­helicene core acts as a π-donor and naphthalimide units appended at 2,25 positions act as π-acceptors, linked by ethynyl moieties which ensure full conjugation. These compounds exhibit several noteworthy chiroptical properties deserving in-depth discussion. The lowest-energy side of the ECD spectra of H8a–b shows intense bands assigned to the ICT states; in particular, for H8b, exciton coupling between ICT states is at play, allowing for a significant increase in ECD intensity with respect to H8a. A modest solvent dependence of the shape and intensity of absorption and ECD spectra was detected, with g _abs of H8b at 435–440 nm ranging from 6 × 10^–3 in CHCl₃ to 9 × 10^–3 in cyclohexane. Conversely, the emission and CPL spectra of H8a and especially H8b were strongly dependent on the solvent, in accord with the ICT nature of the emission band. Both compounds exhibited a pronounced bathochromic shift upon increasing the solvent polarity: passing from cyclohexane to DMF (dimethylformamide), λ_em ranged from 436 to 576 nm for H8a and from 436 to 563 nm for H8b (Figure a). The spectral shape also varied from being narrow and structured in cyclohexane to very broad in DMF. The solvent-dependent fluorescence behavior followed the Lippert-Mataga relationship between the Stokes shift and the solvent orientation polarizability values Δf. This observation demonstrates that the measured spectral shifts are due to nonspecific dipole–dipole solute–solvent interactions. Turning to CPL, while H8a showed almost constant g _lum = 2 × 10^–3 in all solvents, H8b exhibited a pronounced dependence on the solvent, ranging from g _lum = 9.5 × 10^–3 for the structured CPL signal in cyclohexane to g _lum = 3 × 10^–3 for the broad CPL envelope in DMF (Figure a). The complex (chiro)­optical behavior of compounds H8a–b was investigated by TD-DFT calculations run with the 50/50 hybrid BHLYP functional and the SV­(P) basis set. Again, we notice that a relatively small, double-ζ basis set with no polarization functions on hydrogen atoms was employed. More importantly, the authors simulated solvent effects by means of the external iteration PCM method (a state-specific polarization response approach discussed in Section ) for cyclohexane, dichloromethane (DCM), and DMF. Calculated solvent-dependent Stokes shifts reproduced very well the experimental ones. The calculated dipole moments for the S₁ state visibly increased with the solvent polarity, and they were 2 to 3 times greater with respect to S₀ ones. This fact confirms that S₁ is more polar than S₀, which underlies the observed solvatochromism. Calculated g _lum values (Table ) reproduced the observed trends; in particular, the solvent-dependent g _lum for H8b was captured (although the increase upon solvent polarity was underestimated compared to the experiment). The authors provided a model to interpret the solvent dependence, based on exciton theory (Figure b): polar solvents like DMF provide a strong reaction field which suppresses the transition dipole interaction and hence the strength of exciton coupling.

17.

(a) Experimental emission and CPL spectra of carbo[6]­helicene derivatives H8a–b (Chart ) measured in different solvents. (b) Trend of g _lum values measured for H8a–b with an illustration of the solvent field effect. Adapted from ref under a CC-BY-NC 3.0 Unported License.

In a later development of the chiral skeletons, the same authors replaced the naphthalimide moieties in H8a–b with 4-pyridyl and 4-subtituted phenyl moieties in H9a–e (Chart ). The different substituents allowed for a modulation of the push–pull properties of the [6]­helicene core and the appended ethynylaryl moieties: while the pyridyl, 4-cyanophenyl, and 4-nitroophenyl rings in H9a–c act as π-acceptors (A), the 4-amino and 4-N,N-dimethylamino rings in H9d-e act as π-donors (D). Taking carbo[6]­helicene (H1) and its 2,25-bis-ethynyl derivative (H10) as the reference, the authors explored the effect of push–pull design on (chiro)­optical properties, also with the help of TD-DFT calculations run at the CAM-B3LYP/def2-SV­(P) level, estimating vibronic contributions at the FCHT level. First, it was emphasized that the relatively small g _lum values measured for H1 and H10 (Table ) are related to an unfavorable orientation of transition dipoles (Figure ). Passing to substituted H9a–e, frontier orbitals become localized as expected, namely, the HOMO on the D moiety and the LUMO on the A one (Figure ). The S₀–S₁ and S₁–S₀ transitions acquire the character of excitonically coupled ICT transitions. The associated g _lum values increase markedly, reaching a maximum value of 2.8 × 10^–2 for H9b. This corresponds to an almost ideal orientation of transition dipoles, with θ _μm = 5.4° (Figure ). Similar to H8a–b, compounds H9a–e displayed solvatochromism in both emission and CPL spectra, which was not theoretically investigated. The paper also reports the application of a TMS-protected analog of H10 as the active layer in a CP-OLED device, affording electroluminescence g _EL values of (7–8) × 10^–3, almost one-half of the g _lum values measured on thin films.

18.

Top: EDTM (green), MDTM (red), and their angle θ calculated for S₀–S₁ and S₁–S₀ transitions of carbo[6]­helicenes H1, H10, H9b, and H9e (Chart ). Bottom: FMOs of H9b and H9e showing the different localizations of HOMO and LUMO. Adapted from ref under a CC-BY-NC 3.0 Unported License.

A similar strategy was followed by Srebro-Hooper, Berrée, Crassous, and co-workers, leading to carbo[6]­helicenes H11a–b (Chart ) substituted at the 2 and 2,25 positions by boranyl groups. It must be observed that the boranyl functionalities are not embedded in the helicene skeleton, as occurs in other derivatives that we shall encounter below. Like previously described derivatives H8a and H9a–e, compound H11b shows the effect of exciton coupling in the low-energy region of ECD spectra and hence in CPL spectra. The measured g _lum values were 1.3 × 10^–3 for H11a and 2.5 × 10^–3 for H11b. TD-DFT calculations run at the PBE0/SV­(P) level reproduced the 2-fold increase in g _lum passing from H11a to H11b (Table ), which was traced back to a more favorable θ _μm angle.

Turning our attention to pentahelicenes, Zheng, Chen, and co-workers reported in 2022 a systematic study on carbo[5]­helicenes substituted with triarylamino and triarylborane moieties at the 7,14 (H12a–b, Chart ) and 9,12 positions (H13a–c). The substituents, belonging to D and A types, respectively, enable strong emission with tunable properties. All compounds exhibited a major ECD band above 350 nm, whose intensity varied markedly within the set; accordingly, the CPL band above 400 nm (whose sign is in accord with ECD one) displays some variety in intensity and position, keeping its g _lum ≈ 10^–3 (Table ). A notable exception is offered by compound H13b, with a D-H-A skeleton, reaching g _lum = 1.2 × 10^–2. A similar though less pronounced enhancement was also recorded on thin film samples. TD-DFT calculations run with the Minnesota global hybrid functional M08-HX in conjunction with the 6-31G­(d) basis set, and including PCM for hexane, emphasized the role played by the orientation of transition dipoles (Figure ). The θ _μm angle was in fact the minimum for H13b (61°) and led to a calculated g _lum = 2 × 10^–2 in agreement with the experiment (Table ). The pronounced solvatochromism shown by H12b and H13b in emission and CPL spectra was not investigated computationally.

19.

EDTM and MDTM calculated for the S₁–S₀ transition of carbo[5]­helicene derivatives H12a–b and H13a–c (Chart ). Reprinted with permission from ref . Copyright 2022 The Royal Society of Chemistry.

The importance of tweaking the orientation of electric and magnetic transition dipoles becomes apparent when looking at Imai and co-workers’ study on phosphine-derivatized carbo[5]­helicenoid (H14a–b, Chart ). TD-DFT calculations run with the Minnesota MN15 functional (a global hybrid with 44% HF exchange) and a 6-311++G­(3df,2p) basis set predicted that oxidizing H14a to H14b would tilt θ _μm from 90.5 to 99.1°, enhancing the CPL signal by a putative factor of 20. Here we wish to emphasize the use of a triple-ζ basis set with multiple diffuse and polarization functions, which is justified by the predictive nature of the study, calling for special accuracy. As for the functional, MN15 has been sporadically employed for CPL calculations of several kinds of systems, especially by Inoue’s group (vide infra). The calculations offered further insight into the differences between H14a and H14b. The lowest-energy excited state of H14a is localized on the phosphine-substituted naphthalene ring, whereas for the phosphine oxide H14b it has ICT character. In accord with the predictions, H14a was weakly CPL-emissive, whereas H14b showed g _lum = 1.9 × 10^–3 and g _lum/g _abs ≈ 1 in various solvents, in good agreement with calculations (Table ).

A huge amount of experimental and theoretical work has been devoted to heptahelicenes. In 2021, Matsuda and co-workers investigated carbo[7]­helicene (H15a, Chart ) and two derivatives thereof (H15b–c), focusing on the effect of substitution on the CPL properties. A preliminary theoretical analysis found that carbo[7]­helicene (H15a) features quasi-degenerate HOMO–1/HOMO and LUMO/LUMO+1 pairs of orbitals. The authors observed that this situation is ideal for molecular design, as the energy levels (and relative transitions) may be manipulated by chemical substitution. In this case, the introduction of two cyano substituents at 7,12 positions relieved orbital degeneracy by lowering the LUMO (Figure a); however, the S₀–S₁ transition remained only weakly magnetically allowed. Further introduction of two methoxy groups at 2,17 positions flipped the relative energies of HOMO–1 and HOMO, which had different symmetries, thus making the S₀–S₁ transition magnetically allowed with a calculated g _lum = 1.8 × 10^–2. The calculations were run using the B3LYP functional with an empirical dispersion correction of Grimme’s D3 type with Becke-Johnson damping (D3BJ). As the terminal rings of [n]­helicenes with n > 6 undergo significant stacking interactions, the use of a dispersion correction is indeed recommended for DFT functionals like B3LYP, which display incorrect asymptotic behavior and thus are unable to properly describe dispersion forces. The chosen basis set was 6-311G­(2d,p) (i.e., with multiple polarization functions but devoid of diffuse functions). In addition to the novel compounds H15a–c, the calculation approach was tested on other [7]­helicene and [7]­helicenoids whose experimental CPL data were available in the literature. Once synthesized, compounds H15a–c were revealed to be emissive in CHCl₃, with the QYs increasing in the order H15a < H15b < H15c as predicted by calculations. The ECD spectra of H15a–c were apparently consistent in shape, with measured g _abs values of around 2 × 10^–2 (Table ). However, based on the calculation results, it was inferred that the red-most ECD band is due to the S₀–S₁ transition only for H15c; for H15a–b, this transition is allied to a weak rotational strength, and the ECD features are allied to higher-energy transitions. Conversely, in accord with Kasha’s rule, the emission and CPL bands are all associated with the S₁–S₀ transition. In fact, for H15a–b both the g _lum and g _lum/g _abs were much smaller than for H15c (Table ), and for H15b, the CPL band was also opposite in sign to the red-most ECD band. Useful insight into the chiroptical properties was gained by looking at transition dipole vectors and densities for the S₁–S₀ transition. In fact, for H15a–b, the calculated angle θ _μm = 180° was counterbalanced by small absolute values |m ₀₁ | and |μ ₁₀ |, leading to small rotational strength R ₁₀. Conversely, for H15c a less favorable θ _μm = 60° implies halving the geometrical factor of R ₁₀ ∝ cos­(θ _μm ), but this is repaid by larger |μ ₁₀ | and especially |m ₀₁ |, which was well illustrated by transition dipole density plots (Figure b). It must be noticed that the calculated CPL spectrum for H15a had the opposite sign with respect to the experiment, a fact which was not commented on by the authors. The simple carbo[7]­helicene H15a represents one of the rare instances in our survey where the experimental and calculated CPL signs disagree. We infer that the disagreement may be related to the presence of quasi-degenerate frontier orbitals in H15a, requiring a multireference method instead of TD-DFT for a proper description. Finally, the authors also estimated spontaneous emission rates k _f, which are related to emission quantum yields Φ_f. Again, the calculated order was H15a < H15b < H15c, which reproduced the experimental Φ_f values. It was noticed that compound H15c has, in combination with high g _lum, a good Φ_f = 0.17; from the published data, we estimate B _CPL = 155 M^–1 cm^–1, which is indeed among the highest reported for helicene compounds. The same compounds H15a–c were reconsidered recently by Wu and co-workers for vibronic calculations of emission and CPL spectra, for which the CAM-B3LYP/TZVP level of theory was employed in combination with various frameworks for the treatment of vibronic terms. It is worth noting that the calculated CPL envelopes are all negative for (P)-H15a–c, which is at odds with experimental spectra of H15a and H15c. In this case too, the sign discrepancy was not discussed by the authors.

20.

(a) Orbital correlation diagram and composition of the S₁–S₀ transition of carbo[7]­helicene derivatives H15a–c (Chart ). Red and blue frames represent C ₂-symmetric and C ₂-antisymmetric MOs, respectively. (b) Transition moment density analysis of H15b–c, showing the density distribution of EDTM and MDTM vectors for S₁–S₀. Reprinted with permission from ref . Copyright 2021 American Chemical Society.

Katsuda’s group also reported the extended carbo[7]­helicene H16, using a MO correlation diagram (starting from perylene) to evaluate the effect of conjugation extending and ring closing. ECD and CPL quantities were calculated at the B3LYP/6-311G­(2d,p) level (Table ).

Two bromo-substituted carbo­[n]­helicenes (n = 6, 7) will be described in Section devoted to organic triplet emitters.

Turning to hetero[7]­helicenes, in 2020 Wang, Wang, and co-workers reported vibronic ECD and CPL calculations on the aza[7]­helicene H17 at the FCHT level using both VH and AH models (Section ). Overall, the band shapes were nicely reproduced; as it often happens for vibronic calculations, however, quantitative figures such as g _lum and g _abs were not provided. Similar calculations were run on two triarylborane-based [5]­helicenes derivatives.

The two aza[7]­helicenes H18 and H19 were described by Mori, Miura, and co-workers. The calculation approach used by this research group is based on CC2 (Section ; other applications will be discussed below) rather than on TD-DFT. The authors stressed that “a considerable amount (∼15%) of double-excitation character was identified for both of the helicenes, which somewhat justifies our failure of the (conventional) TD-DFT method to reproduce the experimental results”; unfortunately, no data were provided to substantiate this latter statement. We notice that calculated g _lum values (the only reported quantities) sufficiently reproduce experimental ones (Table ).

A pH-sensitive chiroptical switch based on aza[7]­helicene H20 was designed by Pieters, Champagne, Audisio, and co-workers. This compound undergoes a marked evolution of (chiro)­optical properties upon protonation including (a) the emission color changes from blue to orange; (b) the emission band for the neutral form between 410 and 500 nm is structured while the red-shifted band for the protonated form between 500–700 nm is broad and featureless, indicating ICT character; (c) the CPL band changes its sign while keeping its absolute |g _lum| ≈ 10^–3. The above behavior was reproduced and rationalized by TD-DFT calculations run at the B3LYP-D3BJ/6-311+G­(2d,2p) level, including PCM for the solvent (DCM). In particular, it was noticed that upon protonation the LUMO orbital is greatly stabilized, which is responsible for the red shift, and becomes localized on the protonated quinoline moiety; the authors also recognized a common drawback of TD-DFT calculations in underestimating the energies of ICT states. This aspect seems to be compensated for by a similar error in the excited state so that the calculated Stokes shift for H20·H ⁺ is reasonable (Table ). The calculated chiroptical data are also in very good agreement with the experiment, and they were brilliantly substantiated by transition moment plots, NTOs, and transition density plots (Figure ). The pivotal role played by the transition density in boosting CPL has been recently emphasized by Brédas, Yang, and co-workers in a publication we will describe in Section .

21.

Difference in S₁ and S₀ densities for the neutral and protonated forms of aza[7]­helicene H20 (Chart ). A blue (red) area indicates a region where the electronic density is larger (smaller) in the S₁ state than in S₀ state. Reprinted from ref under a CC-BY-NC-ND 4.0 license.

Within the family of carbo[7]­helicenoids, in 2023 the authors of the present review took into consideration the two extended carbo[7]­helicenoids H22a and H23a (Chart ) with terminal phenanthrene moieties. Our attention toward these compounds was justified by exceptionally high g _lum and g _lum/g _abs values measured in CHCl₃, g _lum = 3.0–3.2 × 10^–2 and g _lum/g _abs = 11–12 for both compounds, that is, 1 order of magnitude above those normally found for this class of molecules. , For our computational study, we considered the truncated models H22b and H23b to save computational time. Moreover, we considered the simple carbo[7]­helicenoid H21 to be a model system, which shares the same carbo[7]­helicenoid skeleton as H22a and H23a but displays more common values for g _lum = 1.2 × 10^–3 and g _lum/g _abs = 2.5. We employed two functionals, namely, CAM-B3LYP and PBE0-1/3, a parameter-free hybrid functional obtained by combining the PBE generalized-gradient functional with 1/3 the amount of exact exchange. Both functionals were used in combination with the D3BJ dispersion correction and the 6-311+G­(d,p) basis set. Moreover, we modeled the solvent polarization for CHCl₃ by using PCM in both the linear response and the VEM state-specific polarization scheme (Section ) in its unrelaxed density version (UD). Both calculation methods captured well the ground-state chiroptical properties (ECD spectra and g _abs values) of H21, H22a, and H23a. For reference carbo[7]­helicenoid H21, g _lum and g _lum/g _abs were also reproduced well (Table ), whereas we could not replicate the exceptional g _lum and g _lum/g _abs values previously measured for H22a and H23a. , To verify whether the terminal phenanthrene moieties might undergo intramolecular excimer formation, we looked at various computational data, namely, S₀ vs S₁ geometry, transition moments, NTO, and the related effective electron displacement index Γ_NTO (Section ). In all cases, a substantial consistency was found among the three compounds. For instance, the NTOs associated with the S₁–S₀ transitions and the corresponding Γ_NTO values are all very similar and not much affected by the “outer” benzene rings (Figure ). The inclusion of solvent effects for H22b did not change the situation much, although we evidenced a non-negligible difference between the LR-PCM and VEM-UD models. On the basis of our computational evidence, the difference between the experimental and calculated g _lum and g _lum/g _abs values for H22a/H22b and H23a/H23b was puzzling, and the exceptionally large g _lum values remained unjustified. Therefore, we were forced to admit that the latter could be plagued by instrument or calibration error and suggested their remeasurement. Just 1 week after our manuscript was published online, a contribution by Lu and co-workers appeared, where the CPL spectrum of compound H23a was remeasured in a different solvent (EtOAc/tert-butanol). Quite interestingly, the new recorded g _lum = 3.4 × 10^–3 and g _lum/g _abs = 0.9 were in very good accord with our calculations (Table ) and 1 order of magnitude smaller that the previously reported experimental data.

Recently, Ma, Shen, and co-workers investigated theoretically six [7]­helicenes/helicenoids, including carbo[7]­helicenoid H21 and new compounds H24a–e embedding different group-14 (hetero[7]­helicenoids) and group-15 (hetero[7]­helicenes) elements in the central ring; previously described carbo[7]­helicene H15a was taken as a reference. Experimental CPL spectra were available only for H15a and H21 (Table ). For the latter, a screening of various functionals (B3LYP, PBE0, M06-2X, ωB97X-D, and CAM-B3LYP) and basis sets (6-31G­(d), 6-311G­(d), 6-311G­(2d,p), def-TZVP, and def2-TZVP) was performed. To be noticed, the five functionals performed very similarly in terms of calculated g _lum (all between 4.8 and 5.3 × 10^–3 with the 6-311G­(2d,p) basis set), while basis set convergence was obtained with the 6-311G­(2d,p) basis set; the errors (percent ARE) associated with the smaller 6-31G­(d) and 6-311G­(d) basis sets were 18 and 6%, respectively. Looking at the six analogs with a central five-membered ring, the two hetero[7]­helicenes H24d-e with S and Se atoms featured the highest calculated g _lum (1.4 × 10^–2 for H24d, 1.6 × 10^–2 for H24e), thanks to the more favorable orientation between transition moments (θ _μm = 120° for H24d, 122° for H24e; other values for H15a and H24a–c were 107° and from 102 to 115°, respectively). We also notice that for carbo[7]­helicene H15a, apparently with an (M) configuration, the authors listed a calculated g _lum = +7.8 × 10^–3, in keeping with the negative g _lum = −8 × 10^–3 calculated by Kubo et al. for the (P) configuration. However, the experimental g _lum value quoted by Ma, Shen, and co-workers and taken from Kubo et al. is positive and corresponds to the (P) configuration. The carbo[7]­helicenoid H21 has also been the subject of vibronic ECD and CPL calculations by Liu’s group.

Two carbo[7]­helicenes (H25a–b, Chart ) embedding two naphthalimide rings and endowed with methoxy substituents were reported by Ravat and co-workers. The push–pull design imparts to the S₁ states ICT character for H25b, which is accompanied by the expected solvatochromism due to the large dipole moment difference between S₁ and S₀ states (11.7 D). Quite unusually, the quantum yields and fluorescence lifetimes of H25b increased from nonpolar (Φ_f = 0.11 in hexane) to polar solvents (Φ_f = 0.34 in DMSO). TD-DFT calculations (ωB97X-D/6-31G­(d,p) level) revealed a substantial geometry difference between S₀ and S₁, which explained the broad emission and large Stokes shifts observed even in apolar solvents. The inclusion of a continuum solvent model did not significantly alter the optimized geometries but caused a sizable increase in the dipole moment in more polar solvents, substantiating the dependence of ICT character on solvent polarity. The authors reported ECD and CPL spectra of H25a–b in chloroform, both showing a sign inversion between the red-most ECD band (maximum at 410–420 nm) and the CPL band (maximum at 510–520 nm). Surprisingly, this latter finding was not commented on, and only absolute calculated |g _lum| values are given. Due to this uncertainty, we will not include the data on H25a–b in our statistics.

A few other carbo­[n]­helicenes or hetero­[n]­helicenes with n > 7 have been considered for CPL calculations. Mahato and Panda have described three aza[9]­helicenes (H26–H28, Chart ) with the central benzene ring fused with a 1,4-dihydropyrazine (H27) or a 1,4-dihydroquinoxaline moiety (H28), plus an aza[11]­helicene derived from H26. The aim of this research was maximizing the magnetic transition dipoles (m ₀₁ and m ₁₀ ) and hence the corresponding g _abs and g _lum values with respect to the reference compound H26, for which an experimental g _lum = 2.7 × 10^–2 had been reported. CPL calculations were run at the PBE0-D3/def2-SVP level, including the PCM solvent model for DCM. Based on the calculated data (those for H26 are listed in Table ), it was concluded that H27 would exhibit a chiroptical response similar to that of H26 while H28 would be CPL-silent because of almost perpendicular μ ₁₀ and m ₀₁ moments. The different behaviors were rationalized by looking at NTO and transition dipole density plots. The aza[11]­helicene was also predicted to have a smaller |g _lum| than the parent H26.

3.

Several groups have explored the impact of the (hetero)­helicene size on (chiro)­optical properties. Such systematic studies are especially valued from our viewpoint, as they allow estimating the impact of a specific molecular property on (chiro)­optical performance and all related theoretical parameters. A thorough comparative study of the photophysical and chiroptical properties of over 20 carbo­[n]­helicenes with n = 4, 5, and 6 and different substitution patterns was performed by Pieters, Baudoin, and co-workers. For some compounds of the series, TD-DFT calculations were run (H29a–e, H30, and H31; Table ). In general, the photophysical and chiroptical properties varied considerably. The QY ranged between 0.02 and 0.13, in line with the usual values for carbohelicenes. The cyano group positively affected the QY but drastically reduced the emission dissymmetry (Φ_f = 0.13 and g _lum = 1 × 10^–4 for H29e), while the opposite happened with the methoxy group (Φ_f = 0.056 and g _lum = 3.6 × 10^–3 for H29d). This is a demonstration that high CPL brightness values often originate from a trade-off of emission efficiency (which is higher for more intense μ ₁₀ ) and dissymmetry (which is higher for more intense m ₀₁ ). TD-DFT calculations run at the B3LYP-D3BJ/6-311+G­(2d,2p) level with PCM for DCM efficiently reproduced experimental g _lum values for most compounds (Table ). An analysis of calculation results suggested that the higher g _lum values observed for H26b and H26d in the carbo[4]­helicene series were mainly due to a more favorable orientation between transition moments.

Autschbach, Crassous, and Gujarro described the exo-dithia­[7]­helicene H32 and the two isomeric dithia[9]­helicenes exo-H33 and endo-H34 (Chart ). The first striking difference between the smaller and higher analogs is the CPL sign, which is positive for (M)-H32 and negative for both (M)-H33 and (M)-H34. The experimental |g _lum| was twice as large for endo-H34 than for exo-H33, reaching 1.2 × 10^–2 (Table ). TD-DFT calculations were run with the CAM-B3LYP functional and def2-SV­(P) basis set. Quite surprisingly, electronic-only calculated CPL spectra of exo-H33 and endo-H34 provided the wrong sign. The inclusion of both FC and HT vibronic effects was necessary to recover the correct dominant sign. As is appreciated from Figure , FC terms are allied with a positive progression for exo-H33, responsible for the 0–0 positive band, but they are overcome by the negative progression allied with HT terms, which yields a CPL band shape in good agreement with the experimental envelope.

23.

Vibronically resolved experimental (black curves) and calculated (blue curves) emission and CPL spectra of exo-dithia­[7]­helicene H33 (Chart ). Adapted from ref under a CC-BY-NC 3.0 unported license.

Hirose and co-workers designed three hetero­[n]­helicenes fused to 2,1,3-thiadiazole rings at both ends with n = 5, 7, and 9 (H35–H37, Chart ). The corresponding carbo­[n]­helicenes of the same length were used as models (experimental CPL data are available only for the carbo[7]­helicene H15a, as discussed above). The systematic investigation also included already encountered H15b–c, H17, H32–H34, and maleimide-fused carbo[5]­helicene H38 (Table ). In this case too, the purpose was the enhancement of the magnetic-dipole-allowed character of the first excited state, to maximize g _abs and g _lum, as well as of the quantum yields. The authors postulated that adjusting the energy, shape, and symmetry of FMOs would be a way to achieve the desired properties. TD-DFT calculations were run at the B3LYP-D3BJ/6-311G­(2d,p) level, and the FMOs were carefully examined. Compounds H36 and H37 have HOMOs and LUMOs of different symmetry (C ₂-symmetric or -antisymmetric) and close in energy (Figure ), which is the root for large magnetic transition dipoles in absorption (S₀–S₁) and emission. In fact, they exhibit the largest m ₀₁ moments in combination with θ _μm ≈ 0°; the corresponding experimental g _lum values are the highest for all compounds examined (g _lum = 1 × 10^–2 for H36, 4 × 10^–2 for H37; Table ). Vibronic ECD and CPL calculations were also run on H38 by Liu and co-workers. The 8-Br derivative of H37 will be described in Section .

24.

Orbital correlation diagram and composition of the S₁–S₀ transition of 2,1,3-benzothiadiazole (BT) and hetero­[n]­helicenes H35–H37 (Chart ). Red and blue frames represent C ₂-symmetric and C ₂-antisymmetric MOs, respectively. Reprinted from ref , copyright 2023, with permission from Elsevier.

Mikkelsen and co-workers have reported four hetero­[n]­helicenes with n = 5, 9, 10, and 13 (H39–H42, Chart ). Unfortunately, only experimental emission data were available but not CPL data. While absorption and ECD spectra were calculated by employing an explicit solvent model for DCM using atomistic MD simulations with a polarizable embedding approach, CPL spectra were calculated in vacuo at the B3LYP/def2-SVP level of theory and included vibronic effects. The estimated g _lum values grew with the molecular size, with hetero[13]­helicene H39 reaching the maximum g _lum = 4.3 × 10^–2.

Recently, Saleem, Takizawa, and colleagues have explored the impact of the helical extension on the (chiro)­optical properties of three oxa­[n]­helicenes with n = 5, 7, and 9 (H43–H45, Chart ). A 2-fold increase in g _abs and a 3-fold increase in g _lum were observed on passing from H44 to H45. The g _lum values were reproduced very well by TD-DFT calculations run at the MN15/6-311G­(d,p) level (Table ), which was selected after a screening of functionals (MN15, B3LYP, CAM-B3LYP, BVP86, and B3PW91) and basis sets (6-311G with different polarization and diffuse functions). It is interesting that in terms of best matching with experimental g _lum, the selected basis set (6-311G­(d,p)) performed better than much larger ones (i.e., 6-311++G­(2d,p)), which definitely calls for error cancellation.

Four-coordinate organoboron compounds, which have their most important representatives in BODIPY dyes, are known to be strongly emissive and exhibit outstanding optical properties. , This family of compounds will be described in Section ; the following paragraphs cover helicene derivatives containing boron atoms in their helicene skeletons.

A first example of CPL calculations on three azabora[5]­helicenes (H46a–c, Chart ) was demonstrated in 2018 by Longhi, Pischel, Ros and colleagues. Interestingly, the parent compound (H46a) has the red-most ECD and CPL bands opposite in sign to its derivatives H46b–c for the same configuration; this behavior was correctly reproduced by TD-DFT calculations run at the CAM-B3LYP/def2-SVP level (Table ). A dual band appearing in the CPL spectrum of H46a was not explained. Ikeshita, Imai, Tsuno, and co-workers reported the two carbo[4]­helicenes H47a–b, fused to an ozaxaborinine ring, which can be identified as hetero[5]­helicenoids. These compounds are characterized by modest experimental dissymmetry values in solution, both in absorption and emission (g _lum = 1.1 × 10^–3 for H47a and 3.6 × 10^–4 for H47b). The reason is the coexistence of two diastereomers (i.e., (S,P) and (S,M)), in fast equilibrium in solution, allied with CPL signs of opposite sign which cancel each other, as was demonstrated by TD-DFT calculations (MPW91PW91/6-31+G­(d,p) level).

4.

By comparing the three boron imines H48a–b and H49 (Chart ) with the corresponding carbo[5]­helicene H50 and carbo[6]­helicene H51, Staubitz and co-workers convincingly demonstrated how incorporating B–N groups into helicene scaffolds may be used to boost their (chiro)­optical properties. The B–N helicenes were substantially brighter than the carbon analogues (Φ_f = 0.10 for H48a and 0.17 for H49 vs Φ_f = 0.03–0.04 for H50–H51); cyano substituents further increased the quantum yield (Φ_f = 0.25 for H48b). Compound H49 also showed the largest g _lum = 1.3 × 10^–2, yielding a significant CPL brightness B _CPL = 59 M^–1 cm^–1. TD-DFT calculations were run at the B3LYP/cc-pVDZ and 6-311+G­(d,p) level. Similar to other cases listed in Table , the calculations overestimated both g _abs and g _lum, which counterbalanced each other, providing a reasonable g _lum/g _abs ratio. The authors noticed that a different calculation method, namely, CAM-B3LYP-D3BJ/6-311+G­(d,p), reproduced much better g _lum values, whereas the calculated g _abs was only slightly changed (Table ). A further family of hetero[6]­helicenoids or hetero[8]­helicenoids incorporating the B,N or B,X motif (X = O, S, Se) will be described in Section (Chart ).

34.

Another boron-containing structural motif, the five-membered azaborole ring, was incorporated by Nowak-Król and co-workers into two helicene scaffolds with different extensions (H52–H53, Chart ). CPL dissymmetries were higher than less extended analogs but still around 2 × 10^–3 (Table ) and were reproduced by TD-M06-2X/def2-SVP calculations.

We continue this section by treating a few interesting cases where a helicene unit has been incorporated into a more complex skeleton either by a Csp²-Csp² biaryl junction or by ring fusion. Some of these compounds might well be categorized under macrocycles (Section ) or nanographenes (Section ) but are described here in view of the structural importance of the helicene scaffold. These examples are included in Table only when the chiroptical properties are mainly allied with the helicene unit.

Šolomek and co-workers conceived three helicene carbon nanohoops based on carbo[6]­helicene (H54, Chart ) and on carbo[5]­helicene (H56a–b) on one side and on [6] or [7]­cycloparaphenylene on the other side. , Compound H54 may assume a more stable conformation with Möbius topology (seen in the X-ray structure) and a less stable conformation with Hückel topology. The low-energy region of absorption and the ECD spectra of H54 are consistent with those of carbo[6]­helicene model H55. Accordingly, NTOs obtained after TD-DFT calculations run at the CAM-B3LYP/6-31G­(d) level depicted an S₀–S₁ transition mainly localized on the carbo[6]­helicene unit. In the excited state, the authors anticipated a large structural reorganization for the oligo­(p-phenylene) segment, which is capable of a larger release of molecular strain, thus becoming the main location of the S₁ state. Indeed, the emission spectrum of H54 was very red-shifted and much less structured with respect to H55. Additionally, the photophysical properties, including the high QY (Φ_f = 0.65), matched those of cycloparaphenylenes. The CPL spectrum of (M)-H54 showed a major negative band above 450 nm of opposite sign with respect to the red-most ECD band, suggesting again a substantial structural rearrangement from the S₀ to S₁ state. Curiously, it was flanked by a tiny positive band at 435 nm (Figure ). TD-DFT calculations confirmed that the S₁ state is fully located on the oligo­(p-phenylene) moiety (Figure ); the sign and intensity of calculated g _lum = −2.6 × 10^–3 for (M)-H54 also matched the experimental value g _lum = −2.2 × 10^–3. The small CPL band at 435 nm was assigned to a minor conformer which, although much less populated, had a higher calculated g _lum = 1.2 × 10^–2. A similar bistability was also evidenced for compound H56b, sharing the same [7]­cycloparaphenylene system as H54 though linked to a carbo[5]­helicene. In this case too, the CPL spectrum showed a dual band similar to that of H54, and the minor high-energy component due to a Hückel-type conformer was suppressed in shorter analog H56a.

25.

(a) Experimental CPL spectra of H54 (black curves, left axis; Chart ) and H55 (red curves, right axis). Solid traces for (M) enantiomers, dashed traces for (P) enantiomers. (b) NTOs for the S₁–S₀ transition of H54. Reprinted with permission from ref . Copyright 2022, John Wiley and Sons.

A lateral extension of the helicene backbone was conceived by Würthner and co-workers as a means to increase the intrinsically weak QY of the helicene scaffold. In compounds H57 and H58 (Chart ), a carbo[6]­helicene and a carbo[5]­helicene skeleton, respectively, were fused to two naphthalimide moieties in a symmetrical fashion. In keeping with the expectations, the QYs were very high (Φ_f = 0.73 for H57 and 0.69 for H58), which counterbalanced small-to-moderate emission dissymmetries (g _lum = 5.0 × 10^–4 for H57 and 2.3 × 10^–3 for H58) in reaching relatively high CPL brightness values (B _CPL = 22 M^–1 cm^–1 for H57 and B _CPL = 66.5 M^–1 cm^–1 for H58). Chiroptical data were nicely reproduced by TD-DFT calculations run at the ωB97X-D/def2-SVP level (Table ).

A π-expanded [4]­helicene system, corresponding to the pyreno­[a]­pyrene skeleton, was recently reported by Chalifoux and co-workers (H59a–b, Chart ). Despite the structural complexity, these compounds are relatively rigid and display consistent ECD and CPL spectra. Measured g _lum values were around 10^–3 and were slightly overestimated by TD-DFT-calculations together with g _lum/g _abs ratios (Table ).

A rather unique molecular design is shown by compounds H60a–c (Chart ), which consist of three anthracene units fused in a helical fashion. , They exhibit extraordinarily strong ECD signals in the UV range with Δε ≈ 1500 M^–1 cm^–1 and g _abs = 2.4 × 10^–2 at 351 nm for H60a. The lowest-energy ECD band was observed at 465 nm with g _abs = 1.4 × 10^–2, on par with the CPL band at 470 nm with g _lum = 1.4 × 10^–2. TD-DFT calculations run at the B3LYP/6-31G­(d,p) level revealed an ideal orientation between transition dipole moments for both S₀–S₁ and S₁–S₀ transitions, which are delocalized over the whole helical skeleton. Similar results were obtained for the other two derivatives H60b–c (Table ). The same concept of 1,2-fusion between anthracene rings was further extended to obtain superhelical structures such as H61 (Chart ) with as many as 21 fused benzene rings. This compound showed a prominent ECD band at 400 nm with Δε = 536 M^–1 cm^–1, but the low-energy ECD bands were much weaker. The calculated S₀–S₁ transition (CAM-B3LYP/6-31G­(d,p) level) was in fact associated with very weak MDTM. Conversely, the CPL spectrum was fairly strong, reaching g _lum = 1.2 × 10^–2 at 576 nm, and TD-DFT calculations predicted an even larger value. A relatively large structural rearrangement occurs between calculated GS and ES geometries, which explains the differences seen between S₀–S₁ and S₁–S₀ transitions.

In view of applications of CP emitters in electronic devices, such as CP-OLEDs, CPL TADF emitters may play an important role, , as they are able to harvest triplet excitons overcoming the spin statistics limit imposed on OLED efficiency. This latter implies that, after recombination of holes and electrons in the active layer into excitons, 75% of the excitons will be found in the nonemissive triplet state and only 25% in the emissive singlet state (Figure a). In a TADF molecule, S₁ and T₁ are sufficiently close in energy to allow for a thermally promoted reverse intersystem crossing (RISC), which converts excitons in the T₁ state to the S₁ state, whereafter they can decay back to the S₀ via emission. This is usually achieved by a molecular design consisting of D and A regions, associated with a strong intramolecular charge transfer (ICT), which in turn may lead to reduced S₁–S₀ oscillator strength and low QY. This limitation may be worked around by compounds containing donor and acceptor atoms in ortho/para positions in a condensed polycyclic aromatic skeleton. Such compounds are called Multi-Resonant (MR) TADF (MR-TADF). Given the amounts of parameters at play in such context, a computationally aided molecular design is highly beneficial to achieve suitable CPL-active (MR)-TADF compounds.

26.

(a) Principle of TADF emitters. (b) HOMO (blue isosurface) and LUMO (red isosurface) of hetero[6]­helicene H62b and hetero[4]­helicene H63 (Chart ) with estimated energy gaps (E _g). (c) Singlet/triplet states energy levels with spin–orbit coupling. Reprinted from ref , copyright 2023, with permission from John Wiley and Sons.

In 2023, Cao, Yang and co-workers engineered two hetero[6]­helicenes endowed with narrow and deep-blue circularly polarized emission and MR-TADF behavior (H62a–b, Chart ) similar to the parent hetero[4]­helicene (H63), but with improved photophysical properties and optical stability. The key structural element, typical of MR-TADF emitters, is the presence of electron-deficient B atoms and electron-rich N atoms ortho to each other, which are responsible for intense short-range charge-transfer (SRCT) transitions around 400 nm. Narrow emission bands peak between 440 and 444 nm, corresponding to deep-blue emission, and their fwhm (full width at half-maximum) is 23–27 nm (around 1400 cm^–1). Compounds H62a–b produced CPL spectra in both solution and thin films, with g _lum ≈ 1.5 × 10^–3, allied with B _CPL = 2.35 and 22.4 M^–1 cm^–1, respectively. TD-DFT calculations were run at the B3LYP-D3BJ/6-31G­(d,p) level to gain insight into the electronic and photophysical properties of the emitters. The RISC process benefits from small HOMO–LUMO gaps and large spin–orbit coupling (SOC) between S₁ and T₁ states, while the MR effect is favored by multiresonance FMOs, namely, HOMO and LUMO alternatively localized on adjacent atoms on the aromatic rings, in a unique nonbonding orbital fashion (Figure b). The calculations also predicted much larger SOC constants for H62a and H62b compared to that of parent H63. Accordingly, the experimental rate constant k _RISC for the RISC process increased by 38 and 46%, respectively, passing from H63 to H62a and H62b. Finally, the authors fabricated an OLED device capable of emitting a pure blue color with a remarkable external quantum efficiency (EQE) of 27.5% for H62a and 29.3% for H62b. Apparently, the CP-OLED performance was not tested.

A second helicene system endowed with MR-TADF properties is hetero[9]­helicene H64 (Chart ) also incorporating ortho-oriented B and N atoms. In this case too, the MR effect was easily visualized by plotting FMOs. The calculations, run at the B3LYP-D3BJ/def2-SVP level, predicted parallel electric and magnetic transitions moments, both in absorption and emission; consequently, calculated g _abs and g _lum were both large (1.8 × 10^–2 and 1.3 × 10^–2, respectively; Table ), exceeding the measured values by a factor of ∼2. The OLED device fabricated from H64 combines a bright-orange emission (EQE = 35%) with a decent g _EL = 6 × 10^–3.

A more complex molecular design led to the construction of the two diastereomeric compounds H65 and H66 (Chart ) which combine a hetero[7]­helicene moiety with a 1,1′-binaphthyl moiety. In this case, the acceptor unit in the MR-TADF motif should be replaced by carbonyl groups; however, these compounds exhibited energy gaps between S₁ and T₁ states that were too high to provide the desired RISC property. Of some interest is the fact that (aS,P)-isomer H65 showed relatively larger g _abs and g _lum values than (aR,P)-isomer H66 (Table ), meaning that the 1,1′-binaphthyl moiety is able to influence the chiroptical absorption and emission intensity to some extent. The trend, though not the absolute values, was reproduced by calculations run at the M06-2X/6-311G­(d,p) level (Table ). In this case too, the molecular geometry allows for a parallel orientation of transition moments.

A more successful strategy for combining the TADF-active diketone-triphenylamine unit with a helicene scaffold was conceived by Fuchter, Zysman-Colman, and co-workers. S-shaped double helicene H67 (Chart ; treated here together with other CPL TADF emitters rather than in Section ) has an S₁–T₁ energy gap of 0.24 eV thanks to the effect of twisted donor–acceptor architecture that minimizes the HOMO/LUMO overlap. The measured CPL dissymmetry (g _lum = 4 × 10^–4), reasonably well reproduced by TD-PBE0/6-31G­(d) calculations, falls within the typical range for helicenes but is lower than that of most double helicenes. Other CPL-active MR-TADF emitters will be encountered in Sections and .

The last example in this section concerns a pair of BINOL-[5]­helicene dyads which combine a relatively simple design, and apparently rigid structure, with nontrivial photophysical behavior. Compounds H68 and H69 (Chart ) are both characterized by dual emission with lifetimes of ∼1 ns and 7–8 ns. CPL spectra are rather strong for helicene derivatives, reaching g _lum = 8 × 10^–3 for H69. TD-DFT calculations run at the ωB97X-D/6-31+G­(d,p) level found two ES minima, a C ₂-symmetrical one with calculated g _lum ≈ 8 × 10^–3 and a nonsymmetrical one with calculated g _lum ≈ 3 × 10^–3 for both compounds. An interesting point is that symmetrical minima have small but aligned transition dipole moments while nonsymmetrical minima have large but almost perpendicular transition dipole moments (Figure ). The authors recognized that the S₀–S₁ and S₁–S₀ transitions belong to irreducible representation (irrep) A in the C ₂ group, imposing collinear electric and magnetic dipole transition moments (discussions in Sections and ).

27.

Orientation of EDTM (blue arrows) and MDTM (magenta arrows) calculated for S₀–S₁ and S₁–S₀ transitions of H69. The S₀ structure has C ₂ symmetry, while in the S₁ both a symmetrical (C ₂) and a nonsymmetrical (C ₁) structure were calculated. The figure is original and was constructed using the data available in ref .

3.1.2.2. Statistical Analysis of Helicenes and Helicenoids

The extensive amount of data available on helicenes and helicenoids compiled in Table lends itself to a statistical analysis summarized in Figure . The analysis was built on 74 entries, of which 72 concern TD-DFT calculations and 2 concern CC2 calculations. Two entries from Table were disregarded because one (H33) was a severe outlier and another (H15a) had an inconsistent sign between calculated and experimental CPL. Panels a–c in Figure show scatter plots of g _abs and g _lum values, g _lum/g _abs ratios, and ΔE_S, comparing calculated values with experimental ones; panel d shows box plots recapitulating signed relative errors (REs, eq ) between experimental and calculated data. It can be appreciated that for |g| values below 2 × 10^–3 the performance of calculations is relatively good but worsens for larger absolute values. Both |g _abs| and |g _lum| values tend to be overestimated by calculations, as both best-fitting lines display a positive deviation from ideality and the median in the box plots lies over zero. However, the dispersion of REs appears narrower for |g _lum| than for |g _abs| in Figure d; except for a few outliers, g _lum values seem to be better predicted than g _abs. This result is far from trivial because the accuracy of TD-DFT//DFT calculations was anticipated to be larger for ground-state calculations than for excited-state ones. , Also, the REs on g values tend to compensate for each other, as the median RE for the g _lum/g _abs ratio is close to 0, although there is an overall tendency toward underestimation. The large dispersion of ΔE_S values, which has different explanations, must also be noted: the inherent error associated with TD-DFT calculations of vertical transition energies (0.1–0.3 eV) may cumulate, rather than cancel, upon Stokes shift simulation; the experimental estimation is affected by some uncertainty related, for instance, to the correct identification of 0–0 vibronic bands, band-shape distortion, and so on. Finally, most calculations were run in the gas phase or using LR solvent treatment. Very interesting and again partly unexpected is the comparison between the performance of different calculations methods (Figure e), which we restricted to the simulation of g _lum values. Here, B3LYP outperforms CAM-B3LYP, showing smaller mean relative errors (MREs). This is at odds with the general larger accuracy of CAM-B3LYP over B3LYP in predicting transition energies and oscillator strengths if the typical systematic overestimation of vertical energies is taken into account (Section ). Our observation is, however, biased by the fact that while B3LYP calculations considered in our survey were always run in combination with large basis sets such as 6-311G­(2d,p) and 6-311+G­(2d,2p), most of the CAM-B3LYP calculations are associated with smaller basis sets like def2-SV­(P), def2-SVP, and 6-31G­(d) (Table ). In fact, by restricting the analysis to the largest basis set 6-311+G­(2d,2p), CAM-B3LYP outperforms B3LYP, with MREs of 0.10 and 0.37, respectively. Also significant is that the addition of a dispersion correction of type D3BJ to the B3LYP functional worsens the agreement between calculated and experimental g _lum values. This is again surprising for helicene-like compounds, where the nonbonding interaction between terminal aromatic rings should impact S₀ and S₁ geometry. In this case, the basis set does not seem relevant, as all calculations run at the B3LYP-D3BJ level employed either 6-311G­(2d,p) or 6-311+G­(2d,2p) basis sets (Table ). Only a limited set of calculated and experimental g _abs values are available for this dispersion-corrected functional, so it is unclear if the deficiency concerns more specifically the S₁ state or the S₀ state, as would be expected since dispersion corrections are parametrized for the ground state. Turning to less popular functionals, the Minnesota family appears to perform very well, especially the multiparameter global hybrid MN15 functional which is indeed known to offer superior performance in predicting various chemical properties. More surprising is the accuracy achieved by the old half-and-half hybrid functional BHLYP (also known as BH&HLYP); in this case, however, there are only two entries coming from a single publication. The PBE family is also poorly represented with five entries only; still, we wish to remark on the good outcome of the parameter-free hybrid functional PBE0-1/3. Finally, dispersion-corrected hybrid functional ωB97X-D also performed decently the few times it was employed. Another outcome from our comparison is that the use of the more demanding CC2 method with respect to TD-DFT does not necessarily imply a more accurate simulation of g _lum values of helicenes and helicenoids; again, however, the data set is too narrow to draw a conclusion. Turning back to the role played by basis sets, in Figure e we plot the data limited to B3LYP and B3LYP-D3BJ functionals, which are numerically significant and should offer more consistent results. We can observe an increase in accuracy upon increasing the basis set size, as expected. Still, it is remarkable that the addition of one set of diffuse and polarization functions (i.e., passing from 6-311G­(2d,p) to 6-311+G­(2d,2p)) reduces the MRE by a factor of 3. In summary, our survey demonstrates the good performance of several hybrid functionals such as B3LYP, M06-2X, MN15, and PBE0-1/3, and that the use of large basis sets of the triple-ζ type with diffuse and multiple polarization functions is recommended for accurate CPL predictions of helicene-like systems.

28.

Summary of literature data on CPL calculations for helicenes and helicenoids reported in Table . (a) Scatter plot of calculated vs experimental |g _abs| (black) and |g _lum| (blue) values; the inset zooms in on the area with |g| < 10^–2. (b) Scatter plot of the calculated vs experimental g _lum/g _abs ratio. (c) Scatter plot of calculated vs experimental Stokes shifts (in eV). (d) Box plots of signed relative error between calculated and experimental g _abs (gray), g _lum (blue), and g _lum/g _abs ratios (red); the boxes indicate the 25–75% interval, and the whiskers represent the range (×1.5) of the data outside the interquartile range (IQR). (e) Histogram plot of the MRE on calculated g _lum associated with different calculation methods (any basis set) or with B3LYP­(-D3BJ) and three basis sets. Numbers in italics indicate the number of entries.

3.1.2.3. Double and Triple Helicenes

As introduced above (Scheme ), the terms double and multiple helicenes refer to systems where two or more helicene units are arranged within the same molecule, normally in a symmetrical fashion. A comprehensive review of ECD and CPL properties of multiple helicenes was provided by Mori. This class of compounds is especially interesting from our viewpoint because their complexity and laborious synthesis would particularly benefit from a judicious molecular design; on the other hand, their structure is intrinsically able to solve the issue of inherently small g values found for simple helicenes.

From that respect, an elegant demonstration of the calculation-assisted rational design of CPL emitters was offered in 2018 by Mori and co-workers. Aiming at an improvement of the chiroptical performance of carbo[6]­helicene (H1, Chart ), the authors simulated different combinations of dimers and tetramers of H1 in various geometrical arrangements (Figure ). TD-DFT calculations run at the M06-2X/def2-TZVP level were employed to screen expected g _abs values for the ¹B_b band around 300 nm. The best-performing models, leading to a 2-fold to 4-fold enhancement of g _abs with respect to H1, had the two units facing each other in X-type and S-type arrangements, corresponding respectively to D ₂-symmetric and C ₂-symmetric model dimers. Hence, the authors prepared the corresponding double carbo[6]­helicenes H70 and H71 (Chart ). S-shaped isomer H71 shows an improved QY with respect to H1, whereas X-shaped isomer H70 displays a red-shifted emission and larger Stokes shift thanks to a more effective conjugation allowing for pronounced excited-state relaxation. Moreover, both isomers feature enhanced g _abs and g _lum values, with the latter being maximized (2.5 × 10^–3) for X-shaped isomer H70. Calculations run with the CC2/def2-TZVP method reproduced the sign, shape, and relative intensity of ECD and CPL spectra among the three compounds (Figure ). The most interesting piece of information came from the analysis of transition moments. As discussed above, the limiting factor of absorption and emission dissymmetry values in simple [6]­helicenes is the angle θ _μm being close to 90°. The predicted value for the S₁–S₀ transition of H1 was θ _μm = 112° (Figure ). For the S-shaped and C ₂-symmetric double helicene H71, both μ ₁₀ and m ₀₁ for the ¹L_b transition, belonging to irrep B of the C ₂ point group, lie in the plane perpendicular to the C ₂ axis. However, there is no symmetry restriction of their reciprocal angle, which reaches a more favorable but still not ideal value of θ _μm = 130°. On the contrary, for X-shaped and D ₂-symmetric double helicene H70, symmetry restraints impose parallel (or antiparallel) transition moments for any electric- and magnetic-dipole-allowed transitions; hence, θ _μm = 0 or ±180°. In fact, in the D ₂ point group, any allowed transition belongs to one of the three irreps B₁, B₂, or B₃, which imply electric and magnetic transition dipoles aligned along the z, y, or x axis, respectively. The paper nicely illustrates how the chiroptical properties may be boosted through a rational molecular design, based on symmetry-related reasoning and substantiated by QM calculations.

29.

Design of novel multiple helicenes based on ECD calculations on various arrangements of carbo[6]­helicene H1. The numbers show relative g _abs values, the left column of each pair for dimers and the right column for tetramers. Reprinted from ref under a CC-BY 4.0 international license.

30.

Experimental (solid traces) and calculated (dashed traces) ECD (left panels) and CPL spectra (right panels) of (P)-carbo­[6]­helicene H1 and double carbo[6]­helicenes (P,P)-H70 and (P,P)-H71 (Chart ). Reprinted from ref under a CC-BY 4.0 international license.

31.

EDTM (blue), MDTM (magenta), and their angle θ calculated for the S₁–S₀ transition of H1, H70, and H71. Reprinted from ref under a CC-BY 4.0 international license.

Building on the same concept of symmetry-imposed restraints on the orientation of electric and magnetic dipole transition moments, Hirose, Matsuda, and co-workers explored a few already reported D ₂-symmetric CPL emitters and designed a new helicene-based macrocycle. The previously known compounds were the double carbo[5]­helicene H72a (Chart 3.5), a BODIPY dimer (to be described in Section ), and an oligophenylene macrocycle (to be described in Section ), whereas the new molecule consisted of two carbo[5]­helicene moieties connected through p-phenylene linkers (H73). For the double helicene H72a, the g _lum calculated at the TD-B3LYP/6-31G­(d) level already overestimated the experimental value (3.3 × 10^–2 vs 5 × 10^–3, respectively). For H73, a very high g _lum = 0.24 was predicted, which, if realized, would exceed any known SOM. To investigate the source of such a large dissymmetry, the authors analyzed geometries, MOs, and θ _μm values of H73 in comparison with those of H72b (a simplified model of H72a known as propellicene), carbo[5]­helicene, and 2,13-diphenyl[5]­helicene as partial structures of H73. In addition to the obvious role played by angle θ _μm related to D ₂ symmetry, another key factor was recognized in the degeneracy of FMO, in particular, the character of the S₁–S₀ transition. In fact, for the model molecule the S₁–S₀ transition was the combination of two emissions from the degenerate LUMO pairs to the HOMO, whereas for H72b and H73 a single LUMO–HOMO emission was dominant (Figure ). The major difference between H72b and H73 was found in the twist angle of the two [5]­helicene subunits, which was much more pronounced in H72b. In turn, this was reflected in the ratio between m ₀₁ and μ ₁₀ moments, which led to a calculated g _lum = 7.4 × 10^–2 for H72b, much smaller than H73. When this latter molecule was synthesized and the CPL spectrum was measured, however, the experimental g _lum = 1.5 × 10^–2 was 1 order of magnitude smaller than the prediction. This was traced back to an error in the estimation of μ ₁₀ , which was also reflected in the overestimated calculated radiative rate constant. We highlight the use of the small basis set (6-31G­(d)) and a single functional in these calculations. Considering the relatively small molecular size and limited flexibility, it would have been possible to use larger basis sets or benchmark the functional.

32.

Orbital correlation diagram and composition of the S₁–S₀ transition of double helicenes H72b and H73 (Chart ) and 2,13-diphenyl[5]­helicene (in the middle). Red and blue frames represent C ₂-symmetric and C ₂-antisymmetric MOs, respectively. Adapted with permission from ref . Copyright 2020 American Chemical Society.

The notion that highly symmetrical multiple helicenes outperform C ₂-symmetric simple helicenes has now become widespread. − However, it does not seem to apply to all multiple helicene designs: for instance, Mori’s group has reported a D ₃-symmetric triple helicene (H74, Chart ) whose g _abs and g _lum values are smaller than those of parent simple carbo[5]­helicene. In fact, for D ₃ symmetry, the transitions belonging to the degenerate irrep E do not necessarily have aligned electric and magnetic transition moments (Section ).

Sakamaki, Fujiwara, and co-workers reported two X-shaped double heterohelicenes (H75 and H76, Chart ) for which symmetry-related effects are again at play. The two compounds featured relatively large g _abs and g _lum values above 10^–2; a peculiar property of H75 is the sign inversion between the red-most ECD band and the CPL band. TD-DFT calculations run at the B3LYP-D3BJ/6-311G­(2d,p) level correctly reproduced the signs, including the mentioned inversion, though g _lum values were overestimated by 1 or 2 orders of magnitude. The authors inferred that vibronic effects beyond FC ones could be involved in the observed discrepancy. Rather than running vibronic HT calculations (Section ), they adopted a simplified though efficient procedure by applying low-frequency deformations to the excited-state geometry and calculating electronic transitions thereof. The g _lum values allied with the deformed S₁ geometry were progressively smaller than those obtained from the fully optimized one. Notably, the strong predicted dissymmetry values were associated with a C ₂-symmetric and not a D ₂-symmetric skeleton. However, in the case of H75 and H76, the lowest-energy S₀–S₁ and S₁–S₀ transitions belong to irrep A of the C ₂ point group, meaning that both μ ₁₀ and m ₁₀ are oriented along the C ₂ axis and are necessarily parallel or antiparallel.

Liu and Wang have calculated vibronic absorption, ECD, emission, and CPL spectra of two X-shaped oxygen–boron–oxygen (OBO)-bonded double helicenes H77a–b (Chart ); the experimental spectra of H77b had been reported previously. As is usual for vibronic calculations, the emphasis is put on band shapes rather than absolute intensities, so a comparison of experimental and calculated dissymmetry values is not possible. Calculations were run at the B3LYP/6-31G­(d) level by means of Santoro’s FCclasses code and using all possible vibronic schemes, namely, FC and FCHT for the initial state (FCHT_i) and FCHT for the final state (FCHT_f); TD and TI approaches; and VH and AH models (Section ). As similarly found in previous works, the inclusion of HT effects was key to reproducing the vibrationally resolved experimental spectra and specifically the lack of symmetry between absorption and emission spectra as well as between ECD and CPL spectra. More recently, the pivotal role played by HT effects in shaping CPL spectra of double helicenes has been further emphasized by Mori, who ran vibronic CPL calculations on previously described carbohelicenes H70 and H71.

Hu and co-workers have reported a few X-shaped double [7]­helicenes formally derived from carbo[7]­helicene H78 (Chart ) upon replacing the terminal benzene rings by thiadiazole (H79) and triazole rings (H80) or the inner benzene rings by furan, thiophene, and thiophene 1,1-dioxide rings (H81a–c). In the first series, H79 was not emissive, whereas H80 showed improved dissymmetry and CPL brightness with respect to H78, reaching g _lum = 9.1 × 10^–4 and B _CPL = 30.1 M^–1 cm^–1. The experimental findings were mirrored by TD-DFT calculations (CAM-B3LYP/6-311G­(d,p)), which offered further proof that the boosted g _lum was mainly related to larger transition magnetic dipole m ₀₁ for H80 than for H78, whereas both μ ₁₀ and θ _μm (∼20°) were similar for the two compounds. Larger emission dissymmetry values were found for the second series, with the highest g _lum = 1.2 × 10^–3 observed for H81b. Interestingly enough, sulfone derivative H81c exhibits g _lum = 1.1 × 10^–3 for its red emission above 600 nm. Again, the calculations nicely reproduced the experimental findings.

Swain and Ravat reported compound H84 (Chart ) with two homochiral carbo[7]­helicene moieties fused in an asymmetrical fashion to a pyrene ring, together with the two corresponding single helicenes H82 and H83. The chiroptical properties of H84 resemble more those of [cd]-fused derivative H83 than [e]-fused derivative H82. The experimental g _lum values were 1.27 × 10^–3 at 470 nm for H82, 2.64 × 10^–3 at 491 nm for H83, and 2.47 × 10^–3 at 504 nm for H84. Calculated g _lum values at the ωB97X-D/6-31G­(d,p)//B3LYP/6-31G­(d,p) level were between 1.2 and 1.6 × 10^–3, in decent agreement with the experimental values but without reproducing the observed trend. An analysis of transition moments revealed that for H84, both μ ₁₀ and m ₀₁ lie in the pyrene plane at an angle θ _μm = 49.5°, while for H82 and especially H83 the magnetic moment is almost perpendicular to the pyrene plane with θ _μm = 78–83°. However, H84 suffers from an intrinsically weaker magnetic transition moment for the S₁–S₀ transition.

A D ₂-symmetric helicene dimer with figure-of-eight topology (H85, Chart ), designed by Sun, Wu, and co-workers, exhibited large g _lum = 1 × 10^–2 like other D ₂-symmetric double helicenes discussed above. However, this compound raises several unresolved questions which are worth discussing. First, the reported dissymmetry value for the ECD band at 468 nm, assigned to the S₀–S₁ transition, was g _abs < 2 × 10^–3 (i.e., 1 order of magnitude smaller than g _lum) even though DFT-optimized geometries for S₀ and S₁ states were rather similar. Moreover, the experimental CPL spectra showed two bands of opposite sign: for (M,M)-H85, a positive band centered at 514 nm (g _lum ≈ +5 × 10^–3) and a negative band centered at 680 nm, associated with the larger g _lum ≈ −10^–2. The dual-band appearance of CPL spectra was not commented on by the authors; we suspect that it may be related to anti-Kasha behavior (i.e., that the blue-shifted band arises from an S₂–S₀ emission). Finally, TD-DFT calculations results were also odd. The authors screened four functionals (B3LYP, PBEPBE, ωB97X-D, and M06-2X) in combination with the 6-31G­(d,p) basis set; the triple-ζ basis set 6-311G­(2d,p) with additional polarization functions was also combined with B3LYP. Quite curiously, the authors indicated B3LYP/6-31G­(d,p) as the choice method, although it apparently provided the wrong sign for both the ECD and CPL bands of H85 assigned to S₀–S₁ and S₁–S₀ transitions; on the contrary, M06-2X seems to be the only functional providing fully consistent results. In our view, this compound needs to be re-evaluated from a computational perspective.

As an example of two helicene units connected through an aryl–aryl linkage, Caricato, Avarvari, and co-workers recently reported a family of carbo­[n]­helicenes (n = 4, 5, and 6) linked through a central 2,1,3-benzothiadiazole (BTD) unit. For the carbo[6]­helicene dimer (H86, Chart ), CPL spectra with g _lum = 8 × 10^–4 were reported and simulated at the CAM-B3LYP/aug-cc-pVDZ level.

An original design, which can be described as double hetero[4]­helicenes with two distant rings bridged through a further aromatic moiety, was conceived by Xu, H. Zhang, Z. Zhang, and co-workers. The three bridged compounds H87–H89 (Chart ) are optically stable and feature stacked central rings similar to [2.2]­cyclophanes, with plane-to-plane distances of between 3.2 and 3.4 Å. While H87 was almost nonfluorescent, H89 gave rise to strongly red-shifted emission with λ_em = 656 nm thanks to an ICT promoted by the carbonyl groups, with an associated g _lum = (2.2 ± 0.5) × 10^–3. Compound H88 had the largest g _lum = (2.4 ± 0.1) × 10^–2 for the series. TD-B3LYP-D3BJ/6-31G­(d,p) calculations reproduced the trend of g _lum values, though they overestimated the values by a factor of 3 to 9.

Building upon the structure of double oxa[7]­helicene H81a (Chart ), two triple helicenes were elaborated on, H90 (Chart ) with C ₃ symmetry and H91 with D ₃ symmetry, attaining in both cases g _lum ≈ 2 × 10^–3. , Despite the favorable alignment between transition moments, the main obstacle to larger CPL dissymmetry is represented by relatively small MDTM.

3.1.3. Biaryls

Biaryls are compounds consisting of two aromatic rings connected by a single bond between sp² carbon atoms. If the rotation around the single bond is sufficiently restricted, then optically stable chiral atropisomers may be obtained. Axial chirality has represented for a long time a unique source of chiral species functioning as stereoselective catalysts and other noteworthy architectures. , Together with helicenes and helicenoids encountered in the previous section, biaryl compounds represent the most largely investigated class of CPL-active SOMs. , There is, however, a fundamental difference between the two families which needs to be immediately clarified: the term biaryl refers to many different classes of compounds, having in common only a single (stereo)­chemical element, namely, the stereogenic axis connecting two aromatic rings. Therefore, this family is much more heterogeneous than the helicene/helicenoid one. It is true that most CPL-active biaryls we will encounter are based on the 1,1′-binaphthyl skeleton. , However, while in the case of helicenes and helicenoids the key element of chirality and the chromophore/fluorophore coincide, many CPL-active 1,1′-binaphthyl derivatives emit from a different fluorophoric moiety than the 1,1′-binaphthyl itself. In fact, 1,1′-binaphthyl derivatives have strong, characteristic, and conformation-sensitive ECD spectra in the ¹B_b region around 220 nm, but they emit from the weak ¹L_b state around 280–300 nm with g _lum ≈ 10^–3 and B _CPL ≈ 1 M^–1 cm^–1. A logical way of taking advantage of the 1,1′-binaphthyl skeleton, while improving CPL performance, is therefore by using it as a molecular scaffold to be linked to different emitting moieties, most often aromatic ones, which then assume a stereodefinite spatial arrangement. This section is divided into four subsections: Section describes simple 1,1′-binaphthyl derivatives; Section describes complex 1,1′-binaphthyl derivatives functionalized with emitting moieties; Section describes compounds containing multiple 1,1′-binaphthyl moieties; and Section describes other kinds of biaryls. Finally, a statistical analysis of CPL data is provided in Section .

3.1.3.1. Simple 1,1′-Binaphthyl Derivatives

Before delving into 1,1′-binaphthyls, it is interesting to provide a brief account of a 1,1′-bisanthryl derivative, namely, (1,1′-bianthracene)-2,2′-dicarboxylic acid (B1, Chart ). This compound was reported by Schlessinger and Warshel in 1974 and represents the first example ever of CPL calculation. The authors employed a semiempirical approach based on the Pariser–Parr–Pople method to calculate ground- and excited-state geometries and evaluate g factors for absorption and emission. The calculations predicted for S₀ a single shallow minimum with dihedral angle φ of between 85 and 100° and for S₁ two energy minima with φ = 75 and 113°. (In the following text, we will indicate with φ the aryl–aryl torsional angle (e.g., the C2–C1–C1′–C2′ dihedral angle in 1,1′-binaphthyls, Scheme .)) The predicted g _lum/g _abs ratio (separate values were not given) was around 0.5, in agreement with the experimental findings. To the best of our knowledge, this is not only the first but also the only CPL calculation based on QM semiempirical approaches.

8.

2. (a) Numbering and Definition of the φ Aryl–Aryl Torsional Angle in 1,1′-Binaphthyl Derivatives and (b) Definitions of “Open Chain” and “Chain-Bridged” Geometries.

QM calculations of CPL spectra of simple 1,1′-binpahthyls are relatively rare, if compared with the huge amount of experimental data available for these compounds. , The first report we are aware of concerns N,N-dimethyl-1,1′-binaphthyldiamine (B2, Chart ), which was discussed in the previously mentioned paper by Badala Viswanatha and co-workers, together with several ketones and diketones. Calculations at the CC2 level were applied not only to S₀ and S₁ but also to the T₁ triplet state to investigate room-temperature (RT) circularly polarized phosphorescence (CPP, Section ).

Hasegawa, Imai, and co-workers have employed the 2,2′-dihydroxy-1,1′-binaphthalene unit (also known as 1,1′-binaphthyl-2,2′-diol or BINOL) to build a series of cyclic oligonaphthalenes exhibiting size-dependent CPL properties. In the oligomers, 2 to 5 BINOL units were connected through 7,7′ bonds. Therefore, 2,2′-dimethoxy-1,1′-binaphthalene (B3, Chart ) and 7,7′-dimethoxy-2,2′-binaphthalene (B4) were the obvious choices as model compounds to investigate the impact of key torsions on chiroptical properties. Torsional energy scans were run at the M06-2X/def2-TZVP level, and for model B3 they found a straight dependence of CPL sign on conformation. For (R)-B3, the calculated CPL band was positive for φ ≤ 80°, negligible for φ ≈ 90°, and negative for φ ≥ 100°. This result agrees well with the experimental behavior observed for the cyclic oligomers, where the pentamer has the opposite CPL sign with respect to shorter analogs because of the larger dihedral angle φ. The authors also discussed a typical feature of 1,1′-binaphthyls and more in general of biaryl compounds, namely, the tendency to assume a more planar conformation in the S₁ state with respect to S₀.

As a starting material for the synthesis of binaphthalene-diketopyrrolopyrrole dyads, to be discussed later (Section ), Huang, Liu, and co-workers used 2,2′-dimethoxy-1,1′-binaphthalene (B3, Chart ) and the conformationally locked dinaphtho­[2,1-d:1′,2′-f]­[1,3]­dioxepine (B5) and reported their experimental ECD and CPL values, together with the results of calculations run at the B3LYP/6-31G­(d,p) level with PCM for DCM (Table ). A peculiarity of these two compounds is that while the sign of the red-most ECD band is the same, that is, positive for both (R)-B3 and (R)-B5 around 325 nm, the sign of the CPL band is opposite, that is, negative for (R)-B3 with g _lum = −3.1 × 10^–4 at 345 nm and positive for (R)-B5 with g _lum = +3.8 × 10^–3 at 390 nm. Apparently, an ECD vs CPL sign inversion occurs for B3. TD-DFT calculations could reproduce the observed sign and relative intensity of g _lum values, while their absolute intensities were underestimated by 1 order of magnitude. Curiously, the authors commented on the relative intensity of g _lum values (stronger for the chain-bridged B5 than for B3) and on the fact that they are correctly reproduced by calculations but did not comment on the mentioned sign inversion. The latter must be related to a geometry change from S₀ to S₁ for B3, which was not investigated. A second conformationally locked BINOL derivative with a two-carbon chain, 4,5-dihydrodinaphtho­[2,1-e:1′,2′-g]­[1,4]­dioxocine B6, was discussed by Hasegawa, Imai, and co-workers as a model for their figure-of-eight binaphthyl dimers to be discussed below (Section ). For this compound too, ECD and CPL spectra were consistent, as were the S₀ and S₁ geometries and dissymmetry values calculated for the excited state at the CAM-B3LYP/6-31G­(d,p) and CAM-B3LYP/6-31G+(d,p) levels, respectively (Table ).

3. Collection of Experimental and Computational Data for 1,1′-Binaphthyl Derivatives and Other Biaryls .

	Experiment			Calculation					Number or abbreviation
Type	g abs	g lum	g lum/g abs	g abs	g lum	g lum/g abs	Method	Basis set	This review	Original publication	ref.
BNP	1.9	0.31	0.2	0.09	0.02	0.2	B3LYP	6-31G(d)	B3	1
cb BNP	1.9	3.8	2.0	0.4	0.38	0.9	B3LYP	6-31G(d)	B5	3
cb BNP	2.0	1.6	0.8	1.7	1.3	0.8	CAM-B3LYP	6-31+G(d,p)	B6	3c
BNP		2.6			2.0		CAM-B3LYP	6-311+G(2d,p)	B7a	1a
Funct BNP		5.6			12.9		CAM-B3LYP	cc-pVTZ	B14a	7-PE1
Funct BNP		5.6			9.0		CC2	def2-TZVP	B14a	7-PE1
Funct BNP		1.8			1.4		CAM-B3LYP	cc-pVTZ	B15a	6-PE1
FunctBNP		1.8			1.7		CC2	def2-TZVP	B15a	6-PE1
Funct BNP	0.7	0.6	0.86		1.5		M06-2X	Def2-SV(P)	B16	1
Funct BNP	1.0	1.6	1.6		1.4		M06-2X	Def2-SV(P)	B17	2
Funct BNP		0.24			0.2		CAM-B3LYP	6-31G(d)	B19a	p-BTT
Funct BNP		1.4			1.9		CAM-B3LYP	6-31G(d)	B19b	p-BTB
Funct BNP		0.29			0.5		CAM-B3LYP	6-31G(d)	B19c	m-BTT
Funct BNP		1.1			5.7		CAM-B3LYP	6-31G(d)	B19d	m-BTB
Funct BNP		0.77			2.9		B3LYP-D	6-311G(d,p)	B21	BA34CzBN
Funct BNP		0.35			0.02		B3LYP-D	6-311G(d,p)	B22	BA23CzBN
Funct BNP	0.8	0.7	0.9	0.6	0.8	1.3	LC-BLYP	6-31G(d,p)	B23a	3
Funct BNP	1.3	0.9	0.7	2.3	1.7	0.7	LC-BLYP	6-31G(d,p)	B23b	4
Funct BNP	2.4	2.0	0.8	2.7	2.3	0.8	LC-BLYP	6-31G(d,p)	B23c	5
Funct BNP	2.8	2.2	0.8	2.7	2.8	1.0	LC-BLYP	6-31G(d,p)	B23d	6
Funct BNP		0.46			0.66		CAM-B3LYP	Def2-SVP	B25a	OBN-Cz
Funct BNP		2.1			2.7		CAM-B3LYP	Def2-SVP	B25b	OBN-CzB1
Funct BNP		0.1			0.11		CAM-B3LYP	Def2-SVP	B25c	OBN-CzC1
Funct BNP		0.56			1.5		PBE0	6-31G(d)	B26a	1
Funct BNP		1.4			6.7		PBE0	6-31G(d)	B26b	2
Funct BNP		2.0			1.14		B3LYP	6-31G(d,p)	B27a	RR-ONCN
Funct BNP		1.3			0.11		B3LYP	6-31G(d,p)	B27b	RS-ONCN
Mult BNP	2.9	3.8	1.3	13.4	5.5	0.4	CAM-B3LYP	6-31G(d,p)	B34	1a
Mult BNP	14	14.1	1.0	31.4	9.2	0.3	CAM-B3LYP	6-31G(d,p)	B35a	1b
Mult BNP	9	10.6	1.2	31.0	10.8	0.35	CAM-B3LYP	6-31G(d,p)	B35b	1c
Mult BNP	4.2	3.4	0.8	18.8	4.8	0.3	CAM-B3LYP	6-31G(d,p)	B35c	1d
Mult BNP		0.72			0.092		ωB97X-D	6-31G(d,p)	B36	1
Mult BNP		∼0			0.61		CAM-B3LYP	6-31G(d,p)	B37a	1
Mult BNP		2			1.9		CAM-B3LYP	6-31G(d,p)	B37d	2
Mult BNP		5.5			2.6		CAM-B3LYP	6-31G(d,p)	B37b	3
Mult BNP		4.4			2.3		CAM-B3LYP	6-31G(d,p)	B37c	4
Mult BNP	1.4	0.3	0.2		0.9		B3LYP	6-31G(d,p)	B38	p-BAMCN
Mult BNP	5.0	5.3	1.1		9.1		B3LYP	6-31G(d,p)	B39	o-BAMCN
Biaryl	1.3	9.1	7.0	1.3	9.3	7.1	CAM-B3LYP	TZVP	B40	T4-BT2
Biaryl	1.3	9.3	7.1	1.3	8.0	6.1	CAM-B3LYP	TZVP	B41	2
Biaryl	1.9	8.0	4.2	0.73	8.7	12	CAM-B3LYP	TZVP	B42	3
Biaryl	0.4	0.35	0.9	0.15	0.72	4.8	CAM-B3LYP	TZVP	B43	3
cb Biaryl		1.3			0.18		CAM-B3LYP	6-311+G(2d,p)	B45	3
cb Biaryl		1.0			0.33		CAM-B3LYP	6-311+G(2d,p)	B47	5
Biaryl	1.2	2.5	2.1		21.2		LC-PBE0*	SV(P)	B48	4
cb Biaryl	5	8	1.6		20.3		LC-PBE0*	SV(P)	B49	5
cb Biaryl	5	8	1.6		23.9		LC-PBE0*	SV(P)	B51	7
Biaryl		1			3		LC-PBE0*	SV(P)	B52·2H +	[1,2H2+]
cb Biaryl		2.5			0.1		LC-PBE0*	SV(P)	B53·2H +	[2,2H2+]
Biaryl		1.6			0.8		LC-PBE0*	SV(P)	B48·2H +	[3,2H2+]
cb Biaryl		2			8.2		LC-PBE0*	SV(P)	B49·2H +	[4,2H2+]
cb Biaryl		2			17.6		LC-PBE0*	SV(P)	B51·2H +	[6,2H2+]
cb Biaryl		14			24.3		LC-PBE0*	SV(P)	B54	7
cb Biaryl		14			18.9		LC-PBE0*	SV(P)	B54·2H +	[7,2H2+]
cb Biaryl	6.3	2.1	0.3		3.9		B3LYP	6-311G(2d,p)	B55a	1C
cb Biaryl	5.3	1.0	0.2		12.7		B3LYP	6-311G(2d,p)	B55b	2C
Biaryl		0.4			1.7		B3LYP	6-31G(d)	B57a	BiCz-1
Biaryl		0.6			2.7		B3LYP	6-31G(d)	B57b	BiCz-2
Biaryl		0.1			1.5		PBE0	def2-SVP	B58	BNCz-C6
Biaryl		1.1			1.59		PBE0	def2-SVP	B59	BBNCz-Ph
Biaryl		0.6			0.86		PBE0	def2-SVP	B60	BNCz-Ph
Biaryl		1.6			3.44		PBE0	def2-SVP	B61	BNCz-BN
Biaryl		8.6			0.59		M06-2X	6-31G(d)	B62a	5b
Biaryl		3.5			0.07		M06-2X	6-31G(d)	B62b	5c
Biaryl		5.4			0.38		M06-2X	6-31G(d)	B63a	5e
Biaryl		15			10.4		M06-2X	6-31G(d)	B63b	6
Biaryl	4.7	3.5	5.9		8.4		PBE0	6-31G(d,p)	B64	[2]HA2
Biaryl		5.2			12		CAM-B3LYP	Def2-SVP	B66a	Cz-Ax-CN
cb Biaryl	1.1	1.18	1.1	1.05	1.07	1	PBE0	6-31G(d,p)	B68	C1
Biaryl		1.7			3.3		B3LYP	SVP	B69a	BDBF-BNO
Biaryl		1.8			3.0		B3LYP	SVP	B69b	BDBT-BNO
Biaryl		10			11		B3LYP	6-31G(d)	B70a	1a
Biaryl		<1			4.6		B3LYP	6-31G(d)	B70b	1b
Biaryl		<1			12		B3LYP	6-31G(d)	B70c	1c
Biaryl		1			4.0		B3LYP	6-31G(d)	B71a	2a
Biaryl		<1			0.81		B3LYP	6-31G(d)	B71b	2b
Biaryl		0.55			22		B3LYP	6-31G(d)	B71c	2c

^aData for electronic calculations only (no vibronic or nuclear ensemble calculations). Data for flawed calculations not included (wrong enantiomer, wrong predicted sign). See Charts – are for structure numbering.

^bBNP, open-chain 1,1′-binaphthyl derivative; cb BNP, chain-bridged 1,1′-binaphthyl derivative; Funct BNP, functionalized 1,1′-binaphthyl derivative; Mult BNP, multiple 1,1′-binaphthyl derivative; and cb Biaryl, chain-bridged biaryl derivative.

^c g _abs and g _lum multiplied by 10³.

^dExperimental data in DCM.

^eMultiple conformers present.

^fS₂–S₀ transition.

A systematic investigation of siloxybinaphthyls with different attachments of trialkylsiloxy and methyl groups to the 1,1′-binaphthalene skeleton (B7a–c and B8–B12, Chart ) was reported by Takaishi, Ema, and co-workers. The declared intention of the authors was to find the best binaphthyl dihedral angle φ in the S₁ state to achieve the largest emission dissymmetry. Interestingly enough, compounds B7a–c are liquid at RT and CPL spectra could be recorded on neat samples, achieving a maximum g _lum = +1.6 × 10^–3 for (S)-B7b; solution CPL spectra were consistent with neat samples, although more intense, reaching g _lum = +2.6 × 10^–3 for (S)-B7a and +2.5 × 10^–3 for (S)-B7b. Focusing on compound (S)-B7a, the authors observed that the red-most ECD band had the same sign as the CPL band, though a smaller g _abs = +1.4 × 10^–3, and observed that the difference between the ground- and excited-state geometry would not be so large. Yet, (TD-)­DFT calculations run at the CAM-B3LYP/6-311+G­(2d,p) level found a substantial increase in the dihedral angle φ when passing from S₀ to S₁ geometry, that is, from 96 to 121° (Figure ). This way, the excited state reaches the optimal value for intense emission dissymmetry, as was evidenced by running torsional scans around the angle φ. The highest theoretical g _lum = 2.2 × 10^–3 was predicted for φ = 110°, close to the value found for the optimized S₁ geometry of B7a; the latter yielded g _lum = 2.0 × 10^–3, in good agreement with the experiment (Table ). A similar relationship between g _lum and φ in the excited state S₁ was calculated for the parent 1,1′-binaphthalene, also showing its maximum g _lum for φ = 110°. It is noteworthy that this value of the dihedral angle corresponds to the theoretical value where the exciton couplet between ¹B_b transition dipoles vanishes in the ground state. The other compounds (S)-B8–B11, for which experimental CPL spectra were not available, led to calculated |g _lum| of between 1.1 × 10^–5 and 1.0 × 10^–3, whereas their sign was correlated with the absolute conformation: a negative CPL meant a lowest-energy cisoid conformation (φ < 90°) and a positive CPL meant a transoid conformation (φ > 90°) in the S₁ state.

33.

(a) Relationship between the theoretical g _lum, angle θ _μm between transition moments and the binaphthyl dihedral angle φ (Scheme ) of (S)-7,7′-bis­(triethylsiloxy)-2,2′-dimethyl-1,1′-binaphthyl (B7a, Chart ). (b) GS and ES geometries of B7a. Reprinted with permission from ref . Copyright 2021, John Wiley and Sons.

3.1.3.2. Functionalized 1,1′-Binaphthyl Derivatives

This section is organized following the order of increasing structural complexity, starting from compounds where the 1,1′-binaphthyl moiety is linked to one or two fluorophoric moieties by single bonds or by ring fusion (Chart ), to finish with more complex architectures incorporating two or more 1,1′-binaphthyl moieties (Chart ).

9.

10.

The first reported case of CPL calculations on a chiral fluorophore endowed with axial chirality thanks to a junction with a 1,1′-binaphthyl skeleton is due to Li, Cheng, and colleagues. In this case, the fluorophoric unit was a boron dipyrromethene (BODIPY); many other examples of CPL-active BODIPY derivatives, including biaryl compounds constituted of two BODIPY units linked by a single bond, will be covered in Section . The aims of the research were three polymers containing alternate BINOL-type and BODIPY units; compounds B13a–c (Chart ) were used as model monomeric units in the computational study run at the B3LYP/6-311+G­(d,p) level. For all models, consistent S₀ and S₁ geometries were found with φ = 109–110° for B13a, 78–79° for B13b, and 56° for B13c in both states. The different dihedral angles can be easily understood by looking at the oxygen substituents at 2,2′ positions. The calculated g _lum allied with the BODIPY emission (at 648–659 nm) spanned from g _lum = 1.5 × 10^–4 for B13a to 2.3 × 10^–4 for B13c, highlighting the possibility of a moderate modulation of dissymmetry values by modifying the BINOL unit. Accordingly, the experimental g _lum measured for the polymers at 622–627 nm spanned from g _lum = 1.0 × 10^–3 for the polymer derived from B13a to 2.1 × 10^–3 for the polymer derived from B13c.

A systematic investigation of BINOL derivatives symmetrically substituted with two phenylethynyl groups at all possible positions, from 3,3′ to 8,8′, was performed recently by Tsubaki and co-workers. , Especially interesting is the comparison between 7,7′ and 6,6' disubstituted analogs B14a–e and B15a–e (Chart ). For the same aS axial chirality, in fact, the two families exhibit oppositely signed CPL spectra, featuring a positive band for (aS)-B14a–e and a negative band for (aS)-B15a–e between 350 and 400 nm. Measured g _lum values were on the order of 10^–3, with the largest absolute values for each series belonging to B14a and B15a (Table ). A theoretical analysis was run on the two latter compounds at the CC2/def2-TZVP level, which not only reproduced the observed g _lum values but also allowed interpreting the sign discordance with an electric current flow model: the different curvature of the π-conjugated system generates a counterclockwise current flow for (aS)-B14a and a clockwise current flow for (aS)-B15a, yielding opposite rotational strengths (Figure ). The chiroptical response of the two series is modulated by the tether length, which directly impacts the internaphthyl angle φ ranging from 55° for B14a/B15a to 103° for B14e/B15e. Interestingly enough, compounds B14a and B15a represent two rare examples for which a direct comparison between the results of CPL calculations run with CC2 and TD-DFT is possible, the former outperforming the latter (CAM-B3LYP/cc-pVTZ) in both cases (Table ). Additionally, the authors ran TD-DFT calculations on the complete series of bis­(phenylethynyl) dinaphtho­[2,1-d:1′,2′-f]­[1,3]­dioxepine (i.e., the regioisomers of B13a and B14a); although not reported in Table , these data were included in our statistical analyses (Sections and ).

34.

Electric current flows (red curves) illustrated for the S₁–S₀ transition of (a) B14a and (b) B15a and the orientation of μ ₁₀ (yellow segments) and m ₀₁ (green segments). Adapted with permission from ref . Copyright 2024, Royal Society of Chemistry.

With a similar design, Bettinger and co-workers attached 2,2′-dimethoxy-1,1′-binaphthalene to emissive NBN-benzo­[f,g]­tetracene moieties via ethynyl linkers. The choice of compounds B16 and B17 (Chart ) was made after a computational screening of various candidates, whose details are not reported in the original publication. In light of potentially high flexibility of B16 and B17 (a simplified model was used with n-butyl groups replaced by hydrogen atoms), a systematic conformational search was run with the conformer-rotamer ensemble sampling tool (CREST), based on semisemiempirical xTB methods. Ground-state geometry optimizations were run with r²SCAN-3c, a composite method developed by Grimme and co-workers combining the r²SCAN meta-generalized-gradient approximation functional with a tailor-made triple-ζ basis set and a dispersion correction. In the end, only a single populated conformer was found for B16 and for the truncated model of B17. Excited-state calculations were then run at M06-2X/def2-SV­(P) in vacuo or using state-specific PCM for DCM. The state-specific solvation affected calculated g _lum to a considerable extent, increasing the agreement with experimental values, which were around 10^–3 for both compounds (Table ).

Much higher dissymmetry values, even surpassing 0.1 as single crystals, were obtained by Lai and co-workers by exploiting the phenomenon of twisted intramolecular charge transfer (TICT). This phenomenon commonly involves D–A pairs linked by a single bond which, upon excitation, undergoes intramolecular twisting to yield a relaxed perpendicular structure. The two possible excited-state conformers (planar vs perpendicular) often produce dual emission. Since the relaxation pathways can be modulated by various means, such as substituents, solvent polarity, and steric factors, the TICT process can be exploited for the design of functional molecules. The two o-carborane-1,1′-binaphthyl dyads B18a–b (Chart ) feature chiral BINOL units as chiral electron donors and o-carborane units as achiral electron acceptors. The intramolecular charge-transfer excitation is followed by TICT emission, which is induced by aggregation (aggregation induced emission, AIE; see Section ) in solution, thanks to the suppression of the typical rotational freedom of the carborane. In fact, the addition of water to THF solution caused a progressive increase in emission, accompanied by the insurgence of CPL signals. In the THF/H₂O 5:95 solvent mixture, (aR)-B18a displays g _lum = +9.1 × 10^–4 at 561 nm and (aR)-B18b displays g _lum = −1.8 × 10^–3 at 627 nm. The change in sign, despite the same configuration of the BINOL unit, is apparently related to the different substitution pattern that regulates the preferred binaphthyl conformation, which according to CAM-B3LYP/6-31+G­(d,p) calculations is cisoid for B18a and transoid for B18b. The analysis of FMOs gave only faint evidence of ICT character because the LUMO was fully located on the BINOL moiety as it was the HOMO, apart from a small involvement of the o-carborane C–C bond. In addition to AIE, the compounds also exhibit crystallization-induced emission (CIE) properties. They are both highly luminescent and CPL-active as crystalline powders. The most remarkable g _lum values were observed for single crystals, reaching +0.13 for (R)-B18a and −4 × 10^–2 for (R)-B18b. The analysis was completed by TD-DFT calculations on the crystal lattice using a finite-size cluster model. Regrettably, the paper lacks several computational and experimental details. For example, the strategies employed to mitigate common artifacts that plague solid-state CPL measurements are not documented.

A different possibility for enhancing emission properties of binaphthyl-based skeletons lies in the introduction of donor/acceptor substituent couples. To that end, Zheng, Chen, and co-workers chose triarylamine and triarylborane moieties and a BINOL-derived dioxepine as a source of axial chirality. Substitution at positions 6, 7, 6′, and 7′ tunes both the steric demand of the BINOL moiety and the electronic properties of the resulting compounds (B19a–d, Chart ). All compounds are strongly fluorescent above 420 nm with a QY of up to 100% in DCM and feature solvatochromic emission. Compounds B19b and B19d display dual emission with a broad red-shifted band at 550–580 nm due to ICT from the triarylamine to the triaryborane moiety; accordingly, the HOMO and LUMO are respectively localized on the donor and acceptor groups, as expected. On the contrary, CPL spectra consisted of a single broad band for all compounds, of the same sign as that of the red-most ECD band and g _lum varying from 1 × 10^–4 to 6.3 × 10^–3, also depending on the solvent. The largest dissymmetry value was obtained for one of the two D–A couples, namely, B19d. TD-DFT calculations run at the CAM-B3LYP/6-31G­(d) level were in nice agreement with the experiment for all compounds (Table ).

Tsuji and co-workers designed two copolymers with alternating units based on 5,5,10,10-tetramethyl-5,10-dihydroindeno­[2,1-a]­indene and 2,2′-dimethoxy-1,1′-binaphthalene, whose monomeric units are represented by compounds B20a–b (Chart ). The experimental chiroptical properties were consistent between the polymers and the monomers, though the former exhibited almost doubled g _lum values around 415 nm, attaining ∼10^–3. Calculations were run on B20a and B20b at the B3LYP/6-31G­(d) level and reproduced the experimental trend.

We turn now to 1,1′-binaphthyl derivatives where the partially unsaturated cycle created by a chain between positions 2 and 2′ is fused to a second aromatic cycle. The first example is offered by the positional isomers B21 and B22 where (1,1′-binaphthalene)-2,2′-diamine is fused to CzBN, a known emitter with MR-TADF capabilities (Section ). These compounds exhibited several noteworthy properties, including a QY close to unity, fast intersystem crossing, and CP-OLED applicability. Key CPL parameters were also simulated at the B3LYP-D3/6-311G­(d,p) level and reproduced the relatively higher experimental g _lum found in solution for B21 with respect to B22 (7.7 × 10^–4 vs 3.5 × 10^–4), although the value for B21 was overestimated and that for B22 was severely underestimated (Table ). Thin-film CPL spectra were also recorded, revealing a substantial increase in emission dissymmetry for B21, which reached g _lum = 3.3 × 10^–3. In this case too, we observe that specific details on solid-state CPL measurements were not provided. TD-DFT calculations were also run on π-stacked dimers extracted from the X-ray packing structure, and the most stable dimer arrangement for B22 led to a very large calculated g _lum = 2.5 × 10^–2.

Pieters and co-workers reported two kinds of BINOL derivatives tethered through an eight-membered 1,4-dioxocane cycle to a functionalized benzene ring. In the first case, the latter contained two donor and two acceptor groups, respectively two cyclic alkylamines and two nitrile groups (B23a–d, Chart ). By varying the cycle length, all (chiro)­optical properties could be modulated, including the emission wavelength, QY, lifetimes, ΔE_S, g _abs, and g _lum. The latter reached the highest values for the largest cycle (i.e., B23d (g _abs = 2.8 × 10^–3 and g _lum = 2.2 × 10^–3); Table ). An in-depth theoretical investigation was run at three different levels, namely, LC-BLYP/6-31G­(d,p), B3LYP-D3BJ/6-311G­(d,p), and PBE0/6-311G­(d,p), including the CPCM (conductor PCM) solvent model for toluene; emission properties were evaluated within an equilibrium solvation regime (Section ). The first functional, which was chosen for emission and CPL calculations, is a “pure” (nonhybrid) exchange-correlation functional , with a long-range correction. Calculated g _lum values nicely fit experimental data (Table ). In a subsequent development, Nauborn, Pieters, and co-workers designed a new donor–acceptor tethered again to a BINOL unit as a source of chirality (B24a–c, Chart ). The molecular design provides a system endowed with dynamic chirality and low configurational stability in the indolocarbazole/terephthalonitrile (ICz/TPN) couple, which is controlled by the configurationally stable BINOL moiety (Figure ). Of the three possible conformations of the conjugated moiety, one is heavily favored over the others, although the interconversion barriers are low (around 5 kcal/mol). The conformational bias imposed by the BINOL unit was confirmed by comparing experimental and calculated ECD and VCD (vibrational CD) spectra. The emission properties of B24a–c could be modulated by the solvent and the substituents at positions 3 and 8 on the ICz ring, as is typical for systems showing ICT transitions. For the dimethoxy derivative (B24b) in DMSO, the most red-shifted emission could be observed (λ_em = 632 nm). Regrettably, g _lum remained below 10^–3 in all cases. TD-DFT calculations run at the PBE0/6-311G­(d,p) level revealed a match–mismatch effect between the configuration of the BINOL unit and that of the dynamically chiral fluorophore, which actually favors the observed dissymmetry, because the “best” match occurred for the most stable conformer.

35.

Illustration of the dynamic chirality control concept incorporated in D–A couple B24a (Chart ). Adapted with permission from ref . Copyright 2024, John Wiley and Sons.

The carbazole moiety is a known TADF emitter which found application in several optoelectronic devices. Not surprisingly, then, it has been linked to chiral skeletons to achieve thermally activated delayed CPL. The three isomeric compounds B25a–c (Chart ) derived from 3,3′,4,4′-tetrahydro-BINOL are included in the already mentioned systematic theoretical analysis by Chen et al. They exhibit g _lum values dependent on the substitution pattern, varying from 2.1 × 10^–3 for B25b to 1.0 × 10^–4 for B25c. In this case too, the authors employed stationary point calculations and a nuclear ensemble approach (Sections and ), in combination with TD-DFT calculations at the CAM-B3LYP/def2-SVP level. B25a has two excited-state minima separated by a small barrier (3.4 kcal/mol), and the ensemble-averaged calculated g _lum perfectly reproduced the experimental one (4.6 × 10^–4) while the stationary-point calculation exceeded the experimental absolute value by 30% (Table ). For B25c, the impact of the nuclear ensemble approach is even larger, though in this case the ensemble-averaged calculated g _lum heavily underestimated the experimental one. 4,4′-Biphenanthrene analogs of 25a–b were also recently reported, with g _lum around 2–3 × 10^–3.

Compounds B26a and B26b (Chart ) were recently reported by Qi and co-workers. Despite the apparent similarity, they show opposite sign for the lowest-energy ECD band (negative for (R)-B26a around 340 nm and positive for (R)-B26a around 320 nm) and, accordingly, opposite CPL bands (g _lum = −5.6 × 10^–4 for (R)-B26a at 387 nm and +1.4 × 10^–3 for (R)-B26b at 405 nm). The signs, but not the relative intensities, were reproduced by TD-DFT calculations at the PBE0/6-31G­(d) level (Table ). The differences were attributed to the steric hindrance of the trifluoromethyl group, causing distortion in the structure of B26b and in the “donating ability of the pyridine N unit” with respect to the “attracting group of the trifluoromethyl”; however, the pyridine ring is a known electron-withdrawing group.

Two other CP-TADF compounds, based again on the 1,4-dicyanobenzene acceptor, were described by Zheng and co-workers (B27a–b, Chart ). The combination of the (1,1′-binaphthalene)-2,2′-diamine donor unit with a second axially chiral 2,2′-biphenol unit generates diastereomers (aR,aR)-B27a and (aR,aS)-B27b with distinct, though similar, photophysical properties. In particular, measured CPL dissymmetry values were g _lum = 2.0 × 10^–3 for B27a and 1.3 × 10^–3 for B27b; while the former value was well reproduced by calculations at the B3LYP/6-31G­(d,p) level, the latter was underestimated by 1 order of magnitude (Table ).

3.1.3.3. Multiple 1,1′-Binaphthyl Derivatives

Increased complexity in 1,1′-binaphthalene derivatives can be obtained by coupling two or more 1,1′-binaphthyl units, either directly or through other aromatic rings. A first example of CPL calculations on these kinds of systems was reported by Cheng and co-workers in 2017. The target structures were alternating polymers of 2,2′-diethoxy-1,1′-binaphthalene and various conjugated aromatic rings derived from 1,4-hydroquinone; the monomeric units, employed as calculation models, are represented by compounds B28–B30 (Chart ). While the polymer derived from B28 was CPL-silent, the other two exhibited g _lum ≈ 7 × 10^–4 at around 450–470 nm, in accord with the red-most ECD band. Calculations run at the B3LYP/6-31G­(d) level reproduced experimental dissymmetry values and helped rationalizing the behavior of the polymer derived from B28. For this model, in fact, the HOMO and LUMO were localized respectively on the hydroquinone and the BINOL moiety instead of being fully delocalized on the aromatic skeleton as occurred for B29 and B30. Such a separation affects in the first instance the oscillator strength of the S₁–S₀ transition, which for B28 was ∼18 times smaller than for B29 and B30.

A similar polymeric design but incorporating fluorene moieties was conceived by Quan, Cheng, and co-workers. The monomeric units are represented by B31a and B31b (Chart ). The experimental g _lum values for the polymers were g _lum = 1.3 × 10^–3 for the open-chain derivative and 2.0 × 10^–3 for the chain-bridged one (definition in Scheme ). Theoretical values calculated at the TD-B3LYP/6-311+G­(d,p) level on B31a and B31b reproduced the trend but underestimated both the absolute values and their difference (10% instead of 35%). Interestingly enough, the polymers were used as blends with pyrene-naphthalimide dyes in the active layer of CP-OLED devices, whose g _EL reached 4.8 × 10^–2.

Huang, Liu, and co-workers constructed two binaphthalene-diketopyrrolopyrrole (DPP) dyads B32–B33 (Chart ). The photophysical properties of the two dyes were very similar to each other and also only slightly affected by the solvent. TD-DFT calculations run at the B3LYP/6-31G­(d,p) level on truncated models of B32 and B33 (with octyl groups replaced by methyl groups) revealed in both cases similar geometries for S₀ and S₁ states with planar thiophene-DPP-thiophene moieties, favored by C–H···O interactions. The S₀–S₁ transition at 611–613 nm and the corresponding emission at 642–644 nm were identified as a π–π* transitions delocalized over the whole conjugated skeleton, with partial CT character. By evaluating dissymmetry values, the authors compared dyads B32 and B33 with the parent BINOL derivatives B3 and B5 (Chart ) discussed above. They noticed that in the case of the dyads the conformational locking does increase the emission dissymmetry (g _lum = 5.8 × 10^–4 for B32, 4.5 × 10^–4 for B33). However, the calculation results obtained at the B3LYP/6-31G­(d,p) level raise several doubts because all reported g _abs and g _lum values were much smaller than experimental ones and the theoretical values for enantiomeric couples were not the opposite of each other.

A different structural motif was conceived by Hasegawa, Imai, and co-workers by connecting two BINOL derivatives through 7–7 and 7′–7′ bonds. The obtained compounds had rigid skeletons with a D ₂-symmetric figure-of-eight geometry (B34 and B35a–c, Chart ). The 1,1′-binaphthyl dihedral angle φ is modulated by the tether length, from 52.5° in B35a to 66.4° in B35c (X-ray geometries); consistent results were obtained by DFT geometry optimizations. The absorption and ECD spectra of B34 and B35a–c were consistent with each other and with the typical behavior of BINOL derivatives. The g _abs values are affected by the presence and length of the tether; for B35a, the largest g _abs = 1.4 × 10^–2 was observed for the ¹L_b band at around 330–340 nm (Table ). The same trend was observed, in parallel, in CPL spectra, reaching the maximum g _lum = 1.4 × 10^–2 at 359 nm for B35a. In all cases, the CPL and ECD signs of the ¹L_b band were consistent, and the g _lum/g _abs ratio was between 0.8 and 1.3 (Table ). The dimeric nature of compounds B34 and B35a–c is beneficial in terms of emission dissymmetry; in fact, the monomeric model B6 (Chart ) has a smaller g _lum = 1.6 × 10^–3 (Figure ). TD-DFT calculations run at the CAM-B3LYP/6-31G+(d,p) level clarified that the main reason for such a difference lies essentially in a smaller |m ₀₁ | value and not in a less favorable θ _μm angle obtained for B6. In the ground state, in fact, all compounds B34 and B35a–c display D ₂ symmetry, and the electric and magnetic moments for the S₀–S₁ excitation are forced to be collinear (see discussion below). In the excited state, however, a deviation from D ₂ symmetry occurs (Figure ) which, among other things, causes θ _μm to be tilted away from 0 or 180°. MO plots revealed a large delocalization of FMO especially for the derivatives with smaller dihedral angles φ, such as B35b and B35c, which in turn is responsible for an elongated m ₀₁ moment and larger g _lum values.

36.

Experimental (a) ECD and (b) CPL spectra of figure-of-eight BINOL derivatives B34 and B35a–c (Chart ) and their model B6 (Chart ). (c) Comparison of GS and ES geometries of B35a with relevant dihedral angles. Adapted with permission from ref . Copyright 2021, John Wiley and Sons.

A cyclic 1,1′-binapahthalene trimer (B36, Chart ) was recently proposed by Nozaki and co-workers as a new crystalline cyclonaphthalene derivative. Its g _lum = 7.2 × 10^–4 was lower than that of other oligonaphthalenes found in the literature, such as just discussed B34 and B35a–c. TD-DFT calculations run at the ωB97X-D/6-31G­(d,p) level found a theoretical g _lum = 9.2 × 10^–5 (Table ), whose small value was traced back to the almost perpendicular orientation of electric and magnetic transition moments (θ _μm = 90.5°). Compound B36 has an oppositely signed CPL and red-most ECD band; although this behavior seems to be correctly reproduced by calculations, it was not discussed by the authors, and S₀ and S₁ geometries were not presented. The authors observed that the D ₃-symmetric structure of B36 does not seem advantageous in terms of emission dissymmetry, contrary to the behavior seen for higher D _n symmetry (some examples will be discussed in Section ).

Given the key role played by molecular and transition symmetries in the design of efficient CPL emitters, as already stressed in Section for helicene and helicenoid derivatives and in the present section for 1,1′-binaphthyl derivatives, we recapitulate the situation for 1,1′-binaphthalene (C ₂ symmetry), its D ₂-symmetric dimer analog to compounds B34 and B35a–c, and its D ₃-symmetric trimer B36. In Scheme , we consider the combination of transitions ideally aligned with the long axis (as the ¹L_b transition in naphthalene) and the short axis (as the ¹L_a transition in naphthalene) and depict their lowest-energy excitonic combinations, symmetry-allowed in the various point groups. For the C ₂-symmetric monomer, both combinations belong to irrep B and generate an electric dipole transition moment (EDTM) perpendicular to the C ₂ axis. The orientation of the resultant EDTM with respect to the magnetic dipole transition moment (MDTM) is not fixed by symmetry; therefore, θ _μm may in principle assume any value. For the D ₂-symmetric monomer, the two combinations belong to irreps B₁ or B₂, thus EDTM and MDTM are necessarily collinear (θ _μm = 0° or ± 180°), which is ideal for high dissymmetry factors. Finally, for the D ₃-symmetric trimer, ¹L_b-type transitions combine according to irrep A₂, meaning again collinear EDTM and MDTM (θ _μm = 0° or ± 180°), whereas ¹L_a-type transitions combine according to the degenerate irrep E, for which the orientation between the moments, and hence θ _μm , is not fixed by symmetry. It must be stressed that the analysis applies to symmetric structures, which are more likely to be encountered in the ground state than in the excited state. A conclusive analysis of the role of molecular symmetry in determining CPL properties will be presented in Section .

3. Combination of Naphthalene Short-Axis-Polarized and Long-Axis-Polarized EDTM in 1,1′-Binaphthalene, a D ₂-Symmetric 1,1′-Binaphthyl Dimer, and a D ₃-Symmetric 1,1′-Binaphthyl Trimer .

^a The lowest-energy excitonic combinations are shown. For each combination, the corresponding irrep is indicated according to the point group character table (on the right).

The realization of higher-symmetry macrocycles based on the 1,1′-binaphthyl skeleton may be hampered by their tendency to adopt more stable, lower-symmetry structures than the expected ones. For instance, the potentially D ₄-symmetric 6.6′-linked-1,1′-binaphthyl cyclic tetramers B37a–d (Chart ), developed by Takaishi, Ema, and co-workers, adopt a D ₂-symmetric structure in the S₀ state which is also roughly preserved in the S₁ state. For the parent compound B37a, the D ₄-symmetric ground state geometry was found to be 4 kcal/mol higher in energy than the D ₂ one. The macrocyclic structure forces compounds B37a–c to adopt a cisoid conformation with the dihedral angle around the 1,1′-biaryl junction φ ≈ 70°. In B37d, the bridging chain imposes φ ≈ 50°. Consequently, (S)-B37d has a different ECD spectrum with respect to (S)-B37a–c with an extra negative low-energy band around 340 nm, whereas the latter features a positive band around 300 nm. Accordingly, the CPL spectra of (S)-B37b– c show a positive signal with g _lum = +5.5 and +4.4 × 10^–3, while (S)-B37d shows a negative one with g _lum = −2.0 × 10^–3; B37a is CPL-silent. These observations were well reproduced by TD-DFT calculations run at the CAM-B3LYP/6-31G­(d,p) level (Table ). The negligible CPL of B37a was justified by this compound assuming a conformation with φ ≈ 60° in the excited state, where CPL vanishes, favored by intramolecular hydrogen bonds. However, the CPL was switched on upon the addition of increasing amounts of l-phenylalanine, for which DFT calculations predicted multiple hydrogen-bond interactions with the BINOL OH groups. The observed g _lum = +2.3 × 10^–3 for the 1:1 (S)-B37a/l-Phe complex was also reproduced by calculations run with different functionals and basis sets.

Two compounds featuring two N,N-diphenyl-(1,1′-binaphthalene)-2,2′-diamine moieties fused to a 1,4- or 1,2-dicyanobenzene core (B38–B39, Chart ) were reported by Yan et al. to exhibit TADF and CPL. The “helix-shaped” compound B39 had a stronger dissymmetry (g _lum = 5.3 × 10^–3 at 550 nm in toluene) than the “rod-shaped” analog B38 (g _lum = 3 × 10^–4) and N,N-diphenyl-(1,1′-binaphthalene)-2,2′-diamine (g _lum = 2.8 × 10^–3). Thin film measurements also reached g _lum ≈ 4 × 10^–3 for B39. TD-DFT calculations run at the B3LYP/6-31G­(d,p) level reproduced the observed trend, though overemphasizing the difference between B38 and B39 (Table ). Calculations were also run on a series of model compounds, containing an N,N-diphenyl-(1,1′-binaphthalene)-2,2′-diamine moiety fused to a benzene ring with F and CN substituents in various numbers and positions. The better performance of B39 was ascribed to a more favorable orientation between transition dipoles (θ _μm = 158 vs 99° for B38), depending in turn on the helical shape and on the presence of electron-withdrawing substituents in the position para to the amine groups. A CP-OLED device was also constructed from B38 and B39, with EQE = 27.6 and 20.5%, respectively. Optimization of device architecture provided the best g_EL = 4.2–4.6 × 10^–3, on par with g _lum.

3.1.3.4. Other Biaryls

This subsection collects a range of biaryl systems with diverse skeletons comprising different kinds of aromatic rings and biaryl junctions. The common structural element is the presence of at least one chirality axis, generating optically stable atropisomers.

The first biaryl compound for which CPL calculations were reported, by Longhi and co-workers, was bis­(2,2′-bithiophene-5-yl)-3,3′-bithianaphthene (B40, Chart ). Here, the stereogenic axis links two benzo­[b]­thiophene units appended with bithienyl units. The experimental ECD spectrum of (aR)-B40 (Figure a) displays a clear-cut negative exciton couplet between 300 and 450 nm due to degenerate exciton coupling of π–π* transitions localized on the whole conjugated chromophore. The CPL spectrum displays a single negative band with a maximum at 500 nm. When emission occurs from exciton-coupled states, only the lowest-energy exciton component emits, and its CPL retains the sign of the lowest-energy side of the couplet (Figure b). This is opposite to what happens with the companion spectroscopic technique of fluorescence detected circular dichroism (FDCD), where two bands appear for exciton-coupled systems even when one coupled chromophore is not emissive. However, exciton-split CPL bands may appear due to anti-Kasha emission from the upper excitonic state, provided that it is sufficiently populated at RT as a consequence of small exciton splitting (see an example in Section ). For compound B40, the experimental g _lum/g _abs ratio is as high as 7; this may be partially explained by the observed differences in S₀ and S₁ geometries obtained at the CAM-B3LYP/TZVP level, which reproduced almost perfectly g _abs and g _lum values (Table ). A subsequent development led to designing compounds B41 and B42 (Chart ), embedding respectively 4H-cyclopenta­[2,1-b:3,4-b′]­dithiophene and dithieno­[3,3-b:2′,3′-d]­pyrrole, that is, units commonly employed in optoelectronic organic materials. The new compounds exhibited smaller band gaps than the parent B40, but other photophysical and chiroptical properties were similar, including the sign and intensity of dissymmetry factors. The latter were again well reproduced by CAM-B3LYP/TZVP calculations (Table ). The same group, in collaboration with Villani, Cauteruccio, and co-workers, also reported CPL calculations for the atropisomeric benzo­[1,2-b:4,3-b']­dithiophene derivative B43 (Chart ). In this case, the same level of calculation led to slightly worse agreement in terms of both g _lum and g _abs and their ratio (Table ).

11.

37.

(a) Experimental and calculated ECD and CPL spectra of compound (aR)-B40 (Chart ). Reprinted with permission from ref . Copyright 2014 American Chemical Society. (b) Correlation between ECD and CPL spectra in exciton coupled systems. Gray bars represent GS, and colored bars, ES.

In the four peri-xanthenoxanthene derivatives B44–B47 (Chart ) developed by Takaishi, Ema, and co-workers, the stereogenic element is again a chirality axis of the 1,1′-binaphthalene type. The bridging structure seen in B45 and B47 was found to be crucial to observing CPL activity with g _lum ≈ 10^–3, while open-chain analogs B44 and B46 gave CPL signals below the detection limit. The trend was reproduced by TD-DFT calculations run at the CAM-B3LYP/6-311+G­(2d,p) level (Table ), which underestimated the experimental g _lum values by 1 order of magnitude. The analysis of transition moments revealed for the bridged derivatives more favorable electric/magnetic dipole angles, for instance, θ _μm = 87° for B46 and 78° for B47.

A similar impact of bridged structures on chiroptical properties was observed by Guy, Srebro-Hooper, and co-workers for a series of dibenzo­[c]­acridine derivatives B48–B53. It must be noticed that the chain-bridged compounds B49, B51, and B53 can be categorized among hetero[7]­helicenoids (a class of compounds covered in Section ). In this case too, compounds B49 and B51 showed distinctly stronger g _abs (for the red-most ECD band) and g _lum values (8 × 10^–3) than the corresponding open-chain analogs B48 and B50. Compound B52 was photolabile; moreover, all compounds were weakly emissive, with B49 having the highest Φ_f = 0.014. Calculations were performed using the hybrid PBE0 functional with long-range correction (LC), where the range-separation parameter γ was set to the optimally tuned value (determined to optimize the HOMO and LUMO energies to ionization potential and electron affinity) for helicene systems. Although the g _lum values calculated at the LC-PBE0*/SV­(P)/PCM level exceeded experimental values by 1 order of magnitude (Table ), they allowed explaining the differences seen among the series as due to more delocalized character of the S₁ state in B49 and B51 versus a pronounced flexibility of B48 and B50 also in the S₁ state, resulting in multiple conformers also yielding opposite CPL signs.

The chiroptical properties of the same compounds were re-evaluated a few years later, together with the analog B54 featuring a five-membered central ring, upon acid treatment. The new compound B54 reached g _lum = 1.4 × 10^–2, among the highest observed for SOMs. The enhancement was traced back to an intense m ₀₁ moment. The effect of protonation on the chiroptical properties was very pronounced for the open-chain derivatives B48 and B52, for which the red-most ECD band increased in intensity and even changed sign (for the former compound); consistent variations were observed in the CPL spectra (Table ). For the chain-bridged derivatives B49, B51, and B54, instead, ECD and CPL spectra were almost unaffected by protonation, apart from some intensity decrease (for B51). The only exception is compound B53, which was CPL-silent in the neutral form and reached g _lum = 10^–3 for the deprotonated species.

Putting together all data available on biaryl analogs with open vs chain-bridged structure (definition in Scheme ), it clearly emerges that the latter ones perform better than the former in terms of g _lum values (Figure ). In chain-bridged derivatives, two changes occur with respect to their open analogs: (a) the interaryl torsion assumes a value different than ∼90° commonly encountered in 1,1′-binaphthalene and similar compounds; (b) the conformational flexibility is strongly restricted or totally suppressed. Although it is difficult to disentangle the role played by the two factors, from the analysis of the data summarized in Figure , the conformational restriction appears to be the most crucial one. Although for 1,1′-binaphthyl derivatives a value of φ = 110° corresponds to the largest expected emission dissymmetry, if we compare for instance compounds B3, B5, and B6 (Chart ), the two chain-bridged compounds with φ < 90° have 5- to 10-fold-higher g _lum than the open compound with φ ≥ 90°.

38.

Experimental g _lum values measured for different open-chain and chain-bridged 1,1′-binaphthyl derivatives. See the text and Table for details and references.

Further confirmation of the positive impact of conformational restriction on emission dissymmetry factors came from the work of He and colleagues on a series bidibenzo­[b,d]­furan derivatives with alkyl linkers of different lengths (B55a–d, Chart ). These compounds may also be classified as hetero[7]­helicenoids (Section ). The chain length modulates the interaryl torsion angle φ between 53° in B55a and 72° in B55c (X-ray geometry). Smaller angles correspond to more effective conjugation, which was reflected in red-shifted absorption edge and emission maximum for B55a. This compound also exhibited the largest g _lum = 2.1 × 10^–3 (Table ). Calculations run at the B3LYP/6-311G­(2p,d) level were in poor accordance with the experiment; they overestimated the g _lum of B55b by 1 order of magnitude and predicted the wrong sign for B55c and B55d. Moreover, the data that was claimed to contain g _lum calculations cannot be safely assumed to refer to S₁ structures. A more accurate theoretical description of these compounds is desirable, also in consideration of their very good performance in the solid state where B55a shows g _lum > 0.1 as thin films.

A unique biscarbazole derivative was designed by Autschbach, Favereau, and co-workers, appended with flexible arms ending with phthalimide moieties (B56a–b, Chart ) able to undergo intramolecular exciplex formation upon excited-state interaction between the phthalimide and carbazole moieties. We recall that, according to IUPAC nomenclature, both excimer and exciplex are defined as “an electronically excited complex of definite stoichiometry, ‘non-bonding’ in the ground state”. , However, excimer refers to “a complex formed by the interaction of an excited molecular entity with a ground state partner of the same structure”, and exciplex refers to “a complex formed by the interaction of an excited molecular entity with a ground state counterpart of a different structure”. In other words, the excimer is a noncovalent homodimer, and the exciplex is noncovalent heterodimer, formed in the excited state. Excimers and exciplexes often feature peculiar excited-state properties, including broad and unstructured emission bands with large Stokes shifts whose CPL is not correlated, either in sign or intensity, with the red-most ECD band. Very importantly, chiral excimer emission may produce very strong CPL spectra with amplified g _lum. Both emission and CPL spectra of compounds B56a–b feature a pronounced solvatochromism. In a polar solvent (DCM), the emission spectrum displays a relatively narrow band with λ_em = 417 nm, reminiscent of that of the parent biscarbazole (devoid of pendants); the CPL spectra (Figure ) consist of a single band, with g _lum = 2.0 × 10^–3 and positive signs for (M)-B56a and (M)-B56b, in accord with the lowest-energy ECD band (g _lum/g _abs ≈ 1). In an apolar solvent (methylcyclohexane, MCH), a broad structureless emission band is observed with a maximum at 488 nm for B56a and 478 nm for B56b due to the emission from an exciplex formed between the carbazole donor(s) and the phthalimide acceptor(s). A small emission peak is also found at 400 nm for both molecules, due to residual biscarbazole emission. The CPL spectra also feature new maxima at around 475 nm, which have an opposite sign compared to that recorded in DCM; the recorded dissymmetry values were g _lum = −2.0 × 10^–3 for both (M)-B56a and (M)-B56b (g _lum/g _abs = −0.5 and −0.3, respectively). For (M)-B56b containing a single acceptor unit, the CPL spectrum also shows an extra band at around 400 nm again due to the residual CPL of biscarbazole. An in-depth theoretical analysis was run on B56a using the range-separated hybrid functional ωB97X-D in combination with the def2-SV­(P) and def2-TZVP basis sets, a continuum solvent model for MCH and DCM, and vibronic transitions evaluated with the FC|HT model. The most interesting finding from the calculations is that, while ECD is almost insensitive to the molecular conformation, calculated CPL spectra displayed a remarkable dependence on the molecular structure. Since, however, for a given conformer the rotational strength of the S₁–S₀ transition was calculated to be consistent in DCM and MCH, the observed difference in the two solvents was attributed to a different emitting conformer. Very appropriately, the authors noticed that the energy barriers associated with the interconversion between the various conformers are expected to be low, especially in the excited state; therefore, a thorough exploration of the S₁ potential energy surface (PES) via excited-state molecular dynamics, possibly with the explicit treatment of solvent, would be necessary to treat the system.

12.

39.

Experimental CPL spectra of bicarbazole derivatives B56a–b (Chart ) in various solvents. Inset: calculated CPL spectra for 5 conformers of B56b in the ES state, including LR-PCM for two solvents. Adapted from ref under a CC-BY-NC 3.0 unported license.

CPL calculations for several other biaryls incorporating carbazole rings were reported (from B57a–b to B61, Chart and Table ), all with g _lum within or below 10^–3. , Larger dissymmetries were encountered for compounds B62a–b and B63a–b with multiple stereogenic axes. Especially interesting is B63b with g _lum = 1.5 × 10^–2, apparently arising from an anti-Kasha S₂–S₀ emission favored by quasi-degenerate LUMO and LUMO+1 orbitals. Relatively poor agreement was obtained in most of these latter cases between experimental and calculated g _lum values (Table ), possibly associated with the small split-valence basis sets employed.

Toyota and co-workers have described an anthra­[1,2-a]­anthracene-1-yl dimer B64 (Chart ) and its 10,10′ bis­(mesitylene) derivative, which assume an overall extended helical geometry in the crystalline state. B64 showed g _lum = 3.5 × 10^–3 and, interestingly, a large B _CPL ≈ 120 M^–1 cm^–1, which is quite high among biaryls. Dissymmetry values calculated at the PBE0/6-31G­(d,p) level slightly overestimated experimental values (Table ).

An interesting system composed of a phenanthrene and two biphenyl moieties (B65a–c, Chart ) was designed by Zhao and co-workers. Two different folds are possible for these compounds in both the GS and ES, whose relative energy depends on solvent polarity through dipole–dipole interactions mediated by the substituents. As a consequence, CPL spectra display pronounced solvatochromism and in most cases appear as two bands of opposite sign. Unfortunately, CPL spectra were very noisy due to small QY. TD-DFT calculations with the double-hybrid ωB97X-2 functional confirmed the existence of two distinct foldamers in the S₁ state, associated with opposite CPL signals.

Compounds B66a–d (Chart ) proposed by Fa, Song, and co-workers as putative CP-TADF emitters feature a 1,1′-binaphthyl moiety substituted on each ring with a nitrile acceptor group and a donor group elaborated from a carbazolyl moiety. The donor group modulated the HOMO–LUMO gap between 4.18 and 4.75 eV and the emission maximum from λ_em = 449 nm for B66a to 603 nm for B66c. As expected, the S₁–S₀ transition has major ICT character. Calculated g _lum values at the BMK/6-31G­(d) level were all above 10^–2, thanks to aligned electric and magnetic transition moments. No experimental CPL data were provided. Compound B66a was also reevaluated by Chen et al. Similar to 3,3′,4,4,-tetrahydro-BINOL derivatives B25a–c discussed above (Section ), in this case too, the use of the nuclear ensemble calculation approach led to strongly reduced g _lum (by 70%) with respect to the standard stationary point TD-DFT (CAM-B3LYP/def2-SVP), emphasizing the impact of the conformational disorder generated by molecular vibrations.

A series of D–A dyads composed of a naphthalene donor and a naphthalene imide acceptor, including compounds B67 and B68, were reported by Ouyiang, Lin, and co-workers. The biaryl moiety is in fact reminiscent of BINOL. Not surprisingly then, the photophysical properties depend again on the open vs closed chain structure, which is responsible for different dihedral angles φ (82.1° in B67 and 53.4° in B68, X-ray geometries). Both B67 and B68 show a broad emission band due to the 1,8-naphthalimide chromophore, with significant solvatochromism, from blue emission in low-polarity solvent to yellow emission in DMSO. The quantum yield increased substantially, passing from the parent open chain system B67 (Φ_f = 0.06) to the closed chain one B68 (Φ_f = 0.37), clearly reflecting the influence of molecular flexibility. The same occurred for ECD (>300 nm) and CPL spectra, which were much more intense for B67 (g _lum = 1.2 × 10^–3) than B68. TD-DFT calculations run at the PBE0/6-31G­(d,p) level reproduced well the experimental value for B67 and revealed a 5-fold-enhanced |m ₀₁ | with respect to B68.

A number of CPL calculations on CP TADF emitters based on biaryls appeared recently (from B69a–b to B71a–e, Chart and Table ). , In these latter series, compound B70a exhibited the largest g _lum = 1.0 × 10^–2 while other values were below 10^–3. Once again, we notice that, probably due to the choice of a small basis set, the errors associated with some calculated g _lum values are very large.

3.1.3.5. Statistical Analysis on Biaryls

With respect to the statistical analysis described in Section for helicenes and helicenoids, the validity of a similar operation for biaryls is affected by several issues, such as (1) the more limited number of literature data; (2) the larger structural variability; (3) the fact that several calculations reported in Sections – were run on models for which experimental data are not available; and (4) finally, the observation that some reported calculated CPL parameters are severely flawed; for instance, they have different absolute values for two enantiomers or an inconsistent sign with respect to the experiment. Therefore, we will look at our analysis with a critical eye and include all available data for biaryls, without distinguishing among simple 1,1′-binaphthalene derivatives, 1,1′-binaphthalene derivatives functionalized with extended chromophores, and other biaryl systems.

The impression that TD-DFT calculations of CPL parameters of biaryls are in general less accurate than for helicenes and helicenoids is confirmed by looking at Figure a in comparison with Figure d. The analysis was built on 76 entries from Table , all concerning TD-DFT calculations. Several examples from the literature survey were disregarded because of serious flaws or major inconsistencies, as mentioned above. More than half of the entries (47) lacked experimental and/or calculated g _abs values, and only g _lum values could be considered. The box plots of signed RE between experimental and calculated data display a large deviation and broad dispersion, especially for g _lum values. In particular, it appears that a statistically significant number of calculations overestimated |g _lum| values. On the contrary, g _abs values tend to be better predicted and generally underestimated by calculations. Thus, biaryls behave differently from helicenes and helicenoids, for which g _lum values were more reliably predicted than g _abs. An obvious reason for this discrepancy might be found in the different molecular flexibility of the two classes of compounds. It is likely that in some of the reported calculations on biaryls the correct S₁ minimum may not have been found or that the existence of multiple minima may have been overlooked.

40.

Summary of literature data on CPL calculations on biaryls reported in Table . (a) Box plots of the signed relative error between calculated and experimental g _abs (gray), g _lum (blue), and g _lum/g _abs ratios (red); the boxes indicate a 25–75% interval, and the whiskers represent the range (×1.5) of the data outside the IQR. (b) Histogram plot of the MRE on calculated g _lum associated with different calculation methods (any basis set) or with CAM-B3LYP and various basis sets. Numbers in italics indicate the number of entries.

The comparison between the performances of different functionals is affected by the limited number of available data. Still, it emerges that the B3LYP functional, also with D3 dispersion corrections, performs relatively poorly as far as g _lum predictions of biaryls are concerned. The two best-performing functionals are CAM-B3LYP and, quite surprisingly, LC-BLYP. On the contrary, the “tuned” functional LC-PBE0* appears to be the worst performing functional; it must be stressed, however, that it was applied to a homogeneous class of compounds for which a systematic error is likely. , Another interesting feature which emerges from the analysis is that, for a specific functional, a pronounced substrate-dependent performance is observed even for relatively similar molecular systems. This is in fact a well-known drawback of Kohn–Sham DFT. Considering, for instance, CAM-B3LYP, it provided optimal calculated g _lum values for compounds B19a–c (Chart and Table ), with an average RE < 1, but overestimated by a factor >5 the g _lum of B19d. Similarly, it predicted almost perfectly the g _lum values for compounds B40–B42 (Chart ), whereas it overestimated by a factor >2 the g _lum of B43. Obviously, these comparisons are run for the same basis set.

As for the role of the basis set, in Figure b we analyze the data set for the CAM-B3LYP functional, which is numerically significant. The expected increase in accuracy upon increasing the basis set size is clear-cut only when passing from 6-31G­(d) to 6-31G­(d,p) and 6-31+G­(d,p); the inclusion of a set of polarization functions on hydrogen atoms indeed seems crucial and also computationally not very expensive. The use of triple-ζ basis sets (6-311+G­(2d,p)) is apparently unfavorable in terms of cost/accuracy compromise; however, a larger number of data should be produced before drawing any conclusion.

3.1.4. Pyrene and Perylene Derivatives

The interest in pyrene and perylene derivatives in the field of emission spectroscopy is related to the well-known tendency of these aromatic rings to strongly interact in the excited state via π-stacking, yielding excimer or exciplex structures (for definitions, see Section ). The often intense and red-shifted emission of excimers or exciplexes lends itself to several practical applications. − Not surprisingly then, excimer or exciplex CPL has attracted the curiosity of the scientific community. ,, Probably because of the inherent difficulties in simulating supramolecular systems and more in general of systems held together by dispersion forces, QM calculations of excimer CPL spectra have been relatively rare and are limited to pyrene and perylene derivatives (Chart ). Several benchmark studies in fact highlighted the limitations of TD-DFT in correctly predicting the energies and geometries of excimers of aromatic compounds as well as the beneficial role played by dispersion corrections. −

13.

Takaishi, Ema, and co-workers reported several naphthalene and 1,1′-binaphtahlene derivatives appended with multiple pyrene and perylene moieties undergoing excimer formation and emission. For pyrene derivatives, excimers produce a bright-green emission with a maximum at around 535 nm in DCM, whereas monomeric emission occurs at around 400 nm. As a model of quaternaphthyl derivatives containing eight pyrene rings, the authors calculated the achiral model P1 (Chart ) at the CAM-B3LYP/6-31G­(d,p) level, which displayed in the S₁ state a stacked structure with a plane-to-plane distance of 3.6 Å and a tilt angle of 41° between the major axes of the two pyrene rings (Figure ). Analog achiral compounds with perylene rings substituted at positions 2 and 3 (P2–P3) also yielded excited-state structures with π-stacked and tilted rings (Figure ). Building on these computational results, the authors formulated an original excimer chirality rule to predict the CPL sign of quaternaphthyl derivatives P4a–f containing various aromatic rings, such as 1-, 2- and 4-substituted pyrenes, 2- or 3-substituted perylenes, and 2-substituted anthracene (Chart ). Similar to the exciton chirality rule holding for exciton-coupled ECD spectra, the excimer chirality rule allows one to predict the sign of the excimer CPL band by looking at the clockwise/anticlockwise sense of twist between the aromatic rings interacting in the excited state (Figure ). Later on, the same rule was extended to compounds P4g and P5a–e (Chart ) which contain pyrene and perylene rings in the proper positions on the 1,1′-binaphthyl and quaternaphthyl skeletons to promote intramolecular exciplex formation. In fact, all compounds P4g and P5a–e show broad emission bands above 500 nm (λ_em ≈ 530 nm) in DCM, whereas the monomeric perylene emission occurs at 490 nm. A typical drawback of exciplexes is the low QY compared to monomer emission because of efficient vibrational relaxation due to a loose geometry; for pyrene excimers, typically Φ_f ≈ 0.3. Some of the compounds P4g and P5a–e have instead QY close to unity, possibly because of the sterically hindered environment provided by 4 or 6 large contiguous rings. All compounds displayed CPL spectra with g _lum ≈ 10^–3. Compound P5d exhibited the highest g _lum = 2.7 × 10^–3 accompanied by Φ_f = 0.99. CPL properties were rather insensitive to solvent and temperature, owing to the conformationally rigid exciplex geometry. The aforementioned excimer chirality rule could be generalized for exciplexes, leading to a similar exciplex chirality rule; the two rules correspond, in the context of ECD, to degenerate and nondegenerate exciton coupling, respectively. Additionally, the authors calculated CPL parameters of compounds P4g and P5a–e at the TD-CAM-B3LYP/6-31G­(d,p) level, obtaining g _lum values in fair agreement with the experiments.

41.

Excimer chirality rule illustrated for achiral model compounds P1–P3 (Chart ) and applied to quaternaphthyl derivatives P4a–b. Adapted with permission from ref , copyright 2018, Royal Society of Chemistry, and from ref , copyright 2019, American Chemical Society.

The excimer chirality rule was also applied to a family of derivatives, derived from P6a (Chart ), with pyrene rings sandwiched by two BINOL moieties. These compounds form intermolecular, rather than intramolecular, excimers held together by π-stacking interactions and hydrogen bonds and thus are very solvent-dependent. The prediction made by the excimer chirality rule was substantiated by excited-state TD-DFT geometry optimizations of P6a–b, but calculated g _lum values were not provided.

A similar phenomenon of intermolecular excimer formation mediated by hydrogen bonds holds for proline derivatives P7a–b (Chart ), which display a mechanochromic emission in the crystalline and amorphous (after grinding) states depending on the enantiopure or racemic nature. The CPL signals also varied upon grinding, experiencing a 2-fold increase for P7b from g _lum = 2.2 × 10^–3 to 4.5 × 10^–3. Using a dimeric model for the excimer, generated starting from the X-ray structure, a calculated g _lum = 4.7 × 10^–3 was obtained at the CAM-B3LYP-D3BJ/6-31G­(d,p) level.

In 2022, some of us reported isomannide and isosorbide derivatives functionalized by 1-pyrenecarboxylate and 3-perylenecarboxylate at the secondary alcohol moieties. We aimed at spotting intramolecular excimer emission and CPL using as a chiral scaffold the two isohexides obtained from renewable sources. Of the two epimeric bis­(1-pyrenecarboxylate)­s P8 and P9 (Chart ), only the isomannide derivative P8 with the pyrene rings in a syn orientation gave a broad excimeric emission with λ_em > 500 nm and a Stokes shift of 116 nm, accompanied by strong CPL; the isosorbide derivative P9, instead, showed a structured emission with maxima at λ_em = 395 and 413 nm typical of monomeric pyrene (Figure ). Notably, the recorded g _lum = 9.6 × 10^–3 for P8 was much higher than g _abs = 2.3 × 10^–4, yielding a rather exceptional ratio g _lum/g _abs = 40. It is also worth mentioning that the S₀ → S₁ absorption band is usually very weak in pyrene derivatives, and the main feature observed in the absorption spectra is the S₀ → S₂ transition lying at approximately 350 nm. The common notion that CPL spectra of organic compounds should contain a single band with the same sign and similar intensity as the red-most ECD band is valid only when S₀ and S₁ geometries are similar. In the case of excimers, the excited-state geometry is by definition different from the S₀ geometry, where the chromophores are noninteracting (recall the definition given in Section ). Therefore, excimer or exciplex formation may lead at least in principle to high dissymmetry values. Excited-state calculations were run on P8 using different functionals (B3LYP, B3LYP-D3, BH&HLYP, and BH&HLYP-D2) combined with the def2-SVP basis set. The half-and-half hybrid functional BH&HLYP (also known as BHLYP) was tested because in a benchmark study by Grimme and co-workers on benzene and pyrene excimers it provided accuracy similar to that of higher-level calculations. All functionals yielded consistent excimer structures with multiple conformations; the only exception was B3LYP, which was unable to capture any excimer structure. Reoptimization of the lowest-energy S₁ geometry at the BH&LYP/def2-SVP level with LR-CPCM for DCM provided the excimer input structure used for vertical calculations (Figure ). These were run by combining B3LYP, BH&HLYP, and CAM-B3LYP functionals with def2-SVP and def2-TZVP basis sets, with LR-CPCM for DCM. BH&HLYP and CAM-B3LYP provided consistent results, whereas B3LYP was again unsatisfactory. The calculated g _lum lay between 3 and 4 × 10^–2 (Table ), and the calculated Stokes shifts were 107–113 nm, in fair agreement with experimental data. We wish to remark on the limitations of the B3LYP functional, which failed not only to capture the excimer structure in the excited state (unless a dispersion correction was added) but also to predict correctly its emission, even starting from the correct excimer structure.

42.

Experimental absorption, emission, ECD, and CPL spectra of isomannide and isosorbide derivatives P8 (red traces) and P9 (black traces; Chart ). Inset: ES geometry of P8 emphasizing the intramolecular pyrene excimer. Adapted from ref under a CC-BY 4.0 license.

4. Collection of Experimental and Computational Data for Pyrene and Perylene Derivatives .

	Experiment	Calculation			Number or abbreviation
Type	g lum	g lum	Method	Basis set	This review	Original publication	ref.
Pyrene excimer	9.6	37	BH&HLYP	def2-SVP	P8	3a
Pyrene excimer	9.6	51	B3LYP	def2-SVP	P8	3a
Pyrene excimer	9.6	36	CAM-B3LYP	def2-SVP	P8	3a
Pyrene excimer	10	7.6	CAM-B3LYP	def2-SVP	P12	DC-Py
Pyrene excimer	15	10.4	CAM-B3LYP	def2-SVP	P12	DC-Py DMSO
Pyrene excimer	2	1.5	CAM-B3LYP	def2-SVP	P12	DC-Py TFA
Pyrene excimer	53	72	CAM-B3LYP	6-31G(d,p)	P14a	2
Pyrene excimer	2.4	60	CAM-B3LYP	6-31G(d,p)	P15a	4
Pyrene excimer	22	120	CAM-B3LYP	6-31G(d,p)	P17	3
Pyrene (non excimeric)	10	2.3	PBE0-D3	6-31G(d)	P19	BiPyCO
Pyrenophane	28	17	CAM-B3LYP	6-31G(d,p)	P20	Py
Pyrenophane	25	15	CAM-B3LYP	6-31G(d,p)	P21a	PyS
Pyrenophane	23	12	CAM-B3LYP	6-31G(d,p)	P21b	PySO*
Pyrenophane	33	17	CAM-B3LYP	6-31G(d,p)	P21c	PySO2
Perylene twisted dimer	10	23	PBE0	6-311G(d)	P22	1a
Perylene twisted dimer	10	18	PBE0	6-311G(d)	P22	2
Perylene twisted dimer	22	28	PBE0	6-311G(d)	P23	4
Perylene twisted dimer	30	47	PBE0	6-311G(d)	P24	5
Perylene twisted dimer	9	19	PBE0	6-311G(d)	P25	3
Perylene twisted dimer	13	20	PBE0	6-311G(d)	P26	4

^aData for electronic calculations only. See Charts – for structure numbering.

^b g _abs and g _lum multiplied by 10³.

Starting from 2015, Lacour and co-workers have designed a series of polyoxygenated macrocyclic diamides functioning as chiroptical switches upon binding metal ions. The metal binding/release event is accompanied by off/on switching of intramolecular excimer emission and CPL. In one of the latest developments, disulfide-bridged compound P10 (Chart ) was conceived and studied in collaboration with some of us. Contrary to its analogs devoid of the bridge, P10 is not capable of forming an intramolecular excimer. In pure THF solutions, in fact, only a weak CPL signal is observed around 400 nm, associated with residual intramolecular excimer emission which is visible in the fluorescence spectrum as a tail of the major structure between 350 and 450 nm. Upon addition of an increasing amount of water, however, the CPL spectrum undergoes a first sign inversion and reaches a maximum |g _lum| = 0.022 for the mixture THF:H₂O = 30:70, followed by an intensity decrease and a second sign inversion for the mixture THF:H₂O = 10:90 (Figure ). Evidently, the molecule undergoes multiple aggregation stages in solution. To shed light on these processes, TD-DFT calculations were run using B3LYP and CAM-B3LYP functionals with a D3BJ dispersion correction and TDA (Section ). It is worth noting that this is one of the few applications of TDA to CPL calculations found in the literature, which was chosen considering the size of the system and its lower computational cost with respect to TD-DFT, especially in combination with the def2-SVP basis set with resolution of identity (RI). The input structures for TDA calculations were (Figure ) (a) an intramolecular stacked monomer; (b) an intermolecular stacked dimer obtained from the X-ray geometry of crystals grown from DCM (polymorph 1); and (c) an intermolecular stacked dimer obtained from the X-ray geometry of crystals grown from CHCl₃ (polymorph 2). The input structures led to various low-energy geometries in both S₀ and S₁ states; most of these yielded excimer-like calculated emission and CPL spectra, which represented good models for the aggregated species detected in solution.

43.

Representative excimer structures of compound P10 (Chart ) and its aggregates responsible for the observed CPL signals in various solvent mixtures. Reproduced from ref under a CC-BY 4.0 license.

A unique triphasic CPL switch was designed by Wang et al. based on a catenane structure containing two oligo-ethylene glycol portions and two pyrene rings (P11, Chart ). The molecular switch undergoes a reversible off/on/enhanced CPL signal modulation upon metal binding/release/protonation treatment (Figure ). The computational analysis of this system, run by Bella and co-workers, poses an obvious difficulty related to its dynamic nature and to the expected dependence of the chiroptical response on the conformation. This issue called for an extensive benchmark of computational methods. First, several DFT functionals were tested for ground-state geometry optimization of the parent catenane (devoid of 1-pyrenilmethyl groups), whose X-ray geometry was available. The long-range corrected hybrid functional LC-ωhPBE emerged as the leader functional in this context; therefore, the LC-ωhPBE/6-311G­(d,p) level of calculation was used to the explore the ground-state geometry of the three switchable forms of P11. The same functional was used in ab initio molecular dynamics simulations (Born–Oppenheimer molecular dynamics, BOMD; see Section ) with explicit solvent to describe the excited-state PES of the three forms; TD-DFT calculations were then run on snapshots extracted from BOMD trajectories. For the vertical calculations, a large list of DFT functionals was again employed. This time, the two functionals PW6B95D3 and PBEPBE emerged as the best-performing ones for the neutral closed form and were selected to simulate the remaining forms. The former functional, PW6B95D3, is a six-parameter meta exchange-correlation functional from Truhlar’s group, corrected for dispersion; the second one, PBEPBE (or simply PBE), is the classical GGA functional by Perdew, Burke, and Ernzerhof. The very different nature of these two functionals and the failure of well-reputed functionals such as CAM-B3LYP and ωB97X-D offer a clear demonstration of the very substrate-dependent behavior of DFT, which makes it advisible to test several different functionals on each specific class of compounds and look at the best-performing one on an empirical basis. In the end, the use of PW6B95D3/6-311G­(d,p) calculations with CPM for DCM could reproduce very well the experimental observations for the three switchable forms (Figure ).

14.

44.

(a,b) Possible forms of catenane CPL-switch P11 (Chart ) induced by chemical stimuli. (c) Evolution of CPL spectra due to closed-to-biprotonated and closed-to-open switching. (d) Calculated CPL profiles for the three forms. In each case, positive CPL curves correspond to (R) catenane configuration. Adapted with permission from ref , copyright 2022, John Wiley and Sons, and from ref under a CC-BY-NC 3.0 unported license.

A second switchable mechanically interlocked bispyrene system, based this time on a so-called [c2]­daisy design, was reported recently by Wang and co-workers (P12, Chart ). The system is endowed with dual stimuli responsiveness, that is, to polar solvents (especially DMSO) and anions (trifluoroacetate), both acting by involving thioureas units, which can be reversed respectively by the addition of H₂O and Na⁺. The three possibly states adopted by P12 (state I, DCM solution; II, DMSO addition; and III, CF₃COO^– addition) all display the pyrene excimer emission band between 400 and 500 nm. Chiroptical spectra are, on the contrary, very different: (pR)-P12 has g _lum = −1.0 × 10^–2 in state I, boosted and inverted g _lum = +1.5 × 10^–2 in state II, and reduced g _lum = −2 × 10^–3 in state III; the system acts therefore as a true chiroptical switch (Figure ). Qualitatively, the CPL sign was related to the chirality defined by the two pyrene rings by the already mentioned excimer chirality rule. Quantitatively, the authors run PBE0/def2-SVP calculations which reproduced the experimental trend exceptionally well (Table ) in view of the complexity of the system. The different CPL sign and intensity for states I–III were rationalized on the basis of different twists between the pyrene rings (Figure a), leading to different orientations between electric and magnetic transition dipoles. All states had two excited-state conformers close in energy, thus possibly explaining the presence of multiple CPL bands.

45.

(a) Mechanism of dual-stimuli response of bispyrene-functionalized interlocked compound P12 to solvent and ions. The ball-and-stick structures show the arrangement between pyrene rings in the ES (lowest-energy conformer for each state I–III). (b,c) Evolution of CPL spectra of P12 upon DMSO and trifluoroacetate addition to a DCM solution. In (c), the spectra are shown for both enantiomers. Adapted with permission from ref . Copyright 2025, John Wiley and Sons.

Two further pillar[5]­arene derivatives P13a–b (Chart ) appended with multiple pyrene rings were conceived by Wang and co-workers. While P13a was CPL-silent, the more hindered P13b formed two different excimers with oppositely signed CPL and maximum g _lum = 3.8 × 10^–3 at 433 nm. Interestingly enough, [2]­rotaxanes formed by P13a with bis­(perfluorophenyl) octanedioate and decanedioate were also CPL-active with g _lum ≈ 1.5 × 10^–2 at 455 nm and large B _CPL = 364 M^–1 cm^–1. TD-DFT calculations on the molecularly interlocked species, though run at the B3LYP/6-31G­(d) level, reproduced very well the experimental dissymmetry values.

Takaishi, Ema, and co-workers designed a family of binaphthyl-bridged pyrenophanes (from P14a–c to P18, Chart ) intended to exploit D ₂ symmetry to achieve high dissymmetry values, which manifest excimeric CPL emission with record g _lum values. , Among the oxygenated series, P14a, P16a, and P17 exhibited excimer emission while P15a and P18 showed the characteristic pyrene monomer emission. Thus, the steric and geometrical requirements of the 1,1′-binaphthyl-pyrene linkers control the intramolecular excimer formation. It is interesting that π-stacking already occurs in the ground state for this class of compounds: for instance, the X-ray geometry of P14a features parallel-twisted pyrene rings at 3.6 Å and a twist angle between pyrene long axes of 42.4°. (TD-)­DFT calculations run at the CAM-B3LYP/6-31G­(d,p) level replicated these measurements for the S₀ geometry (distance 3.9 Å, twist angle 46.5°) and predicted that the rings get closer by >10% in the S₁ state (distance 3.4 Å, twist angle 46.1°), thus substantiating excimer formation. The CPL signals for the oxygenated series were all consistent with the red-most ECD band and were also in accord with the excimer chirality rule (see above). The largest g _lum = 5.3 × 10^–2 was observed for P14a, which benefitted from D ₂ symmetry, but the simpler compound P16a also had a remarkable g _lum = 3.2 × 10^–2. TD-DFT calculations on P14a reproduced well the observed g _lum (Table ) and confirmed an almost collinear orientation of transition dipole moments. The sulfurated series, though not improving the dissymmetry values, gave rise to unique characteristics. P14b, P14c, and P16b exhibited excimer emission, while P15b exhibited monomer-type emission. Variable-temperature (VT) CPL spectra of (R,R)-P14b showed a distinct temperature dependence, evolving from a negative signal with g _lum = −1.4 × 10^–2 at −10 °C to a positive signal with g _lum = +6.0 × 10^–3 at 90 °C (Figure a), with little change in λ_em ≈ 485–490 nm and in the QY. The large g _lum variation and the absence of quenching upon heating make P14b suitable as a CPL-switching compound. VT-CPL inversion experiments could be repeated over 10 cycles with small signal loss. To rationalize the CPL-switching behavior of P14b, TD-DFT calculations were run at the CAM-B3LYP/6-31G­(d,p) level. Two low-energy structures were found in both the ground and excited states. In particular for S₁, the two structures were both excimeric, with a plane-to-plane distance of 3.4 Å, but had opposite twist angles between pyrene main axes (+57.7 and −49.2° for the lowest energy and the second minimum of (R,R)-P14b, respectively; Figure b). Furthermore, both structures were D ₂-symmetric and yielded large S₁–S₀ dissymmetry values (g _lum = +6.5 × 10^–2 and −6.9 × 10^–2, respectively). Torsional energy scans confirmed the presence of a double-well PES for both S₀ and S₁; for the two states, the relative stability of structures with negative and positive twist angles is inverted, which ultimately determines the sign of ECD and CPL. The sets of calculations allowed the authors to propose the following mechanism of excitation dynamics (Figure c): in the S₀ state, (R,R)-P14b adopts a left-handed interpyrene chirality (1); after vertical excitation (2), a left-handed excimer is obtained (3); at low temperatures, the latter is kinetically trapped and emits negative CPL (4); and at high temperature, the excimer (3′) overcomes the barrier toward a more stable S₁ conformation with right-handed chirality (4′), which emits positive CPL. Theoretical investigations were also run on other compounds in the series and agreed with the respective CPL behavior. It is also worth mentioning that the calculated relative energy between the S₀ and S₁ conformers was benchmarked over four different functionals (CAM-B3LYP, M06-2X, PBE0, and MN15) and the CCSD method as well as different basis sets (6-31G­(d,p), 6-311++G­(2d,p,) and cc-pVDZ).

46.

(a) Temperature-dependent CPL spectra of binaphthyl-bridged pyrenophane (R,R)-P14b (Chart ). (b) Structures of the two low-energy conformers of P14b in the S₁ state. (c) Suggested excitation dynamics of P14b. Adapted with permission from ref . Copyright 2024, John Wiley and Sons.

Despite a skeleton similar to that of P16a, compound P19 (Chart ), where 1′,1-binaphthyl and pyrene moieties are connected through carbonyl groups, manifested peculiar properties. This compound exists in a highly distorted structure allowing for through-space conjugation between naphthyl and pyrene rings. Emission from the widely delocalized S₁ state reached g _lum = 1.2 × 10^–2 above 600 nm in THF, a value underestimated by TD-DFT calculations run at the PBE0-D3/6-31G­(d) level (Table ).

Rigidly locked pyrene excimers were obtained recently by Zhu, Wang, and co-workers exploiting a pyrenophane design with D ₂ symmetry (P20 and P21a–c). The short contact between pyrene rings is forced in the GS with plane-to-plane distances of 3.32 Å for P20 and 3.41–3.48 Å for P21a–c. The twist angle between pyrene rings is also constrained at around 40–50°. Due to the rigid skeleton, CPL spectra are solvent- and temperature-independent and reach in all cases g _lum > 2 × 10^–2, which were well reproduced by calculations (Table ). Another favorable consequence of rigidity is that collinearity between transition moments is preserved in the S₁–S₀ transition. It is worth noting that calculated S₁ geometries displayed similar plane-to-plane distances as S₀. Strictly speaking, then, the compounds should not be classified as excimers (recall the definition in Section ).

Wang, Jiang, and co-workers have reported two series of double π-helix CPL emitters (P22–P26, Chart ) , which, being based on the double-perylene diimide (PDI) skeleton, will be described in this section but could also be classified among biaryls (Section ). The compounds feature large QYs (maximum Φ_f = 0.7 for P25), high g _lum > 10^–2 (maximum g _lum = 3 × 10^–2 for P24), and extremely high CPL brightness (B _CPL = 573.4 M^–1 cm^–1 for P24). The dissymmetry values are boosted by D ₂ symmetry found in double-heteroannulated compounds P24 and P26, which feature higher g _lum than monoheteroannulated analogs P 23 and P25. TD-DFT calculations run at the PBE0/6-311G­(d) level on truncated models (R₁ = H in Chart ) reproduced satisfactorily the experimental g _lum values, apart from some overestimation (Table ), and confirmed for P22, P24, and P26 the collinear orientation of electric and magnetic transition moments expected for D ₂ symmetry. , Starting from tetrachlorinated precursor P27, the same group further extended the twisted conjugated skeleton to achieve PDI trimer P28 and tetramers P29–P31 which were dubbed π-helical nanoribbons. Again, we describe them in this section coherently with the other PDI derivatives, although they belong to larger organic systems (Section ). Focusing on the tetramers, it must be noted that P29 (with configuration aS, aS, aS) and P31 (with configuration aS, aR, aS) feature D ₂ symmetry, whereas P30 (with configuration aS, aS, aR) features C ₂ symmetry. Compounds P27–P31 display clear size- and symmetry-dependent (chiro)­optical properties summarized in Figure . The best-performing component of the series is D ₂-symmetric tetramer P29, which achieves g _lum = 1.5 × 10^–2 and a record B _CPL = 575 M^–1 cm^–1. Similar to the previous series, TD-DFT calculations were run at the PBE0/6-311G­(d) level on truncated models (R₂ = CH₃ in Chart ), which reproduced the observed trend but slightly overestimated all experimental g _lum values.

15.

47.

Trend of experimental ECD and CPL quantities measured for helical nanoribbons P27–P31 (Chart ). Reprinted with permission from ref . Copyright 2024 American Chemical Society.

3.1.5. [2.2]­Paracyclophanes

[2.2]­Paracyclophane is a widely employed structural motif which, for certain substitution patterns, imparts so-called planar chirality. CPL-active [2.2]­paracyclophane derivatives were reviewed recently. , Surprisingly enough, the number of publications reporting CPL calculations on [2.2]­paracyclophanes is quite small, apart from [2.2]­paracyclophanes substituted with oligo­(phenylene ethynylene) moieties to be described in a following section (Section ). A key parameter to be considered in QM calculations on [2.2]­paracyclophanes is the plane-to-plane or interannular distance, which for the parent [2.2]­paracyclophane in the ground state equals 3.10 Å. Grimme and Mück-Lichtenfeld have demonstrated the large dependence on the functional and dispersion correction of DFT-optimized geometries of [2.2]­paracyclophane derivatives. In particular, B3LYP tends to overestimate the interannular distance, but by adding the D3BJ dispersion correction an accurate value is recovered.

In 2022, Abbate and colleagues reported a series of [2]­paracyclo[2]­(5,8)-quinoliphane derivatives embedding planar chirality (such as C1 and C2, Chart ) as well as both planar and central chirality (C3). All compounds have g _lum ≈ 10^–3 or below. An interesting fact is that while C1 and C2 have a consistent lowest-energy ECD band and CPL band, for C3 the two bands have opposite signs. TD-DFT calculations run at the B3LYP-D3BJ/6-311++G­(d,p) level (Table ) correctly reproduced the CPL sign of the three compounds but failed to predict the correct ECD sign for the S₀–S₁ transition of C3. Nevertheless, they suggested as a possible reason for the odd behavior of the latter compound a large structural rearrangement from the S₀ to the S₁ state.

16.

5. Collection of Experimental and Computational Data for Cyclophane Derivatives .

Exper.	Calculation			Number or abbreviation
g lum	g lum	Method	Basis set	This review	Original publication	ref.
1.04	0.37	B3LYP-D3BJ	6-311++G(d,p)	C1	1
0.61	2.31	B3LYP-D3BJ	6-311++G(d,p)	C2	2
0.042	0.89	B3LYP-D3BJ	6-311++G(d,p)	C3	4
1.4	3.1	CAM-B3LYP	6-31G(d)	C6	10
2.6	7.4	CAM-B3LYP	6-31G(d)	C7	14
1.8	2	MN15	6-31G(d)	C9	4
0.47	0.32	B3LYP	6-31G(d,p)	C11a	1a
0.8	0.025	CAM-B3LYP	def2-SVP	C12	6
0.8	0.713	M06-2X	def2-SVP	C12	6
1.2	2	M06-2X	def2-SVP	C13a	Ph-PCP
1.7	1.8	M06-2X	def2-SVP	C13b	Np-PCP
0.4	1.36	M06-2X	def2-SVP	C13c	Ant-PCP
1.2	0.85	CAM-B3LYP	def2-SVP	C15	CzpPhTrz

^aData for electronic calculations only. Cyclophane-OPE conjugates are listed in Table . See Chart for structure numbering.

^b g _abs and g _lum multiplied by 10³.

Morisaki and co-workers have reported a number of [2.2]­paracyclophane derivatives fused, linked, or bridged with aromatic moieties. Compounds C4 and C5 (Chart ) embed [3]­helicene (phenanthrene) and [4]­helicene units, and in fact they show ECD spectra similar to those of short carbo­[n]­helicenes. CPL spectra were moderately strong with g _lum = 3.5 × 10^–3 for C4 and 2.7 × 10^–3 for C5. Excited-state geometry optimizations were run with the CASSCF method (Section ), and rotational strengths were predicted by CIS­(D) calculations with the 6-31G­(d) basis set, which correctly reproduced CPL signs but overestimated g _lum values by a factor of ∼3. We point out, however, that the claim on CIS­(D) simulations made in this contribution is unclear or even questionable: indeed, this is a perturbative correction to the CIS energy and the transition properties are not involved, so it is not possible to conclude if the rotational strength is corrected at the CIS­(D) level or if it is actually a CIS one. “Hybrid” analogs C6 and C7 exhibited dissymmetry values similar to those of their symmetrical analogs (g _lum = 1.4 and 2.7 × 10^–3 for C6 and C7, respectively). TD-DFT calculations at the CAM-B3LYP/6-31G­(d) level (Table ) clarified that emission occurred in each case from the [4]­helicene units.

A compound exhibiting extremely interesting CPL behavior, reported by Morisaki and co-workers, is C8 (Chart ). This tetra­(p-methoxy)­phenyl-substituted [2.2]­paracyclophane features four anisole units capable of wide rotations around the interaryl C–C bonds. Both ECD and CPL spectra are, in fact, temperature-dependent, but while ECD spectra manifest a mere intensity increase upon temperature lowering (with g _abs moving from 3.6 × 10^–3 at 25 °C to 4.4 × 10^–3 at −120 °C), CPL spectra display a rather unusual thermochromism (Figure ). At 25 °C, a single band with the same sign as that of the red-most ECD band is found, with g _lum = 1.7 × 10^–3; upon temperature lowering, a short-wavelength CPL band arises with a sign opposite that of the major one. With the help of TD-DFT calculations run at the CAM-B3LYP-D3BJ/6-31+G­(d) level of theory, the authors rationalized the collected data as follows. At RT, emission occurs from the S₁ global minimum obtained after a wide structural relaxation and featuring a planarized quinoid-like structure; the predicted θ _μm = 80° for the S₁–S₀ emission is partially compensated for by a large m ₀₁ moment, yielding a calculated g _lum = +1.0 × 10^–3 for (Sp)-C8 in good agreement with the experiment. At low temperatures, a local minimum on the S₁ PES, corresponding to unrelaxed vertical excitation from S₀, becomes emissive too. The estimated θ _μm = 180° for this second emission accounts for a negative g _lum = −2.4 × 10^–2. Although not explicitly discussed in the main text, the first calculated ECD band for (Sp)-C8 is negative, but it is apparently submerged by the second strongest positive band. Thus, the experimental CPL spectrum at RT has only accidentally the same sign as that of the red-most ECD band, and the relaxation to the quinoid structure inverts the sign of the rotational strength of the S₁–S₀ transition with respect to the S₀–S₁ one.

48.

(a) Temperature-dependent CPL spectra of the two enantiomers of [2.2]­cyclophane C8 (Chart ). (b) Suggested emission mechanism of (S _p)-C8 at 298 and 178 K. Adapted with permission from ref . Copyright 2020, John Wiley and Sons.

Morisaki and co-workers also reported a [2.2]­cyclophane derivative tethering an anthracene ring at 1,8-positions via ethynyl bridges (C9, Chart ). Experimental and TD-MN15/6-31G­(d) calculated g _lum were 1.8 × 10^–3 and 2.0 × 10^–3, respectively. The same group synthesized a series of [2.2]­cyclophanes functionalized with oligo­(o-phenylene)­s; the quaterphenyl member of the series is C10. In these compounds, the planar chirality of the [2.2]­cyclophane moiety controls the axial chirality assumed by the biaryl linkages. For instance, (S _p)-C10 exists as a major conformer where all four aryl linkages assume the (aR) absolute conformation, imparting an overall M helicity to the molecule. The helical shape favored high dissymmetry values, especially in emission (g _lum = 1.2 × 10^–2). Calculations run at the MN15/6-31G­(d) level reproduced the CPL sign although they overestimated the g _lum by a factor of ∼2; interestingly enough, they predicted collinear electric and magnetic dipole transition moments, which contributed to the large dissymmetry.

Exploiting the same molecular design as for compound C8, Hasegawa, Mazaki, and co-workers recently reported two [2.2]­cyclophanes, one of which was derivatized with four carbazole units designed for TADF emission (C11a–b, Chart ). The compounds have good QYs but a relatively small g _lum < 5 × 10^–4, which in the case of C11a was reproduced by TD-B3LYP/6-31G­(d) calculations (Table ). Interestingly, these compounds acted as efficient dopants in inducing ECD and CPL activity in blends with achiral polymer F8BT. The thin film of C11a-F8BT with a 3% dopant reached g _lum = 0.01.

Bettinger and co-workers, including one of us, reported a number of [2.2]­cyclophanes derivatized with photoluminescent NBN-benzo­[f,g]­tetracene units. Compound C12 (Chart ) was photostable and moderately emissive (Φ_f = 0.70), with a broad emission band with a maximum at around 500 nm. Dissymmetry values were the same in both absorption and emission (8.0 × 10^–4), and a satisfying B _CPL = 22 M^–1 cm^–1 was obtained. Calculations were run on a truncated model (with n-butyl groups replaced by hydrogen atoms) which theoretically has D ₂ symmetry. After a screening run on ground-state geometry, the M06-2X functional was chosen for the geometry optimization of the S₁ state too, in combination with the def2-SV­(P) basis set. Thereafter, the vertical S₁–S₀ transition was calculated in vacuo with different functionals (CAM-B3LYP, CAM-B3LYP-D3, M06-2X, and ωB97X-D). Surprisingly enough, CAM-B3LYP led to a wrong CPL sign, possibly because of nonrobust θ _μm ≈ 90°. All other functionals reproduced the correct sign, with M06-2X best approaching the experimental g _lum. To complete their study and further verify the performance of the M06-2X functional, the authors calculated the CPL parameters for three other [2.2]­cyclophanes derivatives C13a–c reported by Morisaki and co-workers, finding in all cases good-to-excellent agreement between experimental and calculated g _lum (Table ).

A family of [2.2]­cyclophanes incorporating a boron difluoride Schiff-base moiety was reported by Duan, Gong, and co-workers. Several other chiral boron difluoride Schiff-base complexes will be described in the next section (Section ), dedicated to tetra-coordinated boron systems. Compounds C14a–e (Chart ) display typical emissions of boron difluoride complexes above 450 nm, with moderate to good QYs and g _lum between 3.6 and 6.2 × 10^–3, which increased in the solid state up to 7.6 × 10^–3 for C14c. TD-DFT calculations were run at the PBE0/6-31+G­(d) level and led in some cases to the wrong sign of the CPL band.

Compound C15 (Chart ) has a unique design in the para[2.2]­cyclophane family as it incorporates a donor carbazole unit and an acceptor 2,4,6-triphenyl-1,3,5-triziane unit, thus yielding a D–A dyad. The twisting around the carbazole-phenyl is rather restricted, thus preventing the vibrationally induced loss of the CPL signal. The experimental g _lum = 1.2 × 10^–3 was reasonably reproduced by TD-DFT calculations (CAM-B3LYP/def2-SVP, Table ), using both a single S₁ minimum and the nuclear ensemble approach.

3.1.6. Tetra-Coordinated Boron Compounds

Four-coordinated organoboron compounds have been known for a long time to be strongly emissive and to exhibit several outstanding properties such as aggregation induced emission (AIE, see Section for more details), far-red and NIR emission, solvatochromism, mechanochromism, nonlinear optical properties, and so on. , They found many applications, especially in OLED devices. The most important representatives of this class are 4-bora-3a,4a-diaza-s-indacenes, universally known with their abbreviation BODIPY standing for boron-dipyrromethene. , The applications based on fluorescent BODIPY derivatives are countless and include, for instance, photodynamic therapy. Simple alkyl-derivatized BODIPYs have an absorption maximum at around 500 nm and emit around 520 nm with QY > 0.7. One very favorable property is that the lowest-energy S₁ state is bright, with the absorption and emission oscillator strengths close to unity.

From a computational viewpoint, BODIPYs have represented a real challenge for TD-DFT as for the simulation of excitation energies. In fact, all common DFT functionals tend to overestimate BODIPY absorption energies to some extent, and only the use of electron correlated and multireference methods such as CIS­(D), CC2, and CASPT2 can recover the correct values. − A similar problem occurs for emission energies.

In 2016, we reported one of the first and few calculations of ECD and CPL spectra of BODIPY derivatives. The two compounds BX1 and BX2 (Chart ) are atropisomeric BODIPY “dimers” linked through biaryl linkages at 3,3′ and 1,1′ positions. Both compounds show exciton-coupled ECD spectra between 400 and 640 nm; however, the exciton coupling details are very different from each other. Compound (aR)-BX1 displays a strong negative couplet with a remarkable amplitude (peak-to-trough difference in Δε) A = −575 M^–1 cm^–1, whose sign obeys the exciton chirality rule. Compound (aR)-BX2 shows a weaker positive couplet with amplitude A = +102 M^–1 cm^–1 and a sign that provides an apparent exception to the exciton chirality rule. This inconsistency is due to the fact that the S₀–S₁ in each BODIPY unit is both electric-dipole- and magnetic-dipole-allowed because of the curved geometry of the chromophore. Therefore, besides the standard electric–electric (μμ) exciton coupling mechanism, an additional electric–magnetic (μm) coupling mechanism is operative, which for (aR)-BX2 overcomes the μμ one. The same phenomenon occurs for other biaryls, for instance, 1,1′-bisphenanthrenes. Regardless of the details of the exciton coupling mechanism, the sign and intensity of the lowest-energy component of the exciton-coupled ECD spectra of BX1 and BX2 were consistently reflected in CPL spectra (this concept is illustrated by Figure b in Section ). (aR)-BX1 showed a negative CPL band with g _lum = −3.8 × 10^–3 at 655 nm, and (aR)-BX2 showed a positive CPL band with g _lum = +4 × 10^–4 at 603 nm. As for TD-DFT calculations on compounds BX1 and BX2 (for the latter, a simplified model was used with ethyl groups replaced by methyl groups), the M06-2X functional was chosen after minimal screening, in combination with the def2-TZVP basis set. TD-DFT calculations reproduced well all (chiro)­optical properties of BX1 (Figure and Table ), which features a single ground-state and excited-state conformation, including decent agreement for g _abs and g _lum values. The situation for BX2 is complicated by the conformational flexibility. In fact, the rotation around the C1–C1′ aryl linkage (corresponding to the chirality axis) is largely allowed, with two shallow minima corresponding to C8a–C1–C1′–C8a′ dihedrals around 52° (cisoid conformer) and 128° (transoid conformer) in the ground state S₀ (Figure a). In the excited state S₁, the two corresponding minima were identified for dihedrals around 40 and 136°, respectively, separated by a higher barrier (Figure b). While M06-2X/def2-TZVP calculations reproduced the ECD spectrum of BX2 well (Figure ), they failed in getting the correct CPL sign. The same failure was observed with other functionals (CAM-B3LYP and ωB97X-D) or including a state-specific solvent model. The correct sign could be recovered only by switching from TD-DFT to a coupled-cluster method, namely, the SCS-CC2 level of theory (spin-component-scaled second-order approximate coupled cluster), yet using DFT-optimized geometries (Figure ). The observed failure was related to the intrinsic weakness and nonrobustness of the CPL band of BX2.

17.

49.

Experimental (a,c) and calculated (b,d) absorption, emission, and ECD and CPL spectra of the two enantiomers of BODIPY dimers BX1 (a,b) and BX2 (c,d) (Chart ). Vertical bars represent oscillator and rotational strengths. Spectra calculated with TD-DFT except for the inset, calculated with CC2. Adapted with permission from ref . Copyright 2016, John Wiley and Sons.

6. Collection of Experimental and Computational Data for Tetracoordinated Boron Compounds .

	Exper.		Calculation				Number or abbreviation
Type	g lum	g lum /g abs	g lum	g lum /g abs	Method	Basis set	This review	Original publication	ref.
BF2 bodipy	3.8	0.4	5.6	0.64	M06-2X	def2-TZVP	BX1	1
BF2 Schiff base	0.28	2.2	0.28	1	B3LYP	6-31+G(d,p)	BX3b	1a
BF2 Schiff base	1	1	1.1	3.9	B3LYP	6-31+G(d,p)	BX3a	1b
BF2 Schiff base	0.81	0.2	0.45	0.54	B3LYP	6-31+G(d,p)	BX3c	1b
BF2 Schiff base	0.69		0.84		B3LYP	6-31+G(d,p)	BX4	1a
BF2 Schiff base	0.43		0.27		B3LYP	6-31+G(d,p)	BX5	1b
BF2 Schiff base	0.79		0.56		B3LYP	6-31+G(d,p)	BX6	1c
BF2 Schiff base	0.66		0.83		B3LYP	6-31+G(d,p)	BX8	1d
BF2 Schiff base	1.1	1.5	2.1	0.57	mPW1PW91	6-31+G(d,p)	H47a	1a
BF2 Schiff base	0.36	2.1	4.9	1	mPW1PW91	6-31+G(d,p)	H47b	1b
BF2 Schiff base	0.71	2.9	0.15	0.65	B3LYP	6-31+G(d,p)	BX9	1a
BF2 Schiff base	0.67	6.7	0.56	1.9	B3LYP	6-31+G(d,p)	BX10	1b
BF2 Schiff base	0.47	2.5	0.38	1	B3LYP	6-31+G(d,p)	BX11	1c
BF2 Schiff base	0.4		0.51		B3LYP	6-31+G(d,p)	BX14b	1b
BF2 Schiff base	2.5	0.06	4.0	0.44	B3LYP	6-31+G(d,p)	BX15a	1a
BF2 Schiff base	2.5	0.06	2.9	0.29	B3LYP	6-31+G(d,p)	BX15b	1b
BF2 Schiff base	2.9	0.09	1.6	0.16	B3LYP	6-31+G(d,p)	BX15c	1c
BF2 Schiff base	2.1	0.08	2.5	0.25	B3LYP	6-31+G(d,p)	BX15d	1d
BF2 Schiff base	1.2	0.09	1.1	0.16	B3LYP	6-31+G(d,p)	BX15e	1e
BF2 Schiff base	1.3	0.09	1.1	0.19	B3LYP	6-31+G(d,p)	BX15f	1f
BINOL-O4–B bodipy	0.71	0.84	0.09	0.22	MN15	6-31+G(d)	BX16
BINOL-O4–B bodipy	2.2		2.7		B3LYP	6-31G(d)	BX18	BDTPA
N,N,O,C–B bodipy	3.7		2		PBE0	def2-TZVP	BX19a	3
N,N,O,O–B bodipy	4.7		3		PBE0	def2-TZVP	BX20	4
N,N,O,C–B bodipy	2.3		3.1		CAM-B3LYP	6-311++G(3df,2pd)	BX19b	3b
N,N,O,C–B bodipy	2.6		2.2		CAM-B3LYP	6-311++G(3df,2pd)	BX19c	3c
N,N,O,C–B bodipy	2.1		2.2		CAM-B3LYP	6-311++G(3df,2pd)	BX19d	3d
N,N,O,C–B bodipy	2.2		2.2		CAM-B3LYP	6-311++G(3df,2pd)	BX19e	3e
C,C–B Schiff base	1.5		4		mPW1PW91	6-31+G(d,p)	BX23a	1a
C,C–B Schiff base	1.4		2.9		mPW1PW91	6-31+G(d,p)	BX23b	1b
C,C–B Schiff base	2.3		1.9		mPW1PW91	6-31+G(d,p)	BX24a	2b
C,C–B Schiff base	1.2		2.1		mPW1PW91	6-31+G(d,p)	BX25a	1c
C,C–B Schiff base	2.4		1.8		mPW1PW91	6-31+G(d,p)	BX25b	2c
Tetra BF2	4	1.3	2.9	1.6	CAM-B3LYP	6-31G(d,p)	BX26a	1
Tetra BF2	4	1	2.4	1.1	CAM-B3LYP	6-31G(d,p)	BX26b	4
Tetra BF2	10	1.25	6.3	1.2	CAM-B3LYP	6-31G(d,p)	BX26c	5
diBF2/diO,F–B	2	0.3	2	0.33	CAM-B3LYP	6-31G(d,p)	BX27d	2d
boramidine	0.8	1.23	1.1	1.33	CAM-B3LYP	def2-TZVP	BX28	1
boramidine	1	1.25	1.1	1.36	CAM-B3LYP	def2-TZVP	BX29	2

^aData for electronic calculations only. See Charts and for structure numbering.

^b g _abs and g _lum multiplied by 10³.

^cSee Chart , section .

50.

(a) Overlap of GS (gray) and ES (orange) geometries of BX2 in the transoid and cisoid conformations. (b) Torsional energy scans around the aryl–aryl axis for the ground state (lower black curve) and the first excited state (upper red curve) for BX2. Vertical bars are calculated R ₁₀ values for each angle. Adapted with permission from ref . Copyright 2016, John Wiley and Sons.

Boron difluoride Schiff-base derivatives are much more represented than BODIPY derivatives in the context of CPL calculations. In particular, Ikeshita, Imai, Tsuno, and co-workers reported several compounds endowed with central chirality, helical chirality, axial chirality or containing double BF₂ Schiff-base moieties (Chart ). Starting from the simplest salicylaldehyde imine compounds BX3a–b, they observed that the addition of the diethylamino group imparted solvatochromic properties which, especially in apolar solvents, were beneficial for the emission efficiency (Φ_f = 0.59 in toluene for BX3b, Φ_f = 0.32 in DCM for BX3a). On the other hand, dissymmetry values were reduced in both absorption and emission (g _lum = 2.8 × 10^–4 for BX3b, g _lum = 1.0 × 10^–3 for BX3a, both values in DCM). A computational analysis run at the B3LYP/6-31+G­(d,p) level reproduced the absolute dissymmetry values and the relative intensity observed for the two compounds (Table ). Additionally, it provided the insight that the small g _lum observed for BX3b results from the combination of a large electric transition moment for the S₁–S₀ transition and unfavorable alignment with the magnetic transition moment (θ _μm = 91°). The introduction of PEG chains yielded a derivative (BX3d) liquid at RT with excimeric emission; however, the g _lum = 2 × 10^–3 (454 nm) measured as neat was only slightly higher than solution values for BX3d and its computational model BX3c (Table ).

Building on the results above, the aromatic ring of the Schiff-base moiety was expanded to naphthalene (BX4–BX6, Chart ), anthracene (BX7), and phenanthrene (BX8). These derivatives exhibit multicolor emission, depending on the aromatic framework, and CPL in solution with moderate g _lum values (up to 1.3 × 10^–3 for BX7). Interestingly, the emission was preserved in the solid state as drop-casted films, and g _lum values were boosted by 1 order of magnitude. TD-B3LYP/6-31+G­(d,p) calculations reproduced well the experimental g _lum in solution (Table ) and highlighted again, as the main obstacle to larger dissymmetry values, the unfavorable orientation between transition moments (θ _μm = 89 to 93°). The problem was partially solved by employing a [4]­helicene as the aromatic portion of the Schiff base (H47a–b in Chart , Section ), yet the experimental g factors remained small because of a diastereomeric equilibrium (discussion in Section ). Then, the same authors explored a novel molecular design affording bis­(boron difluoride) Schiff-base complexes BX9–BX11. All three compounds showed weak CPL in solution (g _lum < 7 × 10^–4) but strong CPL in the solid state as KBr pellets (g _lum up to 1.2 × 10^–2 for BX11), though accompanied by some quenching. Again, TD-B3LYP/6-31+G­(d,p) calculations reproduced well the observed trends in g _abs and g _lum values in solution (Table ). In the end, the authors recognized the difficulties of rationalizing CPL properties in the solid state. They referred to KBr-dispersed pellets as an amorphous state, although they are more properly described as a microcrystalline dispersion, where intermolecular exciton coupling may occur between distinct molecules packed in the microcrystals. ,

Interesting solvent- and temperature-dependent CPL behavior is manifested by the two Schiff-base derivatives BX12 and BX13 reported by Ikeshita et al. The CPL signal above 400 nm is positive for (S,S)-BX12 and (S,S)-BX13 in apolar solvents (g _lum = +3.1 × 10^–3 for (S,S)-BX12 and +5.0 × 10^–3 for (S,S)-BX13 in toluene at ∼430 nm) and negative in polar solvents (g _lum = −1.8 × 10^–3 for (S,S)-BX12 and −6.0 × 10^–3 for (S,S)-BX13 in DMF; Figure b). In some solvents, like acetone and acetonitrile (ACN), the CPL sign is at odds with the red-most ECD band. Moreover, the CPL intensity and even the sign (for BX12) depend on the temperature. A conformational equilibrium is at play for compounds BX12 and BX13 between diastereomeric pairs with opposite axial chirality around the N-cyclohexyl bond (Figure a). TD-DFT calculations run at the B3LYP/6-31G­(d,p) level predicted CPL signals opposite in sign for the two diastereomers of both compounds (Figure c). Therefore, the authors concluded that the solvent dependence and temperature dependence of CPL spectra were due to the diastereomeric equilibrium. Although in-solvent TD-DFT calculations were not performed, to evaluate the solvent-dependent relative stability of the two diastereomers, the authors postulated the occurrence of a luminescence deactivation mechanism to occur in polar solvents for the most polar diastereomer.

51.

(a) Illustration of CPL sign switching via axial chirality control of Schiff base bis­(boron difluoride) compound BX12 (Chart ). (b) Solvent-dependent CPL spectra of (S,S)-BX12. (c) Calculated CPL spectra for two diastereomers of BX12. The “c” subscript refers to the central chirality, and “a” refers to the axial chirality. Adapted with permission from ref . Copyright 2024, John Wiley and Sons.

In the latest developments, the same group incorporated two BF₂ Schiff-base moieties into the 1,1′-binaphthyl and the 1,1′-spirobiindane-7,7′-diol (SPINOL) skeletons. In solution, 1,1′-binaphthyl derivatives BX14a–c (Chart ) showed CPL bands with g _lum < 10^–3. The PEGilated derivative BX14a, however, exhibited a 2.5 times larger g _lum (2.1 × 10^–3) of opposite sign with respect to that of the drop-cast film. The compound with the smallest lateral chain, BX14b, was considered for TD-DFT calculations at the B3LYP/6-31+G­(d,p) level (Table ). Like most representatives from the BINOL family (Section ), a larger interaryl dihedral angle φ was found in the S₁ state with respect to S₀ (111.7 vs 92.6°), and the calculated R ₁₀ was found to depend on φ both in sign (which inverted for φ ≈ 90°) and intensity. The authors speculated that, in the thin film, BX14b would adopt a cisoid conformation in the S₁ state (φ < 90°) rather than a transoid one as calculated for the gas phase. The SPINOL series (BX15a–f, Chart ) showed emission colors modulated from blue to orange by the N-substituent, with g _lum ≈ 1–3 × 10^–3. Calculations run at the B3LYP/6-31+G­(d,p) level confirmed the ability of the N-substituent to tweak FMO energies and localization. These compounds are characterized by small g _lum/g _abs ratios, which were decently captured by calculations (Table ) and explained as a consequence of excited-state desymmetrization.

In 2014, de la Moya and co-workers reported an interesting BODIPY-like dye where the two fluorine atoms are replaced by a bridging BINOL unit (BX16, Chart ). The compound features C ₂ symmetry and axial chirality provided by BINOL. The absorption profile is typical of BODIPY dyes, that is, associated with a bright electric-dipole-allowed S₀–S₁ transition, and so is the emission profile with λ_em = 570 nm. The g _lum was relatively small (<10^–3), but interestingly, the sign of the CPL band was opposite to that of the red-most ECD band. The same phenomenon was observed later for a VAPOL (2,2′-diphenyl-3,3′-biphenanthr-4-ol) analog of BX16 and was attributed to the presence of an ICT excited state with opposite rotational strength (S₁–S₀ transition) with respect to the S₀–S₁ transition (Figure a). To investigate the nature of the emission process in BX16, Bloino and co-workers ran extensive simulations of its vibrationally resolved absorption, emission, ECD, and CPL spectra. A screening of various functionals, in combination with the 6-31+G­(d) basis set and IEF-PCM solvent model, was run for the calculation of absorption and ECD spectra. The best-performing functional MN15 was then selected for emission and CPL calculations, but employing energies estimated at the LC-ωPBE/6-31+G­(d) level. For (aR)-BX16, a negative red-most ECD band was calculated with g _abs = −4.0 × 10^–4, in keeping with the experiment (g _abs = −8.4 × 10^–4). By excited-state optimizations, the authors found a crossing between the first and second excited singlet electronic states, which inverted their energy order. The relaxed S₁ state of (aR)-BX16 was calculated to produce positive CPL with g _lum = +9 × 10^–5, in keeping with the experimental sign though underestimating the intensity by a factor of ∼8 (g _lum = 7.1 × 10^–4). While standard electron density plots revealed little difference for S₀–S₁ and S₁–S₀ transitions (Figure b), the electronic transition current density (ETCD) was more informative. ETCD plots help in visualizing the vector field representing the paths connecting two probability densities. ETCD plots of BX16 revealed a more curved flow on the BODIPY moiety for the S₁–S₀ transition (Figure b), which is made possible by a small deviation from planarity occurring in the S₁ state.

18.

52.

(a) Originally suggested CPL-inversion mechanism for compound BX16, involving an ICT state. Reprinted with permission from ref . Copyright 2019 The Royal Society of Chemistry. (b) Electronic density difference of S₀–S₁ and S₁–S₀ transitions (left) and ETCD vector fields (right) of BX16. Reprinted from ref under a CC-BY 4.0 license.

Other BINOL-bridged boron complexes with similar scaffolds, yet containing carbazolyl-appended Schiff base ligands, were designed by Maeda, Ema, and co-workers. Despite being nonemissive in THF solution, they displayed aggregation-induced emission and CPL in the solid state as powders. In particular, compound BX17 (Chart ) attained Φ_f = 0.22 and g _lum = 3.5 × 10^–3 at 595 nm in the solid state. Emission and CPL signals could also be observed from solution aggregates, obtained in THF/H₂O solvent mixtures with a H₂O fraction >80%. In an attempt to simulate the solid-state CPL properties of BX17, the authors ran TD-DFT calculations at the CAM-B3LYP/6-31G­(d) level using different input structures (Figure ): a single molecule of BX17, either (a) fully optimized or (b) with geometry extracted from the X-ray crystal structures or (c,d) two different first-neighboring pairs extracted from the crystal lattice. This two-body approximation has been efficiently exploited for the simulation of solid-state ECD spectra of microcrystalline samples. , Single-molecule calculations (a) and (b) led to predicted g _lum < 10^–3, while two-body calculations led to higher g _lum approaching 10^–2, in accord with the experimental behavior (Figure ).

53.

Crystal packing of compound BX17 showing pairs A and B included in CPL calculations. Bottom data legend: “Opt”, DFT-optimized monomer structure; “X-ray”, X-ray monomer geometry; and “Pair A” and “Pair B”, X-ray extracted dimer geometries. Reproduced with permission from ref . Copyright 2020, John Wiley and Sons.

A further example of the BINOL-bridged boron complex is represented by BX18 (Chart ) reported by Xue et al. and conceived as a D-A-D triad with CP-TADF emission. The compound had g _lum = 2.2 × 10^–3 in solution, reproduced by TD-B3LYP/6-31G­(d) calculations (Table ), and g _EL = 2.6 × 10^–3 embedded in a CP-OLED device.

Hall and co-workers designed a series of novel BODIPY analogs with N,N,O,C-chelated boron atoms. , The first representative of the series, BX19a (Chart ), showed absorption and emission spectra consistent with a highly fluorescent red-shifted BODIPY dye (λ_max = 593 nm, λ_em = 622 nm, and Φ_f = 0.49 in hexane) with increased charge-transfer character. However, when compared to the symmetric N,N,O,O-chelated analog BX20, the g _lum values were similar (3.7 × 10^–3 and 4.7 × 10^–3 for BX19a and BX20, respectively). TD-DFT calculations run at the PBE0/def2-TZVP level confirmed the CT character in the S₁ state of BX19a, with a consequent decrease in the electric transition dipole μ ₁₀ . At the same time, however, they also predicted a decrease in the magnetic transition dipole m ₀₁ and a less favorable angle θ _μm . The authors concluded that the emission dissymmetry of similar compounds could be tuned by proper synthetic design. Yet, the following components of the series (BX19b–e) had experimental g _lum values of between 2.1 and 2.6 × 10^–3, that is, smaller than the parent BX19a. In this case, excited-state calculations were run at the CAM-B3LYP/6-311++G­(3df,2pd) level and yielded good agreement with the experiment (Table ).

Compound BX21 (Chart ) is a dimeric analog of BX20 with a figure-of-eight shape showing significant CPL activity with g _lum = 9 × 10^–3. This compound was included in the already mentioned study by Hirose, Matsuda, and co-workers on D ₂-symmetric CPL-active species. Although the calculated g _lum = 6.9 × 10^–3 at the TD-B3LYP/6-31G­(d) level was similar to the experimental one, the calculation also predicted CT-type character for the S₁–S₀ transition, demonstrating a loss of D ₂ symmetry in the S₁ state, which unfavorably affected the transition dipole orientation. In fact, the estimated angle θ _μm for the S₁–S₀ transition was 61°, considerably deviating from 0°/±180° imposed by D ₂ symmetry.

Recently, Shimizu and co-workers reported a series of pyrrolopyrrole aza-BODIPYs with far-red emission, which include compound BX22 (Chart ). Despite the interesting photophysical properties, CPL spectra were weak and noisy; TD-DFT calculations run at the ωB97X-D/6-31G­(d) level also predicted g _lum ≈ 10^–4 associated with θ _μm close to 90°.

Ikeshita, Imai, Tsuno, and co-workers synthesized and characterized a family of diphenylboron (BX23a, BX24a, and BX25a) and 9-borafluoren-9-yl complexes (BX23b, BX24b, and BX25b) with chiral Schiff-base ligands. The recorded g _lum values were on the order of 10^–3 and of the same sign of the red-most ECD band. TD-DFT calculations were run at the MPW1PW91/6-31+G­(d,p) level and reproduced the observed sign and intensities (Table ). Notice that the calculated θ _μm values were consistently between 94 and 97°, demonstrating little influence of the aromatic rings on the CPL performance.

Two families of compounds containing multiple BF₂ units were recently reported by Ono and co-workers. The first series (B26a–c, Chart ), endowed with axial chirality, are boron 2-picolinoylpyrrole (BOPPY) analogs of BODIPY dimer BX1 (Chart ). Emission is efficiently red-shifted by the terthienyl moieties (λ_em = 514, 534, and 550 nm for B26a–c in toluene, respectively) and additionally displays solvatochromic behavior related to the ICT character of the S₁ state. CPL dissymmetry and brightness were also increased, reaching g _lum = 1 × 10^–2 and B _CPL = 126 M^–1 cm^–1 for B26c. The impact of the presence and position of terthienyl moieties was correctly reproduced by calculations (Table ). The second series are B–O–B bridged dimers of boron acyl-pyridinylhydrazine (BOAPH), whose photophysical and chiroptical properties are again modulated by the aromatic substituent (B27a–d), and, in the case of B27c, also by protonation. Reported data allow for a meaningful comparison of experimental and calculated g _lum only for B27d (Table ). All calculations in the two latter reports were run at the CAM-B3LYP/6-31G­(d,p) level.

Very recently, in collaboration with Lacour and co-workers, two of us reported a pair of axially chiral boramidines incorporating BINOL or 5,5′,6,6′,7,7′,8,8′-octahydroBINOL as the chiral units (BX28–29, Chart ). These compounds are endowed with high QYs, up to 0.95, and g _lum ≈ 10^–3 at 420 nm. Triplet states were detected using transient absorption spectroscopy and computational analysis, which also reproduced observed chiroptical properties (Table ).

Other four-coordinate boron derivatives have already been described among heterohelicenes (H46a–c and H47a–b in Chart , Section ), 1,1-binaphthyls (B13a–c in Chart , Section ), and [2.2]­paracyclophanes (C14a–e in Chart , Section ).

3.1.7. Miscellaneous SOMs

This section collects miscellaneous CPL-active SOMs, emitting from singlet excited states, which do not fit any of the preceding categories. Several SOMs with triplet emission will be described in Section , whereas Section , devoted to supramolecular systems, will also contain calculation results on monomeric systems, often as a reference.

One of the earliest QM calculations of CPL data was run in 2016 by Pecul and co-workers on thioflavin T (S1, Chart ), a fluorophore largely employed in the biophysical characterization of amyloid fibrils. Thioflavin T is achiral but acquires induced ECD (ICD) and induced CPL (ICPL) upon interaction with protein fibrils, for instance, insulin fibrils, which can be generated with either right- or left-handed supramolecular helicity. The opposite handedness is reflected in ICD and ICPL spectra of opposite signs. The induced chiroptical signals are very intense, showing g _lum values of up to 10^–2. They can arise from several effects, including intrinsic chirality (i.e., deviation from planarity) due to the interaction with the protein environment and the coupling with nearby protein groups, especially aromatic rings. The first mechanism was investigated by running torsional energy scans of both S₀ and S₁ geometries around the aryl–aryl axis (curved arrow in Chart ). The CAM-B3LYP/aug-cc-pVDZ level of theory was employed, and various environments were tested (vacuum and PCM with different dielectric constants). It was verified that some twisted structures of S1 corresponding to minima on the S₁ torsional scan could generate CPL signals with g _lum ≈ 2 × 10^–3. Another biologically relevant CPL-active compound is the urobilinoid l-stercolibin (S2). In a wider study on the conformational and chiroptical properties of urobilinoids, Longhi and co-workers reported the ECD and CPL spectra of S2 which showed g _abs = +8.8 × 10^–4 and g _lum = +3.5 × 10^–3. Excited-state calculations were run at the CAM-B3LYP/6-31G­(d,p) level starting from the most representative cluster obtained after MD simulations in explicit solvent and decently reproduced the experimental results.

19.

The benzo­[b]­silole-fused 9,9′-spirobi­(fluorene) S3 (Chart ) reported by Nakano and co-workers showed a broad CPL band above 340 nm with the same sign as the red-most ECD band and g _lum = 7.6 × 10^–4. TD-DFT calculations run at the MN15/6-31G­(d) level demonstrated that emission occurred from one-half of the molecule, which was geometrically relaxed after the emission, while the other half retained the geometry of the Franck–Condon state. Similar to other cases discussed above, structural relaxation following excitation gives rise to a nonsymmetrical excited state. The calculated g _lum = 4.1 × 10^–4 was in sufficient agreement with the experiment. The relatively small dissymmetry value is justified by a quasi-orthogonal arrangement of transition moments for the S₁–S₀ transition (θ _μm = 87°).

CPL calculations have been reported for a few CPL-active natural products. An original application of CPL spectroscopy was proposed by Hu, Lin, and co-workers as a means for stereochemical assignment of the two epimeric talarolactones A and C (S4 and S5, Chart ). The two compounds were isolated from the fungus Ophiosimulans tanaceti which produced, during the isolation process, a fraction displaying strong blue luminescence under UV irradiation. This fraction contained a mixture of S4 and S5, both as racemic compounds. After chiral resolution, moderately intense ECD spectra were recorded for the enantiomeric pairs, which were well reproduced by TD-DFT calculations run at the B3LYP/6-311G+(d,p) level, allowing for the assignment of the absolute configuration. The compounds also exhibited CPL spectra with g _lum ≈ 5 × 10^–4. TD-DFT calculations correctly reproduced the sign of CPL spectra; calculated g _lum values are not given. The authors suggested that CPL calculations might “be a promising unique method for the stereochemical assignment as a supplementary of ECD calculations”. To substantiate their hypothesis, they run CPL calculations on diketone K1 (Chart ), already encountered in our survey, and on spiro compounds S6a–b, whose experimental CPL data had been reported previously. The authors’ conclusion was that their “findings demonstrated that the CPL-calculation method is straightforward and reliable, and it can be applied for the determination of absolute configuration of chiral carbons”. Indeed, the use of multiple chiroptical techniques, such as ECD and VCD, has often been recommended for a safer configurational assignment of natural products. , However, CPL is not expected to offer any additional advantage with respect to ECD, in consideration of their strict relationship and experimental and computational complications that affect CPL.

Rather than as a means for assigning absolute configurations, CPL studies of luminescent natural products are helpful in completing their characterization and exploring excited-state properties. A beautiful example in that this sense is represented by the work by Longhi and co-workers on lycorine (S7, Chart ), narciclasine (S8a), and two of its derivatives (S8b–c). Lycorine (S7) displays consistent ECD (in the low-energy region) and CPL spectra with the same sign, similar g factors <10^–3, and small Stokes shift (ΔE_S = 0.46 eV). Narciclasine (S8a) shows dual emission in EtOH, with maxima at λ_em = 340 and 495 nm; this latter band corresponds to ΔE_S = 1.2 eV and to a CPL band with the same sign as the red-most ECD band at 335 nm. Of the two semisynthetic derivatives S8b–c, the peracetylated one S8c emitted at λ_em = 368 nm, while the triacetylated one S8b featured red-shifted emission at λ_em = 385 nm. At this point, the role of phenolic OH was clear and suggested the occurrence of an ESIPT mechanism. This phenomenon consists of an intramolecular acid/base reaction (intramolecular proton transfer) which occurs only in the excited state but is absent in the ground state, and it is made possible by a change in acidity and/or basicity of the proton donor/acceptor groups. Typically, a phenolic proton is transferred to an adjacent carbonyl group, and the phenol tautomerizes to a quinoid structure (Figure a). The importance of this phenomenon is related to its occurrence in several biological systems and to its many possible applications, especially in biosensing. , The ESIPT behavior of narciclasine was confirmed first experimentally by varying the solvent, pH, and temperature and then computationally by TD-DFT calculations at the M06-2X/TZVP level. In view of the molecular flexibility of compounds S8a–c, a minimal model was first considered in the calculations consisting of the chromophoric moiety responsible for the S₀–S₁ transition (Figure b). The S₀ and S₁ PESs along the proton transfer reaction path were examined for the model fragment, also in the presence of an explicit solvent molecule (MeOH). The two PES get closer to each other along the path, in accordance with the red-shifted emission of the quinoid tautomer. Then, excited-state calculations were run for the most populated conformer of S8a in both phenolic and quinoid forms; the latter reproduced well the observed Stokes shift as well as the CPL sign (Figure c).

54.

(a) ESIPT in narciclasine (S8, Chart ). (b) S₀ and S₁ PES along the proton transfer reaction path of the narciclasine chromophore. (c) Experimental and calculated ECD and CPL spectra of narciclasine. “N” and “T” indicate the normal and tautomeric (after proton transfer) forms, respectively. Adapted from ref under a CC-BY-NC 3.0 unported license.

Three further natural products undergoing ESIPT were described by the same group, namely, mellein (S9a, Chart ) and cis- and trans-4-hydroxymellein (S9b and S9c). The main difference between mellein derivatives S9a–c and narciclasine (S8a) was the absence, for the former, of dual emission; a single red-shifted emission band is encountered for S9a–c with ΔE_S ≈ 1.3 eV, almost independent of solvent. CPL spectra for S9a–c were rather weak with g _lum < 2.5 × 10^–4, but they increased by up to a factor of 1.7 upon deprotonation. An interesting feature of CPL spectra is that the observed signs for S9a and S9c in the neutral form were opposite to the lowest-energy ECD band. DFT and TD-DFT calculations were run at the M06-2X/TZVP level, and for ground-state optimizations, other functionals were explored too. The calculations evidenced the impact of conformational flexibility on the chiroptical properties. For instance, the two possible conformers of neutral mellein (S9a) with equatorial and axial methyl groups led to opposite calculated ECD and CPL spectra, and their calculated relative populations depended critically on the functional employed. The absence of a dual emission was rationalized by evaluating the S₁ PES along the proton transfer reaction path with explicit solvent. Contrary to narciclasine (S8a), the vertical excitation from the S₀ phenolic structure of mellein reaches a steep region on the S₁ PES, which would easily evolve toward the S₁ quinoid structure.

Bedi, Gidron and co-workers have reported three series of tethered twistacenes with multiple stereogenic elements such as helical and axial chirality (S10a–c, S11a–c, S12a–d, and S13a–c; Chart ). They started their investigation by calculating the effects of conformational changes (backbone twist and interaryl torsional angles) on the chiroptical properties at the TD-CAM-B3LYP/6-31G­(d) level. The emissive state is in all cases the ¹L_a excited state of the anthracene chromophore. Because of the rigidity of the backbone, the trends observed for S₀–S₁ and S₁–S₀ transitions were similar. Compounds S12a–d featuring both helical and axial chirality showed the largest g _abs (3–5 × 10^–3) and g _lum values (1.8–2.5 × 10^–3). Both dissymmetry values were found to vary regularly with the backbone twist angles; moreover, the g _lum/g _abs ratio was consistently ∼0.52 for the whole series.

Wu, Chang, and co-workers designed two pairs of D–A dyads S14–S15 (Chart ) which manifest TADF and CPL despite being relatively rigid, at odds with usual TADF emitters. The chiral donor unit is in both cases 12b-methyl-5,12b-dihydroindeno­[1,2,3-kl]­acridine, while the achiral acceptor unit is either 2,4,6-triphenyl-1,3,5-triazine (S14) or N-((p-tert-butyl)­phenyl) naphthalimide (S15). Absorption spectra showed a weak ICT band in the low-energy region, whereas emission spectra displayed a broad band corresponding to blue luminescence for S14 and orange luminescence for S15 with high QYs. ECD spectra in the short-wavelength region were representative of the chiral donor moiety but also displayed weak-to-moderate signals in the ICT region. From CPL spectra, dissymmetry values g _lum = 5.9 × 10^–4 for S14 and 2.0 × 10^–3 for S15 were measured. For both compounds, the CPL sign was opposite to the red-most ECD band, a circumstance which was not noticed by the authors. TD-DFT calculations run at the PBE0/def2-SVP level confirmed the ICT character of the S₁ state and reproduced the experimental CPL sign, although they strongly overestimated |g _lum| values. Interestingly enough, the two dyes were incorporated into CP-OLED devices with efficiencies of 20.3% for S14 and 23.7% for S15.

Another TADF emitter was described by Hirata and co-workers, composed of a triphenyl amine as the donor and naphthacen-5­(12H)-one as the acceptor (S16, Chart ). TD-DFT calculations run at the B3LYP/6-31G­(d) level highlighted the expected CT character for the S₁ state and π–π* character for the T₁ state, localized on the acceptor moiety. The different characters accounted for a small energy separation between S₁ and T₁, which allows for efficient RISC and hence the TADF property (Section ). The broad fluorescence band had its maximum at 513 nm and was paralleled by CPL with g _lum = 1.1 × 10^–3. The authors noticed that the major red-shifted ECD band had the opposite sign to the CPL band and attributed this finding to a different conformation in the S₀ and S₁ states. A minor lowest-energy ECD band, with the same sign as CPL, was instead assigned to a less populated ground-state conformer.

Tasaku, Takikawa, and co-workers recently realized a helicene-fluorescein hybrid functioning as an acid–base-triggered chiroptical switch between a neutral spiro form and an anionic open form (S17a and S17b, Chart ). Thanks to an extended conjugation, the basic form has a broad red-shifted emission between 550 and 850 nm, whereas the neutral form has blue emission with λ_em = 330 nm in DCM. The red shift is accompanied by a CPL sign inversion and a 3-fold increase in |g _lum| from 1.5 × 10^–3 to 4.4 × 10^–3. The sign inversion was correctly reproduced by TD-DFT calculations run at the CAM-B3LYP-D3BJ/def2-SVP level, including a LR-PCM solvent model. On the contrary, the predicted |g _lum| values were both ∼2 × 10^–3.

Compounds S18 and S19 are included in the set chosen by Chen et al. for their investigation of the impact of transition density on CPL, which we already mentioned several times in our survey. Especially interesting is the case of S19, where the configurationally stable spiro center combines with two other stereolabile elements, namely, the axial chirality around the N–phenyl bond and the chirality of the sulfone-containing cycle. Thus, four diastereomers are possible, separated by relatively small activation barriers, the two most stable of them interconverting through a rocking vibration of the SO₂ group. CPL calculations with the nuclear ensemble approach at the CAM-B3LYP/def2-SVP level reproduced the observed g _lum very well.

Wang et al. proposed a theoretical design of a series of chiral ansa compounds composed of a DPP unit and an oligo-methylene belt (S20, Chart ). Based on B3LYP/6-311+G­(d,p) calculations, it was predicted that the chain length would modulate the CPL dissymmetry around 400 nm, for which values of up to ∼10^–1 were calculated. However, the red-most absorption of the DPP core occurs around 600 nm; therefore, the theoretical analysis appears to be seriously flawed.

Though not strictly related to the present review, we conclude this section by mentioning magnetic circularly polarized luminescence (MCPL). This technique is applicable, in principle, to both achiral and chiral emissive substrates and measures CP light emitted under a magnetic field collinear with the light propagation vector. It has been applied to several organic compounds, especially porphyrinoids, condensed aromatics, and helicenes, as a source of photophysical information. , Abbate and Longhi’s group ran the only reported TD-DFT calculation of MCPL of a Si triarylcorrolate (S21, Chart ) in combination with magnetic circular dichroism (MCD) calculations. We recall anyway that MCPL is entirely unrelated to CPL, as they have different origins and are manifestations of completely different phenomena.

3.1.8. Organic Triplet Emitters

Although most of the species emitting CP light from states other than singlets are metal and lanthanide complexes to be discussed in Section , CPL calculations for a few purely organic triplet emitters (organic phosphors) have also been reported. For organic closed-shell compounds, emission from the lowest-energy triplet state T₁ involves two spin-forbidden phenomena: first, intersystem crossing (ISC) from S₁ to T₁ and then T₁ to S₀ emission (phosphorescence). Because of its spin-forbidden character, phosphorescence is usually associated with small efficiencies (low phosphorescence quantum yields, Φ_p) and long lifetimes (τ_p). Spin-forbidden transitions are made possible through SOC. In particular, the transition rate between any two singlet and triple states S _n and T _n depends on SOC matrix elements of the type ⟨Ψ_Tn|H _SO|Ψ_Sn⟩, where H _SO is the spin–orbit Hamiltonian. This latter quantity scales with the fourth power of atomic number Z ⁴, hence efficient SOC is often found for compounds containing heavy elements such as metals. Moreover, it involves a combination of ΔS and ΔL; that is, a change in spin momentum must be accompanied by a change in the orbital angular momentum. As a result, efficient purely organic phosphors often contain elements such as Br, I, and Se and involve transitions with a change in the orbital type and symmetry (El Sayed’s rule), for example n−π* transitions in carbonyl derivatives.

In 2015, Pecul, Coriani, and co-workers developed the first computational protocol of circularly polarized phosphorescence (CPP) based on TD-DFT within a nonrelativistic framework. In this framework, the expression of transition dipole moments for spin-forbidden transitions becomes nonzero if spin–orbit perturbed wave functions for the ground state (S₀) and excited state (T₁) are introduced. Alternatively, transition moments can be obtained directly from the residue of the linear response function when relativistic wave functions are used. Since the spin–orbit operator is a two-electron operator, to reduce computational cost the authors also introduced approximate one-electron operators. Numerical calculations were run on a series of α,β- and β,γ-unsaturated ketones and 1,2-diketones, namely, K1, K3, K5–K7, K16, and K17 (Chart , Section ). The CAM-B3LYP functional was employed in combination with correlation-consistent basis set aug-cc-pVDZ of double-ζ quality. This latter was selected after a broad screening of basis sets of double-, triple-, and quadruple-ζ quality also with the addition of diffuse functions, run on H₂O₂. For nonconjugated ketones such as K3 and K5–K7, in the T₁ state the carbonyl group loses planarity, on a par with the S₁ state (discussion in Section ), and two different local minima were found, often associated with rotational strengths opposite in sign for the T₁–S₀ transition (relevant for CPP). For conjugated ketones such as K1, K16, and K17, on the contrary, T₁ geometry resembles S₀ geometry. No comparison with the experiment was provided.

A few years later, Badala Viswanatha and co-workers re-evaluated the same set of ketone derivatives using CC2 calculations with an RI approximation for the simulation of CPL (Section ) and CPP. In general, CC2 results for CPP agreed qualitatively with TD-DFT results by Pecul, Coriani, and co-workers, with some discrepancies possibly due to an energy reversal of triplet excited states between DFT and CC2. In all cases, calculated rotational strengths for T₁–S₀ transitions were at least 4 orders of magnitude smaller than S₁–S₀ transitions. The authors also considered N,N-dimethyl-1,1′-binaphthyldiamine (B2, Chart ), a metal-free compound for which RT-CPP had been measured on films obtained from a mixture with β-estradiol. By calculations run at the CC2/aug-cc-pVDZ level, two minima were located on each of the S₀, S₁, and T₁ surfaces. The calculated rotational strengths for the S₀–S₁, S₁–S₀, and T₁–S₀ transitions agree with each other, in accord with the experimental findings. We notice that the estimated phosphorescence dissymmetry values g _lum,p for the two T₁ minima are 2–6 × 10^–3, which overestimated experimental values by one order of magnitude. ,

In 2018, Bloino and co-workers calculated vibrationally resolved CPL and CPP spectra of camphorquinone (K1), using a perturbative approach based on a two-component Hamiltonian to evaluate SOC. Since the paper focuses on Ir^III complexes, it will be discussed in more detail in Section .

As a continuation of their studies on CPL-active helicenes (Section ), Autschbach, Favereau, Crassous, and co-workers investigated low-temperature phosphorescence and CPP properties of a family of 2,15-bis-ethynyl-carbo[6]­helicenes H8b and H9a–e (Chart , Section ) endowed with different electron-accepting or -donating moieties. Thanks to their helical topology, helicene derivatives may feature relatively strong SOC. The emission spectra of H8b and H9a–e recorded at 77 K in a frozen 2-methyltetrahydrofuran (2-MeTHF) medium featured two vibronically resolved bands (Figure ): one between 400 and 530 nm due to fluorescence (S₁–S₀ transition) and one between 530 and 700 nm due to phosphorescence (T₁–S₀ transition). The latter component was isolated by means of time-gated emission measurements. In fact, while the lifetime of the fluorescence signals was between ∼1 and ∼20 ns, that of phosphorescence signals was between ∼0.5 and ∼1 s. The compound for which the phosphorescence component was higher is the nitro-substituted H9c because of the spin–orbit enhancement provided by the nitro group. Low-temperature CPL spectra were also recorded, and like emission spectra, they contained contributions from both S₁–S₀ and T₁–S₀ emission phenomena (Figure ). These helicene derivatives offer an almost unique example where single-molecule CPL and CPP signals can be measured under similar conditions and directly compared. The relative intensity of the two signals varied from substrate to substrate; for the nitro-substituted H9c, the highest phosphorescence g _lum,p = 1.6 × 10^–2 was observed, twice larger than the luminescence g _lum = 8 × 10^–3. Surprisingly enough, CPP signals are inverted in sign with respect to their low-energy CPL counterpart (Figure ). To rationalize the observed behavior, vibronically resolved CPL and CPP spectra, including FC and HT effects, were computed using the CAM-B3LYP functional and Slater-type DZP (DZ on hydrogens) basis set. The spin–orbit interaction was treated by employing the zeroth-order regular approximation (ZORA) two-component relativistic Hamiltonian. All calculated g _lum values, for both fluorescence and phosphoresces processes, were >10^–2, overestimating the experimental quantities in some cases. However, the change in sign between S₁–S₀ and T₁–S₀ emission processes was correctly reproduced. Both processes involve the HOMO–LUMO excitation as the major contribution; however, the orientation of electric and magnetic transition dipoles is different, causing the sign inversion (Figure ).

55.

(a) Experimental emission spectra of carbo[6]­helicene H9c (Chart ), both as full emission (solid line, includes fluorescence and phosphorescence) and time-gated (dotted line, phosphorescence). (b) Experimental full emission (top panel) and CPL spectra (bottom panel) of carbo[6]­helicenes H9a–c; (P)-enantiomers, solid curves; (M)-enantiomers, dashed curves. (c) Experimental phosphorescence spectrum of H9a compared with the vibrationally resolved calculated spectrum. (d) EDTM and MDTM calculated for the S₁–S₀ and T₁–S₀ transitions of H9c. Adapted with permission from ref . Copyright 2024 American Chemical Society.

Still within the helicene family, Srebro-Hooper, Crassous, and co-workers recently analyzed three bromo-substituted helicenes, namely, 2,16-dibromo-[6]­helicene, 9-bromo-[7]­helicene, and 8-bromo-[7]­helicene-bis-thiadiazole (H92–H94, Chart ), exploiting the heavy atom effect. In fact, all compounds display vibrationally structured phosphorescence at 77 K in 2-MeTHF, with maxima above 530 nm. CPP was also measured, and it was both inverted in sign and boosted in intensity (g _lum,p > 10^–2) with respect to RT CPL. Triplet states were theoretically investigated by means of TD-DFT and TDA calculations at the PBE0/SVP level, including ZORA-SOC. The agreement between experimental and calculated g _lum,p was satisfactory except for H92, a discrepancy that the authors attributed to the neglect of vibronic contributions. The sign reversal between CPL and CPP was rationalized as due to a change in the character of the underlying transition (S₁–S₀ vs T₁–S₀), as evidenced by MO and density difference plots.

20.

An almost unique example of room-temperature phosphorescence emitters is represented by the pair of compounds S22a,b (Chart ) developed by Qiu, Zhao, Tang, and co-workers. They arise from the derivatization of benzo­[c]­[1,2,5]­thiadiazole dye with a BINOL unit, yielding a D–A dyad endowed with rigidity and, in the case of S22b, heavy elements providing large SOC. The molecular design (Figure a), starting from 5,6-difluoro benzo­[c]­[1,2,5]­thiadiazole (2FBT), was aimed not only at introducing chirality and rigidity but also at creating the premises for a more favorable ISC between the S₁ and T₁ states by the introduction of a low-energy CT state in combination with the heavy-atom effect (Figure b). In fact, the CT-like S₁ state has n−π* character which, according to El Sayed’s rule, favors ISC to the π–π* T₁ state. Solvent-dependent dual emission was observed for S22b at RT, consisting of peaks with maxima at 450 and 640 nm; the latter coincided with the phosphorescence emission in the solid state. In parallel, CPL showed two peaks with opposite signs in the UV and NIR ranges. The NIR CPP band had g _lum,p = 7 × 10^–3, 2–3 times larger than the UV CPL band. TD-DFT calculations run at the B3LYP/def2-SVP level reproduced the sign and relative intensity of the CPL and CPP bands, although the g _lum values were overestimated. SOC terms were computed by relativistic calculations (Section ) and confirmed the existence of a much larger SOC for S22b than for S22a (Figure c). Quite interestingly, by exploiting the sensitivity to O₂ of the phosphorescence emission, S22b was employed for the detection of tumor-related hypoxia in vitro and in vivo by means of confocal microscopy and TEM.

56.

(a) Design strategy of CPP emitter coupling 5,6-difluoro benzo­[c]­[1,2,5]­thiadiazole (2FBT) dye with a BINOL unit. (b) Proposed CPP mechanism: 2FBT exhibits poor ISC and weak phosphorescence; BBT (S22a, Chart ) shows CPL from S₁ state and moderately strong CPP from the T₁ state; and BBTI (S22b) shows strong CPP from the T₁ state after favored ISC due to CT and heavy-atom effects. (c) Energy levels, FMOs, and SOC constants for 2FBT, S22a, and S22b. Reproduced with permission from ref . Copyright 2024 Springer Nature.

3.2. Larger Organic Molecules and Assemblies

In this section, we collect different families of organic compounds and systems, which share the property of not being single small organic compounds. The first three subsections (Sections –) deal with single organic molecules of increasing size and structural complexity; the largest molecules we treat have MW > 2000 uma. In the last subsection (Section ), we describe supramolecular entities ranging from 1:1 host/guest complexes to “infinite” self-assembled systems. The second most common property is that some of the compounds or systems discussed in this section feature the largest observed g _lum values observed for metal-free species, even above 0.1. Despite the challenges posed by the molecular size and structural complexity, we will see that in many cases QM calculations, most often TD-DFT, are able to reproduce with great accuracy CPL data of large organic molecules. In contrast, supramolecular assemblies present a different scenario, where the use of CPL simulations remains underutilized.

3.2.1. π-Conjugated Oligomers

The broad family of π-conjugated polymers and oligomers lies at the core of the well-consolidated yet still developing field of plastic (organic) electronics. In particular, chiral π-conjugated polymers and oligomers have opened the way to novel applications and to the solution of long-standing problems. , Since nearly all optoelectronic devices operate in the solid state, with the active layer in the form of a thin film, relevant chiroptical properties need to be measured on thin films. Obviously, this aspect complicates the theoretical analysis because the simulations run on single molecules are not representative of aggregated states. This premise justifies the relatively narrow number of π-conjugated systems which have been considered for CPL calculations. This section is dedicated to single molecules, which in most cases contain oligo­(phenylene ethynylene) (OPE) moieties; aggregated systems will instead be covered in Section .

Longhi, Miguel, and co-workers reported the first CPL calculations on a small OPE, a twisted stapled OPE with a 1,2-diol moiety (O1, Chart ). The experimental (chiro)­optical properties of O1 and several of its bis-esters were all very similar. Notably, g _abs values for the red-most ECD band at around 370 nm were ∼10^–2. CPL spectra displayed wavelength-dependent g _lum values, reaching 1.0–1.2 × 10^–2 between 390 and 400 nm. Different combinations of functionals (B3LYP and CAM-B3LYP) and basis sets yielded variable differences between S₀ and S₁ geometries. Regrettably, calculated g _lum values were not provided. Later on, the same groups reported the isomers of a double stapled OPE (O2) with various double bond geometries. The two isomers (E,Z)-O2 and (E,E)-O2 had similar g _abs values (1.7 × 10^–2 and 1.5 × 10^–2 at 370 nm, respectively). Also similar were CPL spectra with broad bands and remarkable g _lum = 4.2 × 10^–2 and 3.4 × 10^–2 at 400 nm, respectively; the former value increased to 5.5 × 10^–2 in hexane, among the highest reports for nonaggregated organic molecules. Interestingly enough, these compounds provide rare examples of the g _lum/g _abs ratio above unity for nonexcimeric species, namely, 2.5 for (E,Z)-O2 and 2.3 for (E,E)-O2. TD-DFT calculations run at the M06-2X/def2-SVP level predicted a sizable rearrangement from S₀ to S₁ geometry, especially for (E,E)-O2, favored by the flexible OPE skeleton, which could justify the observed increase in dissymmetry. Calculated g _lum/g _abs ratios were 0.98 for (E,Z)-O2 and 1.2 for (E,E)-O2; with respect to the experiment, g _abs values were overestimated and g _lum values were slightly underestimated.

21.

Starting in 2017, Morisaki and co-workers have reported a large number of experimental and computational CPL data on compounds based on di- or tetra-substitution of the [2.2]­paracyclophane framework with OPE moieties. By varying the substitution pattern of the [2.2]­paracyclophane core and the length of OPE fragments, a wide variety of structures were produced which can be categorized as X-, † (cross)-, ‡ (double cross)-, V60-, and V120-shaped according to the relative directions and angles assumed by OPE units (Chart ). ,− These compounds give rise to nonexcimeric emission with high quantum efficiencies, some of them approaching Φ_f ≈ 1 (O5, O6, and O8). Experimental and calculated g _lum values for all compounds are collected in Table . They exhibit g _lum on the order of 10^–3; the largest values were recorded for V60-shaped molecules, in particular, O4a, O4c, and O4e. In all cases, the CPL spectrum was consistent in sign with the red-most ECD band. TD-DFT calculations were run using various functionals, such as ωB97X-D, B3LYP, CAM-B3LYP, and especially MN15. The basis set was, in most cases, double-ζ basis set 6-31G­(d) with one set of polarization functions on non-hydrogen atoms. This choice is justifiable by the very large molecular size of some systems. In all cases, the calculations correctly reproduced the observed CPL sign and, in general, the experimental |g _lum| values well enough. The choice functional MN15 seems to perform equally as well as B3LYP and CAM-B3LYP. Interestingly enough, B3LYP was used in combination with TDA, still yielding very accurate results. We notice that for each compound a single functional was employed (except for O4b) and no benchmark was run. The calculations highlighted that the most important limitation to higher dissymmetry values in Morisaki’s compounds is the orientation between electric and magnetic transition moments. In many cases, θ _μm lies between 75 and 100°; the only exceptions are the symmetric X-shaped compounds O3b and O10, where larger deviations from 90° are found (Table ). It must be stressed that, at least in principle, these two molecules are D ₂-symmetric. However, MO plots clearly demonstrate that desymmetrization occurs in the S₁ state (Figure ), with the HOMO becoming localized on one OPE moiety alone; the same phenomenon occurs also for C ₂-symmetric systems with V-type geometry. ,

7. Collection of Experimental and Computational Data for Cyclophane-OPE Conjugates .

	Exper.		Calculation				Number or abbreviation
Type	λem	g lum	g lum	θ μm	Method	Basis set	This review	Original publication	ref.
Cyclic	400	11	12	105.0	MN15	6-31G(d)	O1	4
X	427	1.7	9.1	144	ωB97X-D	6-31G(d,p)	O3b	16
X	430	1.5	2.1	68.3	MN15	6-31G(d)	O3c	X
V60	387	3.6	3.3	77.8	MN15	6-31G(d)	O4a	3
V60	401	1.3	0.3	87	ωB97X-D	6-31G(d,p)	O4b	18
V60	401	1.0	2.2	91.1	MN15	6-31G(d)	O4b	8
V60	377	2.4	2.8	78.9	MN15	6-31G(d)	O4c	2
V60	411	1.0	2.3	78.6	MN15	6-31G(d)	O4d	3
V60	377	2.5	2.8	79.5	MN15	6-31G(d)	O4e	V(60)
†	423	0.85	0.22	92.8	MN15	6-31G(d)	O5	8
‡	431	0.5	0.2	91.6	MN15	6-31G(d)	O6	10
V120	387	1.3	2.9	75.7	MN15	6-31G(d)	O7a	5
V120	400	1.2	1.7	99.0	MN15	6-31G(d)	O7b	7
V120	378	1.7	2.6	78.3	MN15	6-31G(d)	O7c	V(120)
V60	426	1.1	1.1	94.4	B3LYP (TDA)	6-31G(d)	O8	5-V
V60	425	0.9	0.5	84.6	B3LYP (TDA)	6-31G(d)	O9	5-X1
X	425	1.2	1.1	57.9	B3LYP (TDA)	6-31G(d)	O10	5-X2
V120	400	1.3	1.5	81.4	CAM-B3LYP	6-31G(d)	O11	D1
Cyclic	460	13	203	0	MN15	6-31G(d)	O12
Cyclic	445	23	20	109.9	MN15	6-31G(d)	O13	3
Cyclic	434	22	69	5.1	MN15	6-31G(d)	O14a	3
Cyclic	469	23	70	0.6	MN15	6-31G(d)	O14b	5
Oligo	391	ND	0.31	91.2	MN15	6-31G(d)	O15	2
Oligo	401	2.5	5.1	101.6	MN15	6-31G(d)	O16	3
Oligo	401	1.8	2.7	95.6	MN15	6-31G(d)	O17	4
Oligo	404	0.38	0.5	88.8	MN15	6-31G(d)	O18	5
Oligo	402	2.8	5.4	80.1	MN15	6-31G(d)	O19	6

^aData for electronic calculations only (no vibronic calculations). See Charts - for structure numbering.

^bX: X-shaped (Chart ); V60: V-shaped with 60° geometry; V120: V-shaped with 120° geometry; †: cross-shaped; ‡: double cross-shaped.

^c g _abs and g _lum multiplied by 10³.

^dExperimental data from ref .

^eExperimental data from ref .

57.

FMOs of X-shaped compound O10 (Chart ) in the S₀ state (left, gray frame) and S₁ state (right, purple frame). Adapted with permission from ref . Copyright 2022, Oxford University Press.

The research on CPL-active [2.2]­paracyclophane-OPE compounds made a significant breakthrough when cyclic derivatives were considered. The first example provided by compound O12 (Chart ) already attained g _lum = 1.3 × 10^–2, 1 order of magnitude larger than for its analog O3a. The same trend was replicated by compounds O13 and O14a–b, where two phenyl rings attached to the opposite sides of [2.2]­paracyclophane were bridged by an ethynyl moiety; all of them had g _lum = 2.2–2.3 × 10^–2. , Moreover, the extended compounds O14a–b display outstanding B _CPL = 356 and 386 M^–1 cm^–1, respectively. TD-DFT calculations, run at the level of choice (MN15/6-31G­(d)), reproduced the CPL sign though overestimating in most cases the g _lum intensity (Table ). More interestingly, the calculations showed that the ethynyl bridge in O13 is able to suppress too wide a relaxation in the S₁ state: the geometry and MOs remain substantially identical to those of S₀, thus preventing the desymmetrization seen above for nonbridged derivatives (Figure ). In the same paper, the authors reevaluated compound O1 reported by Longhi, Miguel, and co-workers (Chart ) as an analog of O13 devoid of the [2.2]­paracyclophane moiety and with 2-fold-smaller g _lum. The significant structural difference between O1 and O13 was recognized in the overlap between the stacked benzene rings. In O13, the [2.2]­paracyclophane scaffold imposes the typical plane-to-plane distance of 2.8 Å, whereas in O1 the two rings lie at 4 Å. Using an acyclic model of O1, where the staple between oxygen atoms was replaced by two methyl groups, the authors ran two series of geometry scans by varying selected angles and recovered high g _lum values when the overlap between benzene rings was maximized. Compounds O14a–b also exhibit symmetrical MOs in the S₁ state and θ _μm angles close to 0° (Table ).

22.

58.

FMOs of compound O13 (Chart ) in the S₀ state (left, gray frame) and S₁ state (right, purple frame). Adapted with permission from ref . Copyright 2022, Oxford University Press.

A series of cyclic oligomeric [2.2]­paracyclophane-OPE compounds was also reported by Morisaki and co-workers (O15–O17, Chart ). Dimer O15 was CPL-silent, while trimer O16 and tetramer O17 attained g _lum > 10^–3 (Table ). TD-MN15/6-31G­(d) calculations revealed that, as for other cases seen above, the relaxed S₁ state was not symmetric and its FMOs were localized on a single OPE moiety. A similar situation occurred for two other cyclic tetramers O18 and O19 (Chart and Table ). Notice that because of the structural analogy with other [2.2]­paracyclophane-OPE derivatives, we preferred to treat cyclic compounds O12–O17 in this section rather than with macrocyclic systems (Section ) to which they also belong.

As a final example of oligo [2.2]­paracyclophane-OPE derivatives, special interest must be paid to compounds O20 and O21 (Chart ). Their experimental ECD spectra, reported by Morisaki and co-workers, showed clear exciton coupling signatures in the low-energy region. CPL spectra were consistent with the red-most ECD band and thus originated from the lowest-energy excitonic level (discussion in Section and Figure ). O20 and O21 have similar g _abs ≈ 3 × 10^–3 and consistent g _lum = 1.8 and 2.6 × 10^–3, respectively. Ehara, Mennucci, and co-workers simulated the (chiro)­optical spectra of O20 and O21 using a fragmentation approach, namely, the Frenkel exciton decomposition analysis (FEDA). TD-DFT was employed to calculate the site energies and transition densities of chromophore units and their exciton coupling potential, and then a matrix method was employed to combine transition moments and energies within the Frenkel exciton model (Section ). The advantage of this method lies not only in the reduced computational time with respect to QM calculations on full systems but also in the physical insight one may gain by looking at the different combinations of transition moments. In the current case, the authors further simplified their systems by replacing n-dodecyl chains with hydrogen atoms and the [2.2]­paracyclophane units in O20 and O21 with pairs of stacked benzene rings. For both compounds, fragment calculations of ECD and CPL spectra were in almost perfect agreement with full calculations run at the CAM-B3LYP/6-31+G­(d) level, including PCM for CHCl₃, and both methods reproduced experimental spectra. Focusing on “trimer” O21, the decomposition analysis of calculated ECD spectra (Figure ) highlighted that the couplet centered around 350 nm is dominated by the coupling interactions between adjacent OPE units (i–j and j–k), whereas the interaction between the distant units (i–k, colored in Chart ) is small. This is true for both electric–electric coupling (μ _i μ _j , etc.) and electric–magnetic coupling (μ _i m _j , etc.), with the first terms being dominant over the second. The rotational strengths due to the intrinsic or intraunit contributions (R _i , etc.) are small. Similar reasoning holds for the calculated CPL band, which originates from an emissive state localized on the central unit j, after a geometry relaxation of the S₁ state (Figure ). This example highlights the utility of Frenkel-type fragmentation approaches not only for larger aggregate systems (to be discussed in Section ) but also for relatively small multichromophoric compounds.

59.

(a,b) Experimental and calculated ECD and CPL spectra of compound O21 (Chart ). Legend: “full”, TD-DFT calculations on the whole molecule; “Exc”, FEDA calculations; “μμ”, electric–electric coupling; “μm”, electric–magnetic coupling; i, j, k, molecular fragments. (c) FMOs of O21 at the S₁ geometry. Adapted with permission from ref . Copyright 2018, John Wiley and Sons.

3.2.2. Macrocycles

The heterogeneous category of macrocyclic compounds is of particular interest for our purposes for a twofold reason. First, some representatives of this family have shown the largest reported emission dissymmetries for metal-free, nonaggregated species. − Second, the chemical synthesis of large macrocycles is especially demanding and costly, and a tailored design of target molecules is compulsory.

The cylindrically shaped macrocycle MC1a (Chart ) reported by Sato, Isobe, and co-workers in 2017 has held for several years the record for the largest measured emission dissymmetry for an organic nonaggregated species with its g _lum = 0.15 at 443 nm, accompanied by high Φ_f = 0.80. With an estimated ε_max ≈ 1.8 × 10⁵ at 365 nm, these figures lead to a record brightness B _CPL ≈ 10⁴ M^–1 cm^–1. The companion compound MC2 has g _lum = 0.10 at 445 nm and Φ_f = 0.74. While CPL calculations were not reported, TD-DFT calculations were run at the B3LYP/6-31G­(d,p) level to simulate ECD spectra. The analysis of the S₀–S₁ transition of MC1a, for which the very high experimental g _abs = 0.17 was measured, revealed a major contribution from the HOMO–LUMO excitation, delocalized over the whole cylindrical π-conjugated system. The giant g _abs value (and the related g _lum) is due to a large m ₁₀ oriented along the cylinder axis and collinear with μ ₀₁ (θ _μm = 180°). The HOMO–LUMO transition in MC1a belongs to irrep A₂ of the D ₄ point group, which is both electrically and magnetically allowed along the z axis. In the nonsymmetrical analog MC2, |m ₁₀ | has similar intensity as MC1a, but the transition dipole orientation is less favorable (θ _μm = 152°). The same year, Du and co-workers reported another series of cylindrical macrocycles based on the oligo­(anthracenes) scaffold. One representative of the series (MC3) reached g _lum = 0.10 at 535 nm.

23.

A subsequent theoretical analysis run by Zhang, Long, and co-workers on MC1b (also indicated as [4]­CC_2,8) and MC3 confirmed for the S₁–S₀ transition a collinear orientation between transition moments and the occurrence of an extraordinarily strong magnetic transition dipole m ₀₁ due to a favorable combination of all atom-centered contributions to MDTM. For compound MC1b, seven different functionals were benchmarked (B3LYP, PBE0, PBE, BMK, M06-2X, CAM-B3LYP, and ωB97X-D), with or without D3BJ dispersion correction. Using the 6-31G­(d) basis set in vacuo, the calculated g _lum values ranged from 0.38 to 0.47, thus overestimating the experimental g _lum = 0.15 in toluene by a factor 2.5–3 (Table ). The best-performing functional was BMK, a hybrid exchange-correlation functional with 42% HF exchange designed for simulating reaction kinetics. With BMK, the authors also tested the impact of the basis set (6-31G­(d), 6-311G­(d,p), and def2-TZVP) and the inclusion of LR-PCM for toluene; the best agreement with the experiment was found at the BMK/6-311G­(d,p)/PCM level (calculated g _lum = 0.34). The quite small dispersion of calculated g _lum values regardless of the variety of the employed DFT methods is interesting. For compound MC3, the same level of calculation provided g _lum = 0.10, in agreement with the experiment. Based on the understanding gained on MC1b and MC3 by theoretical calculations, the authors reasoned that a possible way of further increasing the |m ₁₀ |/|μ ₀₁ | ratio and hence the g _lum might exploit SRCT transitions. Therefore, they conceived compounds MC4 and MC5 incorporating B and N atoms in specific positions (Chart ). FMO analysis confirmed that the HOMO and LUMO were respectively localized on the N and B atoms, creating the optimal requirement for SRCT (Figure ). Finally, TD-BMK/6-311G­(d,p)/PCM calculations predicted theoretical g _lum = 0.56 for MC4 and 0.31 for MC5, representing a significant increase with respect to all-C analogs MC1b and MC3; an experimental confirmation is awaited. A discussion of the role of the macrocycle size on chiroptical properties is postponed to Section .

8. Collection of Experimental and Computational Data for Macrocycles .

Exper.		Calculation				Number or abbreviation
g lum	g lum /g abs	g lum	g lum /g abs	Method	Basis set	This review	Original publication	ref.
150		450		B3LYP	6-311G(d,p)	MC1b	4CC(2,8)
150		440		PBE0	6-311G(d,p)	MC1b	4CC(2,8)
150		470		PBE	6-311G(d,p)	MC1b	4CC(2,8)
150		420		BMK	6-311G(d,p)	MC1b	4CC(2,8)
150		440		M06-2X	6-311G(d,p)	MC1b	4CC(2,8)
150		390		CAM-B3LYP	6-311G(d,p)	MC1b	4CC(2,8)
150		380		ωB97X-D	6-311G(d,p)	MC1b	4CC(2,8)
20		0.059		M06-2X	6-311G(d,p)	MC8	[8]CPP-P[5]A
31		13		PBE38	6-31G(d,p)	MC9	1
11		29		M06-2X	6-31G(d)	MC10a	1BN
0.71		1		M06-2X	6-31G(d)	MC10b	1SBF
1		1.4		PBE0	6-31G(d)	MC11	4
7		10		PBE0	6-31G(d)	MC12	6
22		37		PBE0	6-31G(d)	MC13	8
16		35		PBE0	6-31G(d)	MC14	9
15		19		PBE0	6-31G(d)	MC15	10
2.3	0.82	1.1	0.64	ωB97X-D	def2-SVP	MC17	DATM3
2.3	0.82	1.5	0.64	CAM-B3LYP	def2-SVP	MC17	DATM3
2.4		2.5		B3LYP	6-31G(d,p)	MC20a	2
1.1		1.8		B3LYP	6-31G(d,p)	MC20b	3
0.6		1.3		B3LYP	6-31G(d,p)	MC20c	4
0.6		1.9		B3LYP	6-31G(d,p)	MC20d	5
5.9	0.42	110	1.41	B3LYP-D3	6-31G(d,p)	MC21a	1a
5	0.36	151	1.31	B3LYP-D3	6-31G(d,p)	MC21b	1b
7.6	2.23	12	3.00	B3LYP-D3	6-31G(d,p)	MC21c	1c
38	7.6	101	1.29	B3LYP-D3	6-31G(d,p)	MC21d	1d
0.19	0.01	0.15	0.01	B3LYP-D3	6-31G(d,p)	MC21e	1e
0.9	0.05	16	0.50	B3LYP-D3	6-31G(d,p)	MC21f	1f
1.2	0.1	14	0.13	B3LYP-D3	6-31G(d,p)	MC22a	2a
0.2	0.87	47	1.24	B3LYP-D3	6-31G(d,p)	MC23a	3a
0.75	0.03	8	0.07	B3LYP-D3	6-31G(d,p)	MC22b	2b
0.23	0.01	56	0.50	B3LYP-D3	6-31G(d,p)	MC23b	3b
0.75	0.02	7	0.47	B3LYP-D3	6-31G(d,p)	MC22d	2d
1.3	0.06	13	2.17	B3LYP-D3	6-31G(d,p)	MC23d	3d
1.68		1.17		B3LYP	6-31G(d)	MC24b	TBMC-2

^aData for electronic calculations only (not vibronic). See Charts and for structure numbering.

^b g _abs and g _lum multiplied by 10³.

60.

HOMO, LUMO, HOMO/LUMO overlap, and hole–electron charge-density difference for the S₁–S₀ transition of macrocycles MC3, MC1a, MC5, and MC4 (Chart ). Adapted with permission from ref . Copyright 2024 American Chemical Society.

Excited-state TD-DFT calculations were reported for a few CPL-emitting chiral macrocycles based on the [n]­cyclo­(p-phenylene) scaffold ([n]­CPP), also referred to as carbon nanohoops. A few examples, embedding helicene units, have already been discussed in a previous section (Section ).

In their already mentioned paper about figure-of-eight CPL emitters, Hirose, Matsuda, and co-workers also calculated CPL parameters for MC6. Apparently, the wrong CPL sign was predicted at the TD-B3LYP/631G­(d) level. Jiang and co-workers analyzed a series of chiral carbon nanohoops fused to a triptycene moiety as source of chirality (MC7a–f, Chart ). The photophysical properties of this latter series are size-dependent: the emission wavelength λ_em (from 490 to 440 nm) and the Stokes shift are both reduced upon increasing the size, while both the extinction coefficient and the QY increase. The FMOs in the S₁ state tend to concentrate on the oligo­(p-phenylene) portions of the nanohoops. Both experimental g _abs and g _lum featured an almost linear decrease with size (Figure ); g _lum values varied between 8.1 × 10^–4 for the longest representative of the series MC7f and 3.52 × 10^–3 for the shortest one MC7a while the maximum B _CPL = 100 M^–1 cm^–1 was encountered for the intermediate MC7d. TD-DFT calculations run at the PBE0/6-31G­(d) level clarified that the major obstacle to large dissymmetry for this family of molecules is the orientation between transition dipole moments because the calculated angle θ _μm ranged between 86 and 89°.

61.

(a) CPL spectra and (b) g _lum values recorded for macrocycles MC7a–f showing a dependence on the ring size. Positive CPL signals for (S) enantiomers and negative CPL signals for (R) enantiomers. Reprinted from ref under a CC-BY 4.0 license.

Yam and co-workers conjugated two [n]­cyclo­(p-phenylene)­s (n = 8, 10) to a pillar[5]­arene, yielding two chiral derivatives named [8]­CPP-P[5]­A (MC8, Chart ) and [10]­CPP-P[5]­A. The chiroptical properties of MC8 were measured and simulated at the M06-2X/6-311G­(d,p) level. The red-most ECD band between 350 and 450 nm was localized on the [8]­CPP ring, indicating the transfer of chirality from the P[5]­A ring. The CPL spectrum had its maximum at 488 nm flanked by a low-energy shoulder, which had no parallel in the emission spectra and was not commented on by the authors. A maximum g _lum = 2 × 10^–2 was measured. The calculations provided a much smaller g _lum = 5.9 × 10^–4 for the S₁–S₀ transition (Table ), which led the authors to suggest participation in the CPL signal of the S₂–S₀ transition, for which a larger g _lum = 5.3 × 10^–2 was estimated. However, no further proof for this anti-Kasha behavior was offered.

An original elaboration of the [n]­cyclo­(p-phenylene) motif was provided by Mao, Cong, and co-workers, who designed a helical [8]­cyclo­(p-phenylene) dyad held by 3-buta-1,3-diyne-1,4-diyl linkers (MC9, Chart ). Such a C ₃-symmetric compound reaches g _lum = 3.1 × 10^–2, a value slightly underestimated by TD-DFT calculations at the PBE38/6-31G­(d,p) level (Table ). The latter functional is similar to the commonly used PBE0 but with a higher HF exchange (37.5 vs 25%).

A different conjugated motif, that is, oligo­(p-phenyleneethynylene), was explored by Miki, Ohe, and co-workers in combination with the axial chirality provided by 1,1′-binaphthyl and 9,9′-spirobifluorene units to construct macrocyclic double helicates MC10a and MC10b (Chart ). The two compounds differ in the flexible versus rigid stereogenic scaffold, which modulates the emission and related chiroptical response. In fact, the flexible double helicate MC10a undergoes intramolecular excimer formation with near-IR emission, which is precluded in the rigid analog MC10b that emits in the visible range (Figure ). In absorption, the two compounds gave similar ECD spectra with g _abs = 1.2–1.8 × 10^–3 around 510 nm. In emission, MC10b gave a vibronically split emission band (typical of anthracene) with λ_em = 510 and 545 nm, Φ_f = 0.93, and a single photoluminescence lifetime τ_f = 2 ns. The same features were found in the emission spectrum of MC10a, which also showed an additional broad band between 600 and 850 nm assigned to the excimer. Accordingly, the QY dropped to Φ_f = 0.49 and a second longer lifetime τ_f > 20 ns was measured at 700 nm. CPL spectra were also different for the two compounds: while MC10b gave a weak signal with g _lum = 7.1 × 10^–4 at 490 nm, MC10a showed a broad band extending into the NIR region with g _lum = 1.1 × 10^–2 at 680 nm in CHCl₃ (Figure ). For excited-state calculations, the authors employed the M06-2X functional, combined with the 6-31G­(d) basis set, after noticing that B3LYP did not capture the excimer structure because of an insufficient description of dispersion. With M06-2X, the S₁ geometry of MC10a was clearly excimeric and was associated with calculated θ _μm = 120° and g _lum = 2.9 × 10^–2, in good agreement with the experiment (Table ). For MC10b, the smaller calculated g _lum = 9.9 × 10^–4 was mainly due to an unfavorable θ _μm = 92°.

62.

(a) Illustration of the working principle of flexible double helicates as NIR CPL emitters based on intramolecular excimer formation. (b) Experimental CPL spectra and wavelength-dependent g _lum measured for double helicates MC10a and MC10b (Chart ). Solid lines for (S,S) enantiomers and dashed lines for (R,R) enantiomers. Adapted with permission from ref . Copyright 2019, John Wiley and Sons.

A cyclization strategy starting from easily accessible resorcin[4]­arenes was employed by Xiao, Tong, and co-workers to access a family of highly strained molecular belts with a 2,3-dihydro-1H-phenalene skeleton (MC13–MC15, Chart ). The degree of unsaturation of the belts obviously affected their geometry as well as the optical properties. The intensity of CPL spectra (Figure ) dramatically increased upon passing from the macrocycle M11 and half-belt analog M12 (g _lum = 1 × 10^–3 and 7 × 10^–3, respectively) to the belts MC13–MC15 (g _lum = 1.5–2.2 × 10^–2). TD-DFT calculations run at the PBE0/6-31G­(d) level reproduced g _lum values satisfactorily (Table ) and traced the differences observed between the various compounds mainly back to the orientation between transition dipoles (Figure ). Angle θ _μm was much more favorable for belts MC13–MC15, especially MC14, than for the macrocycle MC11 and the half-belt MC12.

24.

63.

(a) Experimental CPL spectra and wavelength-dependent g _lum measured for molecular belts MC11–MC15 (Chart ). Solid lines for (M) enantiomers and dashed lines for (P) enantiomers. (b) EDTM, MDTM, and their angle θ calculated for the S₁–S₀ transition of MC11–MC15. Adapted with permission from ref . Copyright 2023, John Wiley and Sons.

In collaboration with Górecki, Grzybowski, and co-workers, some of us studied a family of oligomeric macrocycles MC16–MC18 (Chart ) arising from the junction of “minimal” 2,5-diaminoterephthalate fluorophore with the trans-1,2-diaminocyclohexane scaffold. With respect to other macrocycles considered in this section, the synthesis of MC16-MC18 was relatively facile. The three compounds are intensely colored solids with color varying from purple for dimer MC16 to orange red for tetramer MC18. The ωB97X-D/6-311+G­(d,p)-optimized structure of MC16 revealed a cyclophane-like geometry with stacked benzene rings (plane-to-plane distance 3.16 Å). The X-ray and DFT-optimized structures of trimer MC17 display a triangular shape with an internal cavity with a radius of around 2.7 Å. All compounds were emissive in solution, with dimer MC16 showing the most red-shifted λ_em = 605 nm in n-hexane. Interestingly enough, they also retained a large fraction of their QY in the solid state, where the emission profile enters the NIR range especially for dimer MC16 (λ_em = 700 nm). Regrettably, the π-stacking negatively affects QY, which is smallest for MC16 along the series. The ECD spectra for all derivatives present several bands in the UV and vis region, with the latter associated with ICT transitions. VT-ECD measurements were run in an MCH/isopentane 1:3 solvent mixture between +25 and –180 °C. They revealed for trimer MC17 a slight temperature dependence and for tetramer MC18 a much more pronounced temperature dependence, in accord with the considerably larger flexibility of the largest representative of the series, as found by DFT calculations. CPL spectra in the deep-red region of the vis range were obtained, whose intensity increased in the order dimer < tetramer < trimer as a result of the combination of emission efficiency and conformational flexibility. For trimer MC17, the highest g _lum = 2.3 × 10^–3 at 570 nm was recorded. Excited-state calculations were run on MC17 using both ωB97X-D and CAM-B3LYP functionals in combination with the def2-SVP basis set and reproduced the experimental CPL sign, g _lum, and Stokes shift (Table ). The experimental and calculated g _lum/g _abs = 0.82 and 0.80–0.85, respectively, are in agreement with similar S₀ and S₁ geometries (Figure ).

64.

(a) Experimental and calculated ECD and CPL spectra of trimeric macrocycle MC17 (Chart ). (b) Overlap of S₀ (blue) and S₁ geometries (green) of MC17. Adapted with permission from ref . Copyright 2023, John Wiley and Sons.

Amaya and co-workers prepared two shape-persistent oligo­(spirofluorene)­s MC19a,b (Chart ) endowed with D ₃ symmetry and featuring C-to-Si substitution of the spiro centers. The chiro­(optical) properties of the two compounds are impacted by the effects of so-called spiro-conjugation (i.e., the conjugation between hypothetically orthogonal π-orbitals of the fluorene moieties). Furthermore, the D ₃ symmetry splits the FMOs into groups of three, whereas the C-to-Si substitution slightly reduces the HOMO–LUMO gap. Both compounds showed strong violet emission (λ_em = 384 nm for MC19a) with high QY (up to Φ_f = 0.94 for MC19a). ECD and CPL spectra for the pair were moderately intense, with g _abs = 5.2 × 10^–3 at 352 nm and g _lum = 1.2 × 10^–3 at 374 nm for MC19a and similar values for MC19b. TD-DFT calculations run at the B3LYP-D3/6-31G­(d,p) level overestimated both g _abs and g _lum by more than 1 order of magnitude. The overestimation was explained by an effective desymmetrization of the S₁ state, which was not found by B3LYP-D3 calculations in vacuo but was captured by B3LYP-D3BJ calculations run with PCM for DCM. The latter predicted an excited state localized on a single, planarized bisilafluorenyl moiety yielding a calculated θ _μm ≈ 90°, instead of θ _μm = 0° predicted for D ₃ symmetry fully delocalized over the whole skeleton of MC19a.

A series of macrocycles with increasing size from dimer to tetramer, based on the BINOL scaffold, were reported recently by Tang and Cao (M20a–d, Chart ). Observed g _lum values in solution vary inversely with the macrocycle size, from 2.4 × 10^–3 for M20a to 6 × 10^–4 for M20d. The trend was partially reproduced by TD-DFT calculations at the B3LYP/6-31G­(d,p) level (Table ). Not surprisingly, the largest homologue also undergoes the widest structural rearrangement from the GS to the ES geometry. A 1.5- to 2-fold increase in dissymmetry was observed in the gel state.

An extensive investigation of a family of chiral macrocyles, embedding carbo[5]­helicene units linked by various kinds of aromatic spacers, was run by Wu and co-workers. The family comprises D ₂-symmetric figure-of-eight macrocycles (MC21a–f, Chart ), D ₃-symmetric triply twisted Möbius macrocycles (MC22a–c with homochiral (M,M,M) carbo[5]­helicene units), and C ₂-symmetric singly twisted Möbius macrocycles (MC23a–c with heterochiral (M,M,P) carbo[5]­helicene units). For several derivatives, X-ray structures were determined. Absorption and ECD spectra vary a lot depending on the geometry and the aromatic spacer. Dissymmetry values (g _abs) measured in THF spanned 2 orders of magnitude. For the three series, the highest g _abs values for the red-most ECD band were observed for the pyrene-containing macrocycles MC21c (g _abs = 3.4 × 10^–3 at 405 nm), MC22c (g _abs = 3.3 × 10^–2 at 361 nm), and MC23c (g _abs = 1.9 × 10^–2 at 378 nm). Moderately intense CPL spectra were observed for most macrocycles, again with g _lum values spanning 2 orders of magnitude. Pyrene derivative MC21c exhibited both the largest g _lum = 3.8 × 10^–2 at 530 nm and the highest B _CPL = 710.5 M^–1 cm^–1. TD-DFT calculations run at B3LYP-D3/6-31G­(d,p) level confirmed once again the pivotal role played by molecular symmetry and flexibility. For instance, dimeric compounds MC21c and MC21d retain D ₂ symmetry in both the S₀ and S₁ states, hence perfectly aligned electric and magnetic transition moments (θ _μm = 0°) are predicted for both S₀–S₁ and S₁–S₀ transitions. On the contrary, the more flexible trimeric compounds MC22a and MC22b lose their D ₃ symmetry in the S₁ states, hence angle θ _μm reduces from the ideal 180° for the S₀–S₁ transition to 67–78° for the S₁–S₀ transition. Interestingly enough, several representatives had opposite signs between the CPL band and the red-most ECD band, a circumstance which was not noticed by the authors and was also not reproduced by calculations (e.g., MC23b and MC23c). Moreover, in some cases we observe a large discrepancy between experimental and calculated |g _lum| values, up to 2 orders of magnitude (Table ).

A pair of rigid macrocycles based on the Tröger base scaffold (MC24a,b, Chart ), reported by Cai and co-workers, showed g _lum values differing by 1 order of magnitude related to the different substitution pattern (g _lum = 1.68 × 10^–3 for MC24b, <1 × 10^–4 for MC24a) The trend was reproduced by calculations at the B3LYP/6-31G­(d) level, which for MC24a predicted θ _μm ≈ 90°. These compounds feature host–guest binding ability toward dicationic ammonium salts.

In conclusion of this section, we wish to emphasize a few points. First, cylindrical macrocycles with high molecular symmetries may allow for both a maximization of magnetic dipole transition moments and their collinear orientation with electric dipole transition moments, with both requisites being necessary at the same time to observe high dissymmetry values. Second, rigid structures may help prevent actual distortions from the expected symmetric structures, which tend to lower dissymmetry values (discussion in Section ). Third, and most importantly from our viewpoint, TD-DFT calculations with several functionals appear to be accurate enough in the prediction of CPL parameters, including g _lum values. Satisfying agreement was obtained even for the largest macrocycles MC1 and MC3. Many other cases where poor agreement was obtained would possibly benefit from testing multiple functionals, while most often a single functional/basis set combination was employed. Given the extensive effort which must be expended for the synthesis and enantioseparation of chiral macrocycles, robust theoretical calculations may address the targeted design of efficient macrocyclic CPL emitters.

3.2.3. Nanographenes

Nanographenes are extended polycyclic aromatic hydrocarbons with sizes of between 1 and 100 nm. This wide class of compounds has attracted much attention, especially for their multiple applications in materials science. The distortion of typically planar nanographene structure into nonplanar, chiral structures has resulted in a new field of research. Although CPL data on several chiral nanographenes are currently available, calculations of CPL parameters for this family are relatively rare. An obvious reason for this situation is the large molecular size, which increases computational times and affects the accuracy of results.

Wang and co-workers reported one of the first examples of CPL calculations on a nanographene (N1, Chart ), a compound which could also be classified as an X-shaped double carbo[6]­helicene (Section ). Possibly due to the extended π-conjugated structure beyond that of a classical helicene, the authors referred to it as a “nanographene”, which is the reason that it is treated in this section. In fact, almost all chiral nanographenes found in our survey are built around a helicene or helicenoid framework. Compound N1 has consistent ECD and CPL spectra, with similar g _abs = 1.0 × 10^–3 and g _lum = 8.0 × 10^–4; the brightness is B _CPL = 32 M^–1 cm^–1. The authors noticed that these values compared well with other chiral nanographenes available at that time. TD-DFT calculations were run at the B3LYP/6-31G­(d) level (Table ), and the estimated electric/magnetic transition dipole angle θ _μm = 0° was in keeping with the expectation for D ₂ symmetry.

25.

9. Collection of Experimental and Computational Data for Nanographenes .

Exper.		Calculation				Number or abbreviation
g lum	g lum/g abs	g lum	g lum/g abs	Method	Basis set	This review	Original publication	ref.
0.8		2.2		B3LYP	6-31G(d)	N1	1
36		10.3		B3LYP	6-31G(d,p)	N2a	9HBNG
10		5.6		B3LYP	6-31G(d,p)	N2b	10HBNG
8.6		3.7		B3LYP	6-31G(d,p)	N2c	11HBNG
45		37		B3LYP	6-31G(d)	N3	EP9H
2.7	0.69	1.8	1.4	B3LYP	6-31G(d)	N16	oED7H
8.7	0.87	4	0.56	B3LYP	6-31G(d)	N17	mED7H
13.5	0.73	5	0.23	B3LYP	6-31G(d)	N18	pED7H
0.75	0.75	0.89	1.4	B3LYP	6-311G(2d,p)	N20	2C1
1.7	0.81	1.31	0.82	B3LYP	6-311G(2d,p)	N21	2C2
2.3	0.56	4.62	2.5	B3LYP	6-311G(2d,p)	N22	3C2
0.73	0.59	0.22	0.14	B3LYP	6-311G(d,p)	N24	1
0.89	0.63	0.86	0.27	B3LYP	6-311G(d,p)	N25	2
5.5	1.83	8.45	1.22	B3LYP	6-311G(d,p)	N26	3
1.3	0.46	2.3	1	B3LYP	6-311+G(2d,p)	N27	1
8	0.67	10	0.5	PBE0	6-311G(d)	N28	E[10]HAB-C
11	1	12	0.52	PBE0	6-311G(d)	N29	E[10]HAB-B
17	0.71	19	0.54	PBE0	6-311G(d)	N30	E[10]HAB-A

^aData for electronic calculations only. See Charts – for structure numbering.

^b g _abs and g _lum multiplied by 10³.

Three helical bilayer nanographenes N2a–c (Chart ) were developed by Martín and co-workers. The different length of the helicene portion at the bottom of the structures (from a carbo[9]­helicene for N2a to a carbo[11]­helicene for N2c) modulates the extent of the overlap between the benzene rings involved in π-stacking interactions. The maximum overlap is seen for the smaller analog N2a, where as many as 26 benzene rings undergo π-stacking, producing a tightly packed bilayer (Figure ). Accordingly, N2a also produces the most red-shifted emission with λ_em = 575 nm and brilliant orange fluorescence with Φ_f = 0.22. ECD spectra of compounds N2a–c are rich in many bands; however, in all cases the red-most ECD band has the same sign as the CPL band (Figure ). Compound N2a is again the most interesting, yielding the largest g _lum = 3.6 × 10^–2 (at 580 nm) over the series. TD-DFT calculations run at the B3LYP/6-31G­(d,p) level reproduced the experimental CPL sign, although they underestimated the intensities (Table ). The calculated θ _μm angles followed the trend of g _lum values, demonstrating once again the importance of a favorable orientation between transition dipoles.

65.

(a) Structure of bilayer helical nanographenes N2a–c (Chart ) emphasizing the π-overlap. (b) Experimental emission and CPL spectra of N2a–c. Solid lines for (+)-(P) enantiomers and dashed lines for (−)-(M) enantiomers. Reprinted from ref under a CC-BY 4.0 license.

Another chiral nanographene with an exceptionally high dissymmetry value was reported by Gong and co-workers. Compound N3 (Chart ) attains g _lum ≈ 4.5 × 10^–2 at 684 nm and a noteworthy B _CPL ≈ 300 M^–1 cm^–1, among the highest reported for purely organic molecules. Calculations on a truncated model (t-butyl groups replaced by methyl groups), run at the B3LYP/6-31G­(d) level, predicted g _lum = 3.7 × 10^–2 in agreement with the experiment.

Bella, Bruno, and Santoro run an extensive computational analysis of 14 chiral nanographenes (N1, N2a, and N3–N14; Chart ). First, they explored a set of exchange-correlation DFT functionals (APFD, B3LYP, CAM-B3LYP, TPSSTPSS, and ωB97X-D) for their ability to provide the best match between the calculated ground-state geometry with the X-ray one and to reproduce the topology of the nanographene surface in terms of its curvature. It emerged that the functional performance was substrate-dependent, with B3LYP offering the most constant trend and lowest average error. Then, the excited-state potential energy surface was explored by BOMD (Section ), and CPL spectra were calculated as weighted sums of the spectra computed at the TD-DFT level on equispaced snapshots extracted from BOMD simulations. Excited-state geometry optimizations were run with the B3LYP functional, while vertical calculations were run using 10 different functionals (APFD, BLYP, M06L, mPW1PW91, B3LYP, B98, CAM-B3LYP, LC-ωPBE, TPSSTPSS, and ωB97X-D) all in combination with the 6-311G­(d,p) basis set. Once again, B3LYP was demonstrated to be the best-performing functional, better reproducing experimental CPL spectra in terms of both the emission maximum and CPL intensity. In general, LC-ωPBE, ωB97X-D and CAM-B3LYP functionals overestimated the emission frequency maximum; this is a well-known drawback for these functionals also observed for absorption spectra, which can be easily taken into account by means of an empirical linear correction. , B3LYP, together with APFD, B98, and mPW1PW91, provided good performance, whereas TPSSTPSS, M06L, and BLYP underestimated the emission frequency maximum. Unfortunately, in this publication the calculated CPL spectra for the two enantiomers of each nanographene are not distinguished, so it is impossible to judge if the correct CPL sign is reproduced by all functionals in all cases. Moreover, only absolute ΔI intensities are provided, so the estimation of calculated g _lum values is not possible. A similar computational approach was employed by the same authors to deal with C70 fullerene derivative N15.

Three nanographenes based on the double [7]­helicene scaffold were conceived by Niu, Ma, and co-workers (N16–N18, Chart ). They differ in the geometry of attachment of the two [7]­helicene moieties to the central ring. Both g _abs and g _lum values increased when passing from N16 to N17 and then to N18, for which the maximum g _lum = 1.35 × 10^–2 was registered with an ∼5-fold improvement with respect to that of N16. TD-DFT calculations run at the B3LYP/6-31G­(d) level reproduced the observed trend in ECD and CPL intensities along the series (Table ). In this case, the θ _μm angle for the S₁–S₀ transition slightly decreased along the series, passing from 85° for N16 to 82° for N17 to 80° for N18; a major role was instead played by the |m ₀₁ |/|μ ₁₀ | ratio between transition moment intensities, which amounted to 5.1 × 10^–3 for N16 and 6.7 × 10^–3 for N18. The brightness values were also high, with a maximum B _CPL = 176 M^–1 cm^–1 recorded for N18.

26.

A family of nanographenes embedding 1,4-azaborine moieties and aza[7]­heterohelicenes (N19–N22, Chart ) were reported by Ravat and co-workers. These compounds exhibit exceptionally narrow emission and CPL bands around 500 nm, with a fwhm of as small as 423 cm^–1 (10.5 nm), which compares with that of lanthanide complexes. Ultranarrow band emission and CPL are major requirements in OLED and CP-OLED devices with high color saturation. Relatively high QYs were also obtained, though at the expense of g _lum values which remained between 0.75 and 2.3 × 10^–3. TD-DFT calculations were run at the B3LYP/6-311G­(2d,p)//ωB97X-D/6-31G­(d,p) level. Despite the extended conjugated structures, the calculated geometries for S₀ and S₁ states were practically coincident for all compounds. The rigid conjugated core is key to obtaining narrow emission bands because of limited structural relaxation. FMOs were mostly localized on and in proximity to the helicene portion. CT descriptors were evaluated and demonstrated short-range CT character of the first excited state, which agrees with the high oscillator strength. Electronic calculations reproduced the observed trend in g _lum values (Table ). For all compounds, the major limitation to high dissymmetry values seems to come from an unfavorable orientation of dipole moments (θ _μm = 82–84°), whereas the better CPL performance of N22 is related to a larger |m ₀₁ |/|μ ₁₀ | ratio. Vibrationally resolved CPL spectra were calculated at CAM-B3LYP/def2-SVP level using FCclasses code with the FC-HT₁|AH scheme, that is, by including FC and HT effects and computing dipoles and derivatives in the S₁ state and modeling PESs of both states with the AH model (Section ). For compounds N19–N22, the narrow appearance of the main emission and CPL along with the reduced Stokes shift implies that the spectra are dominated by the 0–0 transition. The calculations reproduced well the observed bandwidth for N21 and N22 but underestimated the bandwidth of N19 and N20, which was ascribed to the larger conformational flexibility of these two analogs which, though limited, would lead to spectral broadening.

A series of helical nanographene resulting from the fusion of carbo[4]­helicene and PDI moieties was recently reported by Nuckolls, Santoro, Pittelkow, and colleagues (N23a–d, Chart and Figure ). Their emission and chiroptical properties increase nonlinearly with the oligomer size, reaching maximum values of g _lum = 1.2 × 10^–2, Φ_f = 0.68, and B _CPL = 10³ M^–1 cm^–1 for the largest representative of the series N23d (Figure ). CPL spectra were all similar, with maxima around 550 nm and a second vibronic low-energy band. They were well reproduced by vibronic TD-CAM-B3LYP/6-31G­(d) calculations with the FC|AH model. The calculations could also capture the increase in CPL intensity along the series, although the predicted trend was almost linear and the calculated g _lum values overestimated the experimental ones for N23c–d (Figure ). A key role is played by the orientation between transitions dipole moments, with θ _μm progressively decreasing from 84° for N23a to the more favorable value of 40° for N23d. Another decisive factor was highlighted by means of difference density plots between S₁ and S₀ states and transition density plots. Whereas the excitation tends to be localized on the central PDI core, it becomes more delocalized toward the neighboring PDI units as the system’s size increases. Because of the nature of the excited state, each system N23a–d can in principle emit from multiple excited-state minima, each localized on one PDI unit but close enough in energy to be all populated at RT. The interstate coupling would be therefore mediated by vibrational motions triggered by electronic excitations, originating a manifold of vibronic nonadiabatic excited states whose description requires an appropriate treatment of electronic–vibrational coupling (Sections and ).

27.

66.

CPL spectra and related quantities measured for helical nanographenes N23a–d (Chart ) as a function of the molecular size. Adapted with permission from ref . Copyright 2025, John Wiley and Sons.

Very recently, Kumar, Babu, and co-workers elaborated the nanographene motif based on the carbo[7]­helicene already found in N5 by embedding a phenazine or [1,4]­diazocine moiety, yielding compounds N24–N26 (Chart ). Compound N26, incorporating a 1,1′-binaphthyl-2,2′-diamine, displays 10-fold-increased chiroptical properties with respect to the other two analogs, reaching g _lum = 5.5 × 10^–3 and B _CPL = 146 M^–1 cm^–1. TD-DFT calculations run at the B3LYP/6-311G­(d,p) level reproduced very well the experimental data (Table ) and demonstrated that the reasons for g _lum enhancement in N26 were both a larger |m ₀₁ | and more favorable θ _μm = 52°. A similar design, incorporating a central carbazole ring, led to compound N27 by Gong and co-workers, which combines helical chirality and hindered rotation around the N–hexabenzocoronene junction. The compound exhibited g _lum ≈ 3–4 × 10^–4 in toluene and up to 1.3 × 10^–3 in ACN. TD-DFT calculations at the B3LYP/6-311+G­(2d,p) level were closer to the latter value (Table ). The almost perfect alignment between transition moments was apparently not sufficient to achieve larger dissymmetry values.

Liu et al. inserted within a helical nanographene scaffold multiple BN pairs able to generate MR-TADF properties. Three analogs with a variable number of terminal fused benzene rings were prepared (N28–N30, Chart ) showing g _lum values of between 0.8 and 1.7 × 10^–2 and B _CPL of up to 583 M^–1 cm^–1, the highest reported for BN-doped helicenes. Calculations run at the TD-PBE0/6-311­(d) level reproduced the experimental trend (Table ) and confirmed the consistent behavior across the three compounds.

3.2.4. Supramolecular Systems, Self-Assemblies, and Aggregated States

This section represents a distinctive part in our survey for a variety of reasons. From an applicative point of view, all devices mentioned in the Introduction based on CP light emission are composed of materials in their solid state, where aggregation unavoidably takes place. This fact may be a drawback for many chemical species whose emission occurs in dilute solutions but is quenched in aggregated phases because of aggregation-caused quenching (ACQ). Especially in the presence of distinct solid-phase domains, exciton migration occurs toward local energy traps which relax through nonradiative pathways. On the other side of the spectrum of possible photophysical phenomena, aggregation induced emission (AIE) concerns molecules which are nonemissive in the dispersed state but become emissive in the aggregate form (solution aggregates and solid state). Most commonly, the so-called AIE luminogens (AIEgens) contain flexible molecular portions, for instance, aryl–aryl torsions allowing for libration modes, by which they dissipate excited-state energy in a nonradiative way; these motions become restricted in the aggregated state. Not surprisingly, AIEgens have attracted enormous attention for their possible applications in optoelectronic devices, fluorescent bioprobes, and chemosensors. Several chiral AIEgens have been reported to emit CP light with high degrees of dissymmetry. − In general, the search for organic compounds which retain or exhibit CPL activity in the solid phase is probably the most active subfield of research in the field of CPL-active materials. ,,,, As a matter of fact, the highest g _lum values (>0.1) reported in the literature for metal-free compounds have been recorded for aggregated phases (even excluding cholesteric phases). ,,,− Obviously, the control of the morphology and hence of the fabrication methodology, including deposition and postdeposition processes, is key for achieving high dissymmetries.

From an experimental viewpoint, CPL measurements on solid samples need some special care as photoselection phenomena due to linear anisotropies may take place and significantly affect or even overwhelm the true CPL signal in the case of low or moderate g _lum. Usually, experimental CPL spectra of thin films are recorded with 0 or 180° geometry between excitation and collection. It is advisible to confirm the genuine nature of the signal by checking the enantiomeric compounds, whenever available, or at least rotate the sample around the optical axis to confirm that the CPL spectra recorded in each position are not significantly dependent on the rotational angle. From a computational viewpoint, aggregate systems pose a manifold of different issues, stemming from poorly defined aggregate structures to the existence of multiple aggregated states, accompanied by large sizes and molecular complexity. This explains why such crucial systems are heavily underrepresented in the literature of CPL calculations. This section first addresses discrete assemblies, such as host/guest complexes, followed by small models, like dimers, used to represent larger aggregates and concludes by considering larger systems.

A chiral exciplex formed in the solid state by an octahydro-1,1′-binaphthyl-2,2′-diol derivative as an acceptor (A1.1, Chart ) and a commercially available hole-transport material based on triphenylamine (A1.2) as a donor was described by Su and co-workers. The thin film prepared from the 1:1 mixture of A1.1 and A1.2 displays a broad exciplex emission band with λ_em = 520 nm and a red shift of >120 nm with respect to its separate components. The films were also CPL-active with dissymmetry values (g _lum ≈ 2–3 × 10^–3) slightly dependent on the composition. Although some calculation results are reported in the paper, including the simulated CPL data for A1.1 and the A1.1+A1.2 1:1 complex, the calculation method is not disclosed and any detail is missing.

28.

Guo, Fang, and co-workers designed a macrocycle (A2.1, Chart ) with AIE properties featuring a large chiral cavity for the fluorometric and CPL detection of ursolic acid (A2.2). The fluorescent host selectively binds ursolic acid with respect to many other analytes (amino acids, nucleobases, tartaric acid, citric acid, and others) producing a blue shift in its emission band from 478 to 467 nm, which is manifest by a color change from cyan to blue under UV-light irradiation. More interestingly, while both A2.1 and A2.2 are CPL-silent, the CPL signal is turned on upon binding, with g _lum = 2.5 × 10^–3. When taken alone, A2.1 occurs as a mixture of conformational enantiomers, whose g _lum values calculated at the B3LYP/6-311+G­(2d,p) level amounted to ±7.3 × 10^–2. The inclusion complex between (R,R)-A2.1 and A2.2, optimized in the S₁ state, showed a structure with multiple hydrogen bonds between the guest O atoms and the host NH groups. The calculated g _lum = 2.5 × 10^–3 agreed with the experimental value. Although the authors recognize that a stereoselective binding process occurs between (enantiopure) ursolic acid and the two enantiomers of the guest, the process was not studied either experimentally or computationally. For instance, the complex between (S,S)-A2.1 and A2.2 was not simulated.

Two metal–organic hexahedral cages with a Pd₆L₁₂ framework, where the ligand includes an axially chiral 1,1′-biphenol moiety and a K⁺-binding group such as 18-c-6 or MOM, were reported by Liu and co-workers to form inclusion complexes with BODIPY derivatives A3a–b, respectively, with host-to-guest 1:2 and 1:1 stoichiometry. The complexes, characterized by X-ray diffractometry (Figure ), were CPL-active in solution and especially in the solid state. On KBr pellets, g _lum = 2.4 × 10^–2 and 1.0 × 10^–2 were measured for the 1:2 complex of A3a and the 1:1 complex of A3b, respectively, in correspondence with the bound BODIPY emission around 530 nm. TD-DFT calculations were run on the guest molecules extracted from the X-ray geometries, freezing the carbon atoms in closest contact with the guest molecules to reproduce a rigid chiral environment. The calculated g _lum reproduced qualitatively the observed trend, although the absolute values were far from the experimental ones. Li, Zhou, and co-workers described a chiral MOF based on Zn-imidazolate able to host various halogenated benzenes and vary its g _lum values according to the guest structure, which, according to M06-2X/def-TZVP calculations, modulated the arrangement between transition dipole moments.

67.

X-ray structures of CPL-active inclusion complexes between metal–organic hexahedral cages and BODIPY derivatives A3a–b (Chart ). Color code: green, Pd; yellow, K. Reprinted from ref , copyright 2021, with permission from Elsevier.

Taking advantage of the favorable electronic properties of the tetraphenylene (TPE) scaffold, Cao and co-workers designed an ionic cage (A4, Chart ) capable of chiral adaptive recognition of several biologically relevant analytes such as nucleotides, amino acids, oligopeptides, and even polypeptides and proteins. , The cage may assume three conformations upon rotation of the two TPE units (an achiral “meso” MP conformation and two enantiomeric MM and PP conformations), which are biased toward one enantiomeric form upon binding a chiral substrate. The binding generates ECD and CPL signals associated with the TPE transitions. CPL bands above 500 nm were measured with g _lum values of up to 5 × 10^–4 with thymidine monophosphate dimer and 6 × 10^–4 with TyrTyr dipeptide. A theoretical investigation of the chiral recognition phenomenon involving A4 was performed by Bella and co-workers. Using a similar protocol to that encountered for nanographenes (Section ), they first spotted the DFT functional best reproducing the X-ray geometry of A4, which was found to be ωB97X-D in combination with the 6-311G­(d,p) basis set. Then, they focused on the complexes between A4 and several nucleotide pairs, running BOMD at the ωB97X-D/6-311G­(d,p) level with explicit water molecules for the S₁ state, allowing the collection of a statistically relevant conformational ensemble on which vertical excited-state calculations were run; several DFT functionals and basis sets were benchmarked (M06L, MN12L, B3LYP, CAM-B3LYP, LC-ωhPBE, mPW3PBE, ωB97X-D, BMK, SOGGA11X and HSEH1PBE functionals; 6-311G­(d,p), LanL2DZ, def2-SVP, and DZP basis sets). The final CPL spectra were obtained as a weighted sum of spectra calculated for temporally equispaced MD snapshots. Almost all functionals provided the correct CPL sign, with mPW3PBE and B3LYP best matching the line shape and the CPL maximum. The impact of the basis set (combined with B3LYP) was found to be minimal. No assessment of calculated CPL intensity in terms of g _lum is possible from the published data. A similar investigation was conducted on the complexes between A4 and two dipeptides (TyrTyr and AlaAla). The authors also explored the recognition behavior of a TPE-derived triangular macrocycle bearing three crown ethers, developed by Zheng and co-workers as a probe for chiral carboxylic acids.

Rotaxanes and catenanes are mechanically interlocked systems, which have fascinated chemists in the last 30 years thanks to the possibility of controlling their conformation by stimuli-induced switching. CPL properties of rotaxanes and catenanes have received scarce attention, despite the potential for a switchable CPL response. Two examples of a rotaxane and a catenane containing two pyrene rings arranged in a favorable way to form excimers upon excitation are discussed in Section (P11 and P12, Chart ). , A system similar to P12 had been reported previously by the same authors, including an AIEgen 9,10-distyrylanthracene unit (A5, Chart ). In this case too, the addition of an acetate or sodium ion induced reversible switching between two states with different CPL properties as cast films. The acetate-bound state has a 6.4-fold-enhanced CPL signal, a trend which was reproduced by calculations. The latter were run using a three-layer ONIOM­(QM:MM) scheme (Section ), which consisted of partitioning the system into three portions treated at different levels: the fluorophoric core with TD-DFT, the remaining guest portion and two AcO^– ions with DFT, and the two host molecules with a force field.

When coming to the quantitative prediction or simulation of CPL spectra of nanosized aggregated phases, the system size, complexity, and heterogeneity represent obvious obstacles to full atomistic calculations. Although computational protocols for the QM simulation of chiroptical properties of complex systems are available (Section ), many calculation approaches relied on the use of small systems, especially dimers. This two-body approximation may capture first-neighbor interactions in molecular aggregates or crystals, as demonstrated by a few examples already discussed in Sections and ; , however, it overlooks the collective nature of chiroptical properties, which often depend on long-range interactions. Therefore, researchers in the field face the situation of having limited tools available, precisely for the most promising systems.

As a first example of this simplified approach, Shi, Liu, and co-workers obtained luminescent and CPL-active nanotubes from the self-assembly of substituted 1,4,5,8-naphthalene diimide A6 (Chart ) in the presence of trifluoroacetic acid (TFA). Because of their appearance, the tubes were dubbed nanobamboos (NB). The NB morphology varied as a function of the TFA amount (Figure ), and so did emission and CPL spectra. The maximum g _lum ≈ 7–8 × 10^–2 was observed for an intermediate concentration of TFA. Dimers of A6 with face-to-face parallel stacking were considered for TD-DFT calculations starting from a geometry of S₁ optimized at the B3LYP-D3/6-31G­(d) level and systematically varying either the horizontal offset or the twist between naphthalene diimide long axes. Vertical calculations for each structure were then run at the same level of calculation and without reoptimization. Not surprisingly, the calculated g _lum values showed significant variation, spanning a range that aligns with the experimental observations.

29.

68.

(a) TEM images of bamboo-like π-nanotubes (nanobamboo, NB) obtained from the self-assembly of substituted 1,4,5,8-naphthalene diimide derivative A6 (Chart ) under different growing conditions (from NB-1 to NB-4). (b,c) Experimental emission spectra of NB and the starting nanosheet (NS). (c) Measured g _lum values for NB, NS, and a mixed nanobamboo (MNB). (d) Model dimers of A6 used in the calculations showing the displacement (top) and the rotation (bottom) between aromatic rings. (e) Calculated g _lum values for the dimers as a function of the geometry between the aromatic rings. Adapted with permission from ref . Copyright 2021, John Wiley and Sons.

An example of a TD-DFT calculation on oligomeric models of larger aggregates is offered by BF₂ diketonates A7a–b (Chart ). Compound A7a is CPL-silent in solution but emits green fluorescence with very large g _lum = 0.11 as an aqueous dispersion of microcrystals. The presence of anisotropic scattering was excluded by changing the sample orientation. Single-crystal X-ray diffraction revealed the existence of a supramolecular helix held together by multiple noncovalent interactions between the BINOL units, whose axial chirality dictates the supramolecular helical chirality. Starting from the X-ray lattice, TD-DFT calculations were run with the QM:MM approach, where the QM layer contained a single molecule, two different dimers, a trimer, and a tetramer, while the MM layer included the surrounding molecules in the crystal lattice. TD-DFT calculations run at the PBE0/6-31G­(d) level showed an increase in g _lum values upon increasing the model size; however, the highest values obtained for the trimer and the tetramer remained significantly lower than the experimental data. The authors recognized the possible causes for such a discrepancy in the finite nature of the adopted model and the neglect of nuclear motions. Apart from g _lum values, the calculations clearly reveal the impact of delocalized exciton on transition dipole moment orientation, which seems to be at the root of enhanced dissymmetry. Analog A7b, whose t-butyl groups prevent efficient intermolecular packing, was CPL-silent in solution and exhibited g _lum = 7.6 × 10^–3 in the dispersed microcrystalline state, though showing a helical but less compact supramolecular arrangement in the X-ray structure.

A similar approach was employed recently by Niu et al. to rationalize solid-state CPL properties of a 1,1′-binaphthyl-based AIEgen. The solvent polymorphs of A8 (Chart ) crystallized from THF/CH₃OH and CDCl₃/CH₃OH exhibited different packings and oppositely signed solid-state CPL spectra, which were reproduced by TD-DFT calculations (B3LYP-D3BJ/6-31G­(d,p)) run on two molecules of A8 and two solvent molecules comprising the unit cell. CDCl₃ molecules, in particular, were responsible for a large induction ability in the transition dipole moment of the AIEgen.

Liu’s group characterized stimuli-responsive supramolecular systems composed of phenylalanine derivative A9.1 (Chart ) as a chiral inducer and various fluorescent moieties, including achiral tetraphenylethene-fluorene derivative A9.2 and triphenyl-1,3,5-triazine derivative A9.3. In both cases, the binding between the chiral inducer and the achiral fluorophore occurs through hydrogen bonding, and then aggregation is mediated by π-stacking, resulting in helical stacks with specific properties. The 1:2 mixture between A9.1 and A9.2 forms spherical nanoparticles which evolve into larger assemblies after water addition, with a complex dynamic process occurring over 24 h. Since the nanoassemblies are emissive (λ_em = 470–480 nm) and CPL-active, the dynamics could be followed by CPL, which reached its maximum g _lum ≈ 0.11 for the latest aggregation stage, namely, μm-sized crystals. The 1:1 mixture between A9.1 and A9.3 also gives rise to multiple aggregated phases, depending on the water content of H₂O/DMSO solvent mixtures. A gel phase and a crystalline phase displayed g _lum = −0.04 and +0.16, respectively. Single molecules and dimers of A9.2 and A9.3 were considered for TD-DFT calculations at the B3LYP/6-31G­(d) and B3LYP-D3BJ/6-311G­(d) levels, respectively, but these minimal models did not produce the expected large dissymmetry values. In helical supramolecular aggregates of (fluorescent) dyes with long-range order, large dissymmetry values are often associated with a large coherence length (exciton delocalization). ,

Wang, Liu, and co-workers reported an intriguing phenomenon of wavelength-dependent emission and CPL spectra for a series of tartaric acid acylhydrazones (the parent compound is A10, Chart ) which self-assemble into helical nanoribbons forming stable gels in DMSO/H₂O solvent mixtures. Even to the naked eye, the gels were either blue- or green-emissive depending on the irradiation wavelength. The same behavior was reproduced by fluorescence spectroscopy on gels and suspended fibers: with 280–300 nm excitation, the main emission band was centered at >500 nm, whereas with 365–380 nm excitation, the band was blue-shifted to around 450 nm (Figure ). When the gels of A10 derived from l-tartaric acid were excited at 300 nm, they displayed a positive CPL band centered at around 520 nm (extrapolated g _lum ≈ 7 × 10^–3), but when the excitation was shifted to 375 nm, a weaker negative CPL band was observed at around 470 nm (Figure ). CPL activity was certainly related to supramolecular chirality, as DMSO solutions of A10 were CPL-silent. The photophysics of A10 and related compounds is rather complicated, as it depends on the combination of an ESIPT process (which is triggered by excitation of <370 nm and emits at 518 nm; Scheme ) and an excimer emission (which is triggered by excitation of >375 nm and occurs at 467 nm; Figure ). TD-DFT calculations at the ωB97X-D/6-31G­(d) level were carried out to characterize the three relevant excited-state phenomena (absorption by the GS phenol tautomer, emission by an ES keto tautomer obtained after ESIPT, and emission by an ES excimer pair). The resulting CPL signs for the two emission processes were predicted to be opposite. However, a single geometry was employed, neglecting the possibility of different conformations and/or different aggregate geometries.

69.

(a,b) Emission and CPL spectra measured on gels of A10 (Chart ) with two excitation wavelengths. (c) Proposed mechanism of excitation-dependent emission of the A10 gel involving GS and ES intramolecular proton transfers. Adapted with permission from ref . Copyright 2022, John Wiley and Sons.

4. Illustration of Ground-State and Excited-State Proton Transfer Processes (ESIPT and GSIPT) in Tartaric Acid Derivative A10 .

^a The transferred H is shown in bold.

A complex macromolecular system made of hyperbranched chiral poly­(fluorene-2,4,7-triylethene-1,2-diyl) derivatives such as A11 (Chart ) was described by Nakano and co-workers. The polymers act as chirality inducers toward small aromatic molecules such as naphthalene, anthracene, pyrene, and so on in mixed thin films obtained by drop casting. While the thin films of pure polymers displayed moderate CPL activity with g _lum = 1 × 10^–3 at 500 nm, the blends showed stronger additional CPL activity at shorter wavelengths, in correspondence with the transitions of the aromatic dyes. Regrettably, the energy transfer from dye to polymer was not efficient or complete. The most efficient film was prepared from the blend of A11 and anthracene, showing g _lum = 3 × 10^–2 at 450 nm. This was then modeled by a system composed of a model trimer (A12) and two anthracene molecules. BOMD was then run on the S₁ state at the ωB97X-D/6-31G level. While the model is rather small compared to the real system, it yielded the expected CPL sign, while the calculated g _lum values were not provided.

A CPL-active AIEgen based on two TPE moieties linked to a pillar[5]­arene was designed by Han, Tang, and co-workers. The monomeric species (A13, Chart ), which emits around 500 nm but is CPL-silent, forms a supramolecular coordination polymer by binding Ag⁺ ions, which shows a 5-fold increase in emission and a CPL signal with g _lum = 1.3 × 10^–4. However, the most pronounced effects were observed upon aggregation, which boosted both the QY (Φ_f ≈ 0.5) and the CPL, reaching g _lum = 3.7 × 10^–3 in thin films cast from the Ag⁺-coordination polymer and 3.9 × 10^–3 for solution aggregates in DMSO/H₂O 2:8 solvent mixtures. The polymer brightness was estimated as B _CPL = 11.2 M^–1 cm^–1. TD-DFT calculations at the ωB97X-D/def2-SVP level were run on monomer A13 and on an Ag⁺-bound dimer, in both cases using a single conformation. For the S₁–S₀ transition, the dimer showed a 1.8-fold increase in the calculated g _lum (from ∼1.4 × 10^–3 to 2.44 × 10^–3). The simplification inherent in this approach is evident.

More recently, Wang et al. rationalized CPL properties of blends of an achiral polyfluorene with limonene as a chiral inducer and 5,10,15,20-tetrakis­(4-bromophenyl)­porphyrin as the emitter moiety by running TD-DFT calculations on simplified aggregate models.

Theoretical investigations aiming at the description of collective chiroptical properties of large nanosized aggregates must clearly abandon atomistic simulations in favor of more computationally efficient approaches. As discussed in Section , Spano and co-workers provided a theoretical background for the simulation of ECD and CPL spectra of aggregated systems within the vibronic Frenkel exciton theory valid for the intermediate or strong electronic–vibrational coupling regime. This approximate treatment offers, among other things, a picture of the dependence of dissymmetry values on the exciton coherence size in chiral aggregates, spatial correlation length, site-correlated disorder, and long-range interactions. In all cases, the comparison with experiment focused on aggregates of a chiral oligo-(p-phenylene-vinylene) (A14, Chart ), a system well known to form helical stacks of hydrogen-bonded dimers in solution, extensively studied by Meijer and co-workers, and showing a wavelength-dependent emission dissymmetry with maximum g _lum = 3.5 × 10^–2 at 512 nm.

Although Frenkel exciton theory can be invoked in several circumstances concerning aggregate CPL emission, we are aware of only a few explicit calculations of aggregate CPL spectra within this theoretical framework. Krausz and co-workers recorded CPL spectra at 2 K for the CP43 proximal antenna protein of photosystem II, which showed a broad negative band centered at 683 nm with g _lum ≈ 8 × 10^–3. The CPL intensity decreased by increasing the temperature to 120 K, as a consequence of the change in excited-state populations. The protein contains 13 chlorophyll a (Chl-a) pigments, which are chemically identical but interact differently with the protein environment, causing a dispersion of absorption peaks (site energies) in the Q _y -band region between 650 and 690 nm. ECD and CPL spectra are therefore the results of multiple exciton couplings between Chl-a units. By means of Frenkel exciton theory including exciton–vibrational coupling, the authors could reproduce the sign, shape, and temperature-dependent intensity of the CPL signal. Morisaki, Chujo, and co-workers applied a qualitative exciton model to rationalize the chiroptical response to the self-assembly of two X-shaped [2.2]­paracyclophane-OPE derivatives (O3b and O10, Chart ). Thin films of these compounds, prepared by various deposition techniques, were decently emissive (maximum Φ_f ≈ 0.55 for drop-cast, nonannealed films of O10) and exhibited extraordinarily large dissymmetry values, especially after annealing: the thermally annealed drop-cast films reached g _lum = 0.25 for O3b and 0.27 for O10, among the largest observed for self-assembled organic compounds. The Frenkel exciton decomposition analysis (FEDA) introduced in Section was employed by Shiraogawa and Ehara for a theoretical study of 2,5-diphenyl-1,4-distyrylbenzene in its crystal form (A15a, Chart ). Compound A15a is achiral and crystallizes in the achiral P2₁/c space group; therefore, crystals are ECD- and CPL-silent. However, the crystal lattice contains pairs of skewed chromophores. FEDA calculations were run on partial structures using CAM-B3LYP/6-31+G­(d,p) to evaluate the QM parameters. They reproduced zero-CPL for the crystal but nonzero terms for specific couplings. With the guidance of these results, the authors conceived a modification of the skeleton of A15a into A15b and A15c, able to provide “asymmetric” multichromophoric systems. By substituting a molecule of A15a with A15c in the crystal packing, a strong CPL signal was predicted.

The FEDA approach was further employed by Genet, Painelli, and co-workers to rationalize chiroptical properties of an achiral cyanine dye (A16, Chart ) capable of forming chiral CPL aggregates in thin films via a multistep process. First, a herringbone packing produces a tube characterized by a single red-shifted ECD band above 600 nm and an almost superimposed CPL band of the same sign, allied with the π–π* cyanine transition whose EDTM is oriented along the long axis (Figure a). Second, the tubes arrange themselves into bundles showing ECD and CPL couplets due to the exciton coupling between helically packed tubes (Figure b). After a quantum modeling of the chromophore with TD-DFT methods, FEDA calculations reproduced both spectral shapes and the large g _lum ≈ 8 × 10^–2 measured for the bundles. The unusual bisignate appearance of the CPL spectrum was explained by invoking an anti-Kasha emission from the upper excitonic state, separated by only 10^–3 eV from the lower state and then populated at RT. Such small excitonic splitting is compensated for by the large EDTM typical of cyanine dyes in producing large exciton-coupled ECD and CPL signals.

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Calculated absorption (Abs), emission (Emi), ECD, and CPL spectra for tube (a) and bundle (b) aggregates of cyanine A16, using FEDA calculations. Reprinted with permission from ref under a CC BY-NC 3.0 license.

Although the last examples demonstrate how computational simulations may drive crystal and material engineering toward efficient CPL emitters, at the end of this subsection we wish to observe that in the crucial context of CPL-active supramolecular aggregates, calculation approaches appear underrated both from the viewpoint of the interpretation of experimental data and as predictive tools. While the complexity and size of supramolecular systems are certainly major obstacles in this context, they alone do not fully account for the limited number of publications, especially considering the availability of efficient calculation methods like FEDA. We believe that a paradigm shift is needed in the field of CPL-active materials, moving away from the commonly encountered trial-and-error approach toward a more rational methodology based on quantitative predictions.

3.3. Metal Complexes

As mentioned in the Introduction, metal coordination compounds are among the most efficient CPL emitters as nonaggregated single molecules. This is true for several transition-metal (TM) complexes, especially Ru^II, Ir^III, Pt^II, and Cr^III, but it is dramatically evident for Ln^III (Ln = lanthanide) complexes. ,, For the two families, intraconfigurational d–d and f–f transitions are possible, which correspond to electric-dipole-forbidden and magnetic-dipole-allowed transitions according to the Laporte rule (valid for centrosymmetric coordination geometries). Thus, large magnetic transition moments m ₀₁ together with relatively small electric transition moments μ ₁₀ are observed, yielding in some cases huge g _lum values. As a matter of fact, the highest reported g _lum for a single molecule belongs to an Eu^III complex, namely, Cs–Eu^III tetrakis­((+)-3-heptafluoro­butylyrylcamphorato), with its g _lum = 1.38 for the ⁵D₀ → ⁷F₁ transition, meaning that ∼85% of the emitted photons are left circularly polarized and 15% are right circularly polarized. The downside of metal-based CPL emitters is that the prediction of CPL parameters by QM calculations is, generally speaking, much more complicated than for SOMs. − In open-shell systems, spin-forbidden transitions and excited states other than singlets are involved, often posing the problem of spin contamination. Moreover, these systems have multireference character which is hard to describe with single-reference methods like DFT. Because of the strong electron–electron interaction between unpaired electrons, an accurate description of electron correlation is mandatory, and a marked dependence on the exchange-correlation functional is expected. Heavy metals, including lanthanides, require large and flexible basis sets and the use of effective core potentials (ECP) and relativistic wave functions. Relativistic effects also give rise to significant SOC, whose explicit inclusion into transition dipole moments is necessary for accurate spectra predictions. For the above reasons, the number of reported CPL calculations on metal coordination compounds, and especially lanthanide complexes, is very limited in comparison to the amount of available experimental data. Moreover, practically all papers described below on d-block TM complexes concern closed-shell systems (Scheme ), which implies that some of the most interesting metal ions, such as Cr^III, are missing among those considered for QM calculations of CPL properties.

5. Scheme of d-Block Orbitals for Selected d ⁿ Configurations.

In both closed-shell and open-shell TM complexes, the effect of SOC is to remove the degeneracy of nonrelativistic triplet states (e.g., T₁) into three magnetic sublevels T₁(0) and T₁(±1) (corresponding to M_S = 0, ±1), which combine into three SOC states |Ψ_1–3⟩ in the relativistic limit. Emission from the three states to the S₀ state follows a selection rule analog to the Faraday C-term in MCD: CPL arises from (±1) components of the parent nonrelativistic T₁ state with opposite signs, while the (0) component is CPL-silent (Scheme ). This section covers first metallohelicenes (Section ) and other transition-metal complexes (Section ) and then lanthanides (Section ).

6. Splitting of the Triply Degenerate T₁ State into Three Sublevels under the Action of SOC.

3.3.1. Metallohelicenes

Metallohelicenes are a family of compounds with especially interesting properties, including strong chiroptical signals, enhanced emission, CPL activity, redox switching, magnetic properties, and so on. The first calculation of the CPL properties of a metallohelicene was reported in 2014 by Autschbach, Crassous, and co-workers. They investigated a series of platinahelicenes including the monoplatina[6]­helicene M1 (Chart ). , Square-planar Pt^II complexes have a closed-shell d ⁸ configuration (Scheme ). Due to the heavy atom effect, SOC and hence ISC are boosted and allow for triplet states to be populated and give rise to phosphorescence emission. For compound M1, the emission maximum was observed at λ_em = 644 nm at RT, to be compared with λ_em = 430 and 453 nm for the ligand L1 (Figure ; notice that this ligand is a [6]­helicene). Calculations were run on L1 and M1 at the BHLYP/SV­(P) level, using the TD-DFT method on S₀ and S₁ states and the TDA method on T₁ states. For these, the energy was reproduced satisfactorily well, while the S₁ energy of M1 was severely overestimated. The structural similarity between the optimized geometries of S₁ and T₁ states favors a rapid intersystem crossing (ISC) in the complex. The CPP spectra of the enantiomers of M1 showed the expected mirror-image relationship and had g _lum,p = +1.3/–1.1 × 10^–2 (Figure ), with a >10-fold increase with respect to that of ligand L1 (g _lum = ±8 × 10^–4). This trend was not reproduced by TD-DFT calculations, which provided a much larger g _lum value for the S₁–S₀ transition of L1 than for that of M1. It must be stressed that a more recent measurement yielded a smaller g _lum,p = 3 × 10^–3 for M1, more in line with calculated values by Autschbach and Crassous. The authors also ran spin–orbit CASSCF and CASPT2 calculations on the T₁ state, which predicted for M1 nonzero rotational strength for the T₁–S₀ emission.

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Experimental total luminescence (bottom panels) and CPL (top panels) measured for ligand L1 and its complex M1 (Chart ). Red dots, (M)-configuration; black dots, (P)-configuration. Reprinted with permission from ref . Copyright 2014 The Royal Society of Chemistry.

One year later, the same research groups investigated a second monoplatina[6]­helicene (M2, Chart ) obtained from (2-pyridyl)-4-aza[6]­helicene ligand (L2). Complex M2 displayed red phosphorescence at RT (λ_em = 547 nm, τ_p = 8.2 μs) and no fluorescence emission, accompanied by CPL with g _lum,p = 1 × 10^–3, 1 order of magnitude smaller than the initially measured value for M1 but also smaller than that of ligand L2, having g _lum ≈ 3 × 10^–3 at 426 nm. TD-DFT calculations were run at the BHLYP/SV­(P) level on M2, L2, and their protonated forms, evaluating both S₁ and T₁ excited states. Because of the missing SOC terms, intensities and rotational strengths could not be evaluated for the T₁–S₀ emission. However, the rotational strengths calculated for S₁–S₀ parallel the observed g _lum trend.

More recently, Autschbach and co-workers ran a further theoretical investigation on monoplatina[6]­helicene M1 together with analog monoplatina[7]­helicene M3 plus other Ir^III complexes to be described below, extending TD-DFT theory to treat spin-forbidden transitions. This was possible by means of a two-component relativistic TD-DFT approach (2c-TD-DFT) , based on ZORA. With respect to the perturbative approach introduced by Ruud, Coriani, and colleagues (Section ), 2c-TD-DFT includes an electron spin contribution to the magnetic dipole transition moment, which is expected to be crucial for compounds containing heavy metals. The calculations employed TDA with the PBE0 hybrid functional and a composite basis set from the Dunning-Hay family (TZ2P for Pt, DZ for H, and DZP for C, O, and N). A PCM solvent model for DCM was also included. The choice of TDA over TD-DFT was justified by the more accurate prediction of T₁ energies (Section ). Calculated g _lum,p values for (P)-M1 and (P)-M3, including SOC, were both around +2 × 10^–3, in very good agreement with experimental data (g _lum,p = +4 × 10^–3 for (P)-M3). Neglecting the electron spin contributions to the magnetic dipole transition moment resulted in smaller g _lum,p values with the wrong sign.

Using the nuclear ensemble approach, which averages calculated quantities over allowed vibrational modes, Chen et al. obtained a further calculated g _lum,p = 5 × 10^–3 for monoplatina[6]­helicene M1, which was, however, in worse agreement with the experimental values than stationary point calculations at the same B3LYP/def2-SVP level.

3.3.2. Other Transition-Metal Complexes

This section is organized according to the metal ion incorporated into the complex and will treat in order Pt^II, Ir^III, Re^I, and Cu^I species; the electronic configuration of d-block orbitals is summarized in Scheme .

Among square-planar d ⁸ complexes, two Pt^II complexes with 1,3-di­(2-pyridyl)­benzene) ligands were described by Shinozaki and co-workers (M4a–b, Chart ). Because of the small chiral perturbation exerted by the pinene moiety, these compounds displayed weak ECD spectra. Interestingly enough, emission spectra showed the contribution of monomeric species as sharp bands below 550 nm and of trimer and excimer species contributing to the broad band between 550 and 800 nm. In correspondence with this, a weak CPP signal was detected with g _lum,p ≈ 5 × 10^–4. Supramolecular assemblies mediated by Pt–Pt bonds give rise to highly luminescent networks ranging from 1D to 3D architectures. In monomeric Pt^II complexes, the lowest-energy transition is either ligand-centered (e.g., a π–π* transition) or a metal-to-ligand charge transfer transition (MLCT) which can relax to a triplet state (³MLCT). On the contrary, aggregate species feature Pt–Pt bonding involving Pt–Pt 5d _{z ²} antibonding orbitals as the HOMO (Figure ), which generate metal–metal–ligand charge transfer (MMLCT) transitions as the lowest energy ones. The same applies to the monomeric and dimeric species of M4a–b (Figure ). Dimers and trimers of model compounds M5a–b were built by placing 2 or 3 monomers parallel to each other at a Pt–Pt distance of 3.4 Å and scanning the X–Pt–Pt–X angle by 10° steps. Based on the variation of the calculated R ₁₀ and f ₁₀ with the dihedral angle (B3LYP/6-31G­(d,p)/LanL2DZ­(Pt) level), it was inferred that in the excited state the aggregates feature a twist of 160°.

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(a) HOMO and LUMO of dimer and trimer (rotated by an angle θ = 60°) of model complex M5a (Chart ). (b) Dependence of calculated oscillator and rotational strengths of the S₁–S₀ transition of the dimer and trimer of M5a. Adapted with permission from ref . Copyright 2016, John Wiley and Sons.

Two red phosphors based on platinabinaphthalenes (M6a–b, Chart ), reported by Liu, Chen, and co-workers, displayed emission and CPP both in solution (g _lum,p ≈ 10^–3 at around 640 nm) and as thin films (maximum g _lum,p = 2.6 × 10^–3). A CP-OLED device was also fabricated with EQE ≈ 2% and g _EL ≈ 10^–3. Nonrelativistic TD-DFT calculations were run at the B3LYP/6-31G­(d)/LanL2DZ­(Pt) level. Calculated g _lum values for the S₁–S₀ transition were around 10^–3. The authors stressed the correspondence with experimental g _lum,p values, which, however, refer to the T₁–S₀ transition.

C ₂-symmetric, helically shaped Pt^II complex M7 (Chart ) reported by Morisaki, Inoue, and co-workers displayed bright-red phosphorescence as a broad, structured band between 600 and 800 nm. The corresponding CPP signal had g _lum,p = 2.6 × 10^–3 of the same sign as the red-most ECD band; the dissymmetry value was 1 order of magnitude larger than for smaller analog M8. For the quantitative prediction of CPP, a three-function model for the T₁ state was employed that allowed for all possible interconversions among the three sublevels and their emission to S₀ (Figure ). Calculations were run with the CASSCF method with SOC, using a relativistically contracted (RC) Ahlrichs’ def2-SVP basis set for C, H, and N atoms and a segmented all-electron RC basis set (SARC) for Pt. The estimated relaxation rate of the highest-energy SOC sublevel (SOC₃) was much higher than for the other two sublevels, thanks to the contribution of the singlet component, closer in energy. Hence, SOC₃ was the dominant species in phosphorescence and CPP (coefficients labeled “C _n” in Figure ). The averaged calculated g _lum,p = 2.3 × 10^–3 was in exceptionally good agreement with the experiment. The smaller analog M8 had a similarly strong m ₀₁ as M7 and consistent θ _μm ≈ 88° but an ∼10-fold-stronger μ ₁₀ , which is again in accord with the experimental trend of g _lum,p values.

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Left: HOMO and LUMO involved in the T₁–S₀ transition of complex M8. Right: energy levels, SOC couplings, and contribution to the T₁–S₀ transition of the three T₁ sublevels with respective calculated g _lum values. Reprinted with permission from ref . Copyright 2022 American Chemical Society.

Another interesting Pt^II complex from the same research group is M9, obtained from 2-azatriptycene ligand L9 (Chart ; the numbering of ligands follows that of the complexes, and it is not progressive). The latter exhibited solvent-dependent fluorescence spectra with λ_em shifting by 172 nm when passing from cyclohexane to ACN. The large Stokes shift (ΔE_S = 1.25 eV) was consistent with a through-space ICT (TS-ICT) nature of the lowest-energy transition. CPL spectra were also solvent-dependent and reached their maximum g _lum = 3.3–3.5 × 10^–3 in DCM and ACN, of the same sign as the red-most ECD band. TD-DFT calculations run at the MN12-SX/6-31+G­(d) level with PCM for ACN afforded a calculated g _lum = 5 × 10^–3 in good agreement with the experiment. The functional employed is a screened-exchange (SX) hybrid functional of the Minnesota family, with 25% HF exchange in the short range and 0% HF exchange in the long range. This functional was also employed by Bella and co-workers in their screenings on nanographenes described in Section . Chiral Pt^II complex M9 exhibited green phosphorescence in DCM, with τ_p = 5.5 μs, accompanied by CPP with g _lum,p = 1.3 × 10^–4. A theoretical analysis like that described for M7 and M8 (Figure ) was run, leading to consistent results, that is, the leading role played by the highest-energy sublevel SOC₃ of the T₁ state. The small g _lum,p value was due to an unfavorable arrangement between transition moments, leading to θ _μm ≈ 91.4°.

Morisaki and co-workers also reported dinuclear Pt^II complex M10 (Chart ) embedding two [2.2]­cyclophane moieties. In toluene solution, it emitted as a broad band with λ_em = 656 nm and a decent CPP dissymmetry g _lum,p = 1.0 × 10^–3. Relativistic TD-DFT calculations were run using the LC-BLYP functional combined with the RC-def2-SVP basis set for C, H, N, and O atoms and the SARC-SVP basis set for Pt. Because of the C ₂ symmetry, two degenerate triplet states (T₁ and T₂) are possible, splitting into a manifold of six SOC sublevels. The brightest sublevel had a predicted g _lum,p = 3.3 × 10^–3, consistent with the measurements.

Among octahedral d ⁶ complexes, Ir^III-based phosphors have attracted much attention especially because of their applications in optoelectronics and biological and medicinal contexts. − CPL data and calculations were reported for a few cyclometalated Ir^III complexes, that is, compounds containing one or more metallacycles whereby each ligand chelates the Ir^III core via a heteroatom and a carbon atom.

The first calculation of CPL data of a series of Ir^III phosphors with a tris­(bidentate) coordination frame dates back to 2008 by Bernhard and co-workers (M11a–d and M12, Chart ). As mentioned in Section , this is regarded by some publications as the first example of a CPL calculation. In fact, the authors calculated rotational R ₀₁ and oscillator strengths f ₀₁ for the S₀–S₁ transition at the B3LYP/LanL2DZ level and extrapolated them to the T₁ state by means of an energy correction factor ΔE = (E_T1 – E_S1). The assumption beyond such an approximation was that the geometries (“and as a consequence the electronic structure”) should not be significantly different for the S₁ and T₁ states, hence the rotational and oscillator strengths should also be the same, apart from the energy correction. The provided list of calculated ΔE·R ₀₁/f ₀₁ values correlates reasonably with the observed trend in g _lum values of M11a–d and M12, all between 0.6 × 10^–3 and 2 × 10^–3 at around 500–545 nm.

By means of the same relativistic 2c-TD-DFT method described in Section , Autschbach and co-workers investigated the three cyclometalated Ir^III complexes M13–M15 (Chart ), two of which are a diastereomeric pair containing [5]­helicene N-heterocyclic carbene (NHC) ligands. The experimental CPP spectra had been reported before and showed g _lum,p = +3.7 × 10^–3 and +1.5 × 10^–3 at 530 nm for (P,Λ)-M13 and (P,Δ)-M13, respectively, and −9 × 10^–4 at 493 nm for Δ-M15, revealing that the ligand configuration controls the sign of the CPP signal. Using TDA in combination with PBE0 and the DZP/DZ­(H)/TZ2P­(Pt) basis set, g _lum,p values could be well reproduced both in sign and intensity. Similar to the case of M1 and M3 discussed above, neglecting the electron spin contributions to the magnetic dipole transition moments resulted in much smaller g _lum,p values for (P,Λ)-M13, while for the other two species the difference was smaller. A [6]-azairidahelicene Ir^III complex was also reported recently by Cossìo, Freixa, and co-workers, and its CPP was simulated by SOC-TD-DFT calculations.

Bloino and co-workers calculated vibrationally resolved phosphorescence and CPP spectra of anionic Ir^III complex M16 (Chart ), for which experimental CPP data are not available. It is interesting to notice that five DFT functionals were benchmarked within the 2c-TD-DFT approach, which led to discordant signs of g _lum,p: range-separated functionals such as CAM-B3LYP and ωB97X-D predicted the expected negative CPP sign for Λ-M16, while global hybrids such as B3LYP, PBE0, and M06-2X predicted a positive sign. The discrepancy was traced back to the MLCT character of the T₁–S₀ transition, which is not accurately described by global hybrids.

Both helicenic and nonhelicenic NHC ligands can form octahedral d ⁶ complexes of Re^I. Neese and co-workers studied a family of compounds, namely, M17a–b, M18a–b, and M19a–b (Chart ), in a work which delves into the photophysics of triplet emission in the presence of SOC and in the mechanisms of CPP intensity. The phosphorescence spectra of M18a–b and M19a–b appear as a structured band with a maximum at around 520 nm, while those of M17a–b display a broad band peaked at 510 nm (Figure ). CPP spectra varied substantially among the series, showing the most intense signals for (P,A) isomers M19a–b (g _lum ≈ 5 × 10^–3) while being almost negligible for M17a and M18a. The computational protocol was based on the following steps: (a) geometry and excited-state optimizations were run at the PBE0-D3/def2-TZVP level with def2-TZVP/J density fitting for the resolution of identity (RI); (b) TD-DFT calculations on the S₀ geometry were run with several different functionals of the GGA, global hybrid, hybrid meta-GGA, range-separated hybrid, and double hybrid type, which indicated PBE0 as the functional best matching the experimental ECD spectra; (c) relativistic TD-DFT calculations were run with PBE0, including vibronic effects within the FC and HT schemes; (d) excited-state energies were evaluated at the CCSD­(T) level; and (e) relativistic effects were computed in the framework of the second-order Douglas-Kroll-Hess correction (DKH2), using the DHK-def2-TZVP basis set for the main elements and SARC-DHK-TZVP for Re and I. The frontier MOs of compounds M19a–b, providing the strongest CPP signals, are shown in Figure . The T₁ state has major ³ILCT (intraligand CT) character of the π–π* type, mainly located on the helicene moiety. In the ideal C _3v ligand field geometry, this transition belongs to irrep A₂, and it is electric-dipole-forbidden and magnetic-dipole-allowed along the z axis (OC–Re–X direction). In the relativistic limit, the nonrelativistic T₁ state (with ideal A₂ symmetry for M19b) splits into three magnetic sublevels T₁(0) and T₁(±1) (Scheme ), which combine into three degenerate SOC states |Ψ_1–3⟩, each with its own (0, ±1) components. The emission spectrum is due to |Ψ₃⟩ → |Ψ₀⟩ and |Ψ_1,2⟩ → |Ψ₀⟩ emission processes, but only the former one makes a detectable contribution to the CPL signal because of a favorable orientation of electric and magnetic dipole transition moments, both of which have components along the x, y, and z axes and define an angle θ _μm ≈ 38° (Figure ). The calculated |g _lum| = 0.13 is the highest among the series, in keeping with the observed trend; however, it overestimates the measurement by a factor of ∼30 and apparently provides the wrong sign for (P,A)-M19b. We notice that the experimental g _lum values reported by Neese and co-workers differ by 1 order of magnitude with respect to the original source. For the CPL-silent M17b, the emitting state is the ideally degenerate ³E (T₁,T₂) state with mixed MLCT/XLCT (halogen-to-ligand CT) character. The two (quasi)­degenerate transitions contribute to the emission spectrum with opposite polarizations, and the resulting CPL signal is negligible, in accordance with the experiment.

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Experimental emission (PL) and CPL spectra and GS structures of complexes M17a–b, M18b, M19a (two isomers), and M19b (Chart ). Descriptors A and C refer to the chirality at the metal. Reprinted with permission from ref . Copyright 2023, AIP Publishing.

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Energy levels and MO diagram of complexes M19a–b adapted in an “ideal” C _3v symmetry around the Re centers. MOs shown for M19a; acronyms for transitions are defined in the text. Reprinted with permission from ref . Copyright 2023, AIP Publishing.

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(a) Experimental and calculated phosphorescence spectrum of M19a with magnetic sublevel contributions. (b) Sublevel diagrams and computed relaxation times. Reprinted with permission from ref . Copyright 2023, AIP Publishing.

In 2025, Srebro-Hooper and Crassous investigated a different helicene-type Re^I complex M20 (Chart ), hosting a [5]­helicene-pyrazino-phenanthroline ligand, which exhibited moderate CPL activity at RT (g _lum = 2.3 × 10^–3 at 507 nm) and much stronger CPP activity at 77 K (g _lum,p = 2.9 × 10^–2 at 560 nm). Similar to compounds M18a–b and M19a–b described above, compound M20 may exist in two isomeric forms (P,C)-M20 and (P,A)-M20 with similar calculated populations. The ligand and its Re^I complex exhibit similar photophysical properties, including a structured phosphorescence emission above 550 nm at 77 K. The computational analysis employed TDA both for geometry optimizations run at the LC-ωPBEh/def2-TZVP level and for vertical calculations run at the LCY-PBE0*/TZP level with a tuned γ* value. The letter “Y” in LCY denotes the use of a Yukawa potential (a Slater function) in the long-range correction scheme. A continuum solvent model was included in all calculations, and SOC was estimated for triplet modeling. The calculation confirmed a consistent S₁ state for the ligand and the complex, mainly of a ¹ILCT nature (from the helicene to the pyrazino-phenanthroline), mixed with minor helicene ¹ππ* and ¹MLCT (for the complex) components. The T₁ state has geometry and character similar to those of the S₁ state. Interestingly, the isomeric triplet structures of (P,C)- and (P,A)-M20 yielded opposite calculated CPP values, with the (P,C) isomer matching the same sign as the experiment, though with 1 order of magnitude smaller g _lum,p.

Tetrahedral d ¹⁰ complexes, with filled d-shell (Scheme ), include Cu^I complexes, which are efficient CPL emitters especially as multinuclear species and small clusters. , Tsubomura and co-workers reported three mononuclear Cu^I compounds M21–M23 (Chart ) with 2,2′-bipyridine and DIOP (2,3-O-isopropylidene-2,3-dihydroxy-1,4-bis­(diphenyl­phosphino)­butane) ligands. Experimental g _lum values for M22 and M23 were relatively weak (∼1 × 10^–4 and 2 × 10^–4, respectively). The most sterically hindered analog (R,R)-M23 showed dual CPL, with g _lum = −2 × 10^–4 at 500 nm and +9 × 10^–4 at 600 nm. Time-resolved fluorescence measurements demonstrated the presence of two emissive species for M23, a short-living one emitting at longer wavelengths and a long-living one emitting at short wavelengths. The two species were identified as two diastereomers, namely (R,R,P) and (R,R,M). Nonrelativistic DFT and TD-DFT calculations were run using the PBE0 functional and a composite basis set (6-31G­(d) for C and H, 6-31+G­(d) for N and P, and 6-311G for Cu). The authors optimized the geometries of S₁ and T₁ states, and for the latter, the paper also reports oscillator and rotational strengths. However, the transition they refer to is unclear, as SOC was not included and the T₁–S₀ transition should have negligible f and R in the nonrelativistic limit.

Zn^II complexes also belong to the closed-shell d ¹⁰ family. Helicene-type Zn^II complex M24 (Chart ), reported by Mori, Hasobe, and co-workers, combines a C ₂-symmetric helicate structure with the presence of dipyrromethene moieties. The ECD spectrum of M24 displays an exciton couplet feature between 500 and 680 nm, with a remarkable g _abs = 0.20 at 615 nm. Far-red emission was also observed with λ_em = 658 nm and Φ_f = 0.23, accompanied by a CPL signal with g _lum = 0.022 at 660 nm, meaning an unusually small g _lum/g _abs ≈ 0.1. Calculations run at the M06-2X-D3/def2-QZV level were not able to reproduce this observation, as they predicted larger g _lum than g _abs.

Ikeshita, Imai, and Tsuno reported three C ₂-symmetric Zn^II complexes with Schiff-base ligands (M25–M27, Chart ). , An equilibrium between diastereomeric species with different chirality at the metal (e.g., Λ-(S,S) and Δ-(S,S)) was detected by NMR. CPL spectra were weak in solution, with g _lum < 10^–3 for all compounds, but increased to up to 3.2 × 10^–3 for M26 when measured in the solid state as KBr pellets. For M25, an ECD/CPL sign inversion occurred both in solution and in the solid state, while for M27 the excimeric solid-state CPL had an opposite sign to that of the nonexcimeric solution one. By running TD-DFT calculations at the B3LYP/6-31+G­(d,p) level, the peculiar chiroptical behavior was attributed to the diastereomeric equilibrium. ,

As a rare example of linear d ¹⁰ complexes, Gong and co-workers reported an interesting analysis of carbene-Cu^I-amide complex M28 (Chart ). This compound is CPL-silent in solution but acquires nonnegligible CPL in the solid state, both as a powder and in the two polymorphic crystalline forms, one of which (“crooked crystal”) attained g _lum = 2.7 × 10^–2; this latter measurement, however, was not validated for the two enantiomers. TD-DFT calculations were run at the M06L/def2-SVP level following torsional energy scans along the dihydroacridine–Cu junction, revealing a large variation of calculated g _lum values both in sign and intensity, up to a maximum of 3.5 × 10^–2. The authors concluded that M28 is CPL-inactive in solution due to free ligand–metal rotation, while the restricted solid-state conformation enables strong CPL activity.

CPL-active metal nanoparticles represent a rapidly advancing field with significant ongoing development. ,, The structural complexity, collective nature of photophysical properties, and often elusive characteristics of metal-based nanostructures often hamper QM calculations. On the contrary, small metal clusters capped with chiral ligands exhibit molecular-like properties, including CPL, which can be tackled with standard computational approaches. Two examples of TD-DFT CPL calculations of Au₁₃ and Cu₄Au clusters with NHC carbene ligands appeared recently, , both employing the PBE0 hybrid functional.

3.3.3. Lanthanide Complexes

Chiral lanthanide­(III) complexes are among the most efficient CPL emitters in terms of both g _lum and B _CPL, with emission ranging from UV to NIR. ,− ,, They also offer the possibility of being tailored by changing the ligands and the metal center, often maintaining isostructurality along the lanthanide series. Thanks to these properties, chiral lanthanide complexes have found applications as bioprobes, cell trackers, and emitters in CP-OLEDs. The transitions of interest have an intraconfigurational nature between f levels (f–f). Besides the interelectronic repulsion, separating the L levels (10⁴ cm^–1), the J-term splitting is on the order of 10³ cm^–1 due to the strong spin–orbit coupling. Moreover, depending on the nature and symmetry of the ligands, crystal field splitting takes place among m _J levels (1–10 cm^–1). This is at odd with what happens with d-block metals, where the optical transitions occur among d orbitals whose degeneration is removed by crystal field interactions. A theory and selection rules for the optical activity of lanthanides were proposed by Richardson in 1980, and they have been successfully applied to predict the most promising transitions for CPL and other chiroptical properties in a qualitative way. Despite this, quantitative ab initio calculations are very involved, as they require at least a good description of the spin–orbit coupling and the use of a suitable multireference approach.

For such reasons, the first ab initio calculations of chiral lanthanide complexes were reported only in 2019 by Autschbach, Le Guennic, et al. These authors proposed a complete and restricted active space (CAS/RAS) self-consistent field (SCF) multireference wave function framework (Section ) together with SOC treated by state interaction (RASSI) to compute the CPL spectrum of [Eu­(DPA)₃]^3– complex (Ln1, Chart ; DPA = dipicolynic acid). Here, we shall focus on the two most prominent chiroptical features of Eu^III complexes, namely, ⁵D₀ → ⁷F₁ and ⁵D₀ → ⁷F₂ bands, giving rise to two and three nondegenerate transitions, respectively, in a D ₃ geometry (Figure ). The calculation method was able to correctly predict the emission energies, with only a minor overestimation with respect to the experimental ones. Different active spaces were tested for CPL calculations. CAS­(6,7)­PT2-SO was used at first, yielding two opposite contributions for the ⁵D₀ → ⁷F₁ intraconfigurational transition. A strong modification of the CPL spectra was observed when introducing either ligand-based π/π* orbitals (by using RAS­[12,2,2,3,7,3]­CI/PT2-SO) or Eu 5d orbitals in the active space (RAS­[6,0,2,0,7,5]­CI/PT2-SO) (Figure ). As expected, the hypersensitive transition ⁵D₀ → ⁷F₂ is heavily affected by the chosen active space, with its magnitude (EDTM intensity) significantly increasing with respect to the CAS­(6,7) approach (∼18 and ∼60 times, respectively, when π/π* and 5d orbitals are considered). Because Ln1 is dynamically racemic, no direct CPL data is present in the literature. However, the comparison of the calculated CPL data for Ln1 with the spectra of an adduct of Ln1 with dimethyl tartrate allowed the authors to conclude that better agreement is reached when 5d orbitals are considered. This would imply a more prominent role of static versus dynamic coupling in determining the CPL of Ln1, even though a larger active space would be necessary for quantitatively more accurate results.

33.

77.

Major electronic level and emission pathways for selected Ln^III ions.

78.

Calculated emission and CPL spectra of Λ-[Eu­(DPA)₃]⁻ (Ln1, Chart ) for the ⁷F_J ← ⁵D₀ (J = 0–4) transitions obtained at various calculation levels. Reprinted with permission from ref . Copyright 2019 American Chemical Society.

Shortly afterward, Le Guennic et al. employed a similar approach to calculate the CPL of an Yb complex with dipicolinate-type ligands (Ln2a, Chart ). Yb^III has a f ¹³ electronic configuration, giving only one possible intraconfigurational transition (²F_5/2 → ²F_7/2, Figure ), falling in the 950–1050 nm region. As the complex is slightly distorted from ideal D ₃ symmetry, the emitting and ground states are split into three and four m _J levels, respectively. The calculations, carried out on model Ln2b at the RASPT2 level, which takes into account the 5d electron mixing, were able to qualitatively reproduce the sign alternation observed in the experimental CPL spectrum. An analysis of the calculated spectrum also allowed the authors to highlight the contribution of the transitions stemming from higher m _J levels (1′ and 2′) of the excited ²F_5/2 state.

More recently, CPL calculations were carried out on heteroleptic complexes, where usually the luminescence sensitization role is performed by an achiral ligand (typically a β-diketonate) and chiral induction is obtained with a chiral neutral ancillary ligand. In an example, Le Guennic’s group calculated the CPL of Eu complex Ln3 (Chart ) at the SA-CASSCF level. In this case, the calculations yielded a major overestimation of the energies, but some qualitative agreement between the experimental and calculated CPL spectrum was obtained for the magnetic character ⁵D₀ → ⁷F₁ transition. The CPL associated with other transitions is much weaker, not allowing for a comparison between the experimental spectrum and calculated spectrum.

An interesting example is provided by the calculations on two Yb complexes, Ln4 and Ln5 bearing helicene-derived ancillary ligands. Ln4 is found as a monomeric compound in the solid state, while Ln5a–c are simplified structures of the synthesized complex which forms an extended 1D coordination polymer. SA-CAS­(13,7)­PT2/RASSI-SO calculations provided reasonable agreement with the main feature of the experimental CPL spectra at 975 nm. The computational analysis of the transition shows that at 298 K the transitions stemming from the 0′ m _J level contribute 59% to the overall CPL spectrum. The same calculations provide a 1.3-times more intense rotational strength for Ln4, in qualitative agreement with the experimental results. On the other hand, the effects of the supramolecular packing of the coordination polymer cannot be simulated at the state of the art.

Similarly designed Eu complexes (Ln6–Ln8, Chart ) were studied by Sigoli et al. with a CASSCF/NEVPT2+QDPT approach in two different solvents (ACN and DCM). The solvent effect was introduced by using the universal continuum solvation model (SMD). Despite this approximate treatment, surprisingly the solvent effect on the calculated oscillator strengths was very pronounced (up to ∼236 times in the case of the ⁵D₀ → ⁷F₁ transition of Ln8). On the other hand, both the experimental and calculated g _lum values showed a much less pronounced variation with the solvent. We note that for all of the compounds the correlation between experimental and calculated g _lum values is excellent, which is rather unusual for lanthanide complexes for several reasons outlined above. This outcome may partially come from error compensation in the calculations of the oscillator and rotational strengths. A decomposition of the electric (μ ₁₀ ) and magnetic (m ₀₁ ) transition moment suggests, as expected, a large predominance of the latter in the ⁵D₀ → ⁷F₁ transition. In the case of ⁵D₀ → ⁷F₂ transition, the m ₀₁ contribution is instead small but still significant, in line with the low-symmetry coordination geometry, especially for Ln6, which would justify its rather large g _lum (−0.091) for such a transition type.

Three tetrakis camphorate complexes of Eu^III (Ln9a–c, Chart ), with tetraalkyl ammonium counterions, were reported by Hasegawa and co-workers. The complexes exhibit extremely large g _lum values for the ⁵D₀ → ⁷F₁ transition, up to 1.54 for Ln9a. Curiously, the paper also reports the simulation of “CPL properties [by] TD-DFT calculations [···] performed using structures optimized at the ground state”. Apart from the inadequacy of TD-DFT calculations in the case of lanthanide complexes, vertical excitations computed on S₀ geometry would obviously provide absorption and ECD spectra rather than emission and CPL spectra.

The CPL properties of the ⁴G_5/2 → ⁶H_J (J = 5/2, 7/2, 9/2; Figure ) transitions of Sm^III complexes Ln10 and Ln11 (Chart ) were studied by Grasser and Le Guennic. Several active spaces were tested with rigid cryptate complex Ln11 by using SA-CAS and SA-RAS SCF/RASSI-SO, and eventually SA-RAS­[7,1,1,1,7,1]­SCF/RASSI-SO, mixing 4f with ligand π/π* orbitals, was chosen, even though it was not possible to disentangle dynamic versus static coupling contributions to the total CPL spectrum. In all of the calculations, it was necessary to consider all of the transitions stemming from the three close-in-energy Kramers doublets (m _J = 0′, 1′, 2′) of the excited ⁴G_5/2 state, which are all significantly populated at room temperature (51, 35, and 14%, respectively, for Ln11). The CPL transition to the ⁶H_5/2 state is well reproduced in the case of Ln11, while less satisfactory agreement is found for the transition to ⁶H_7/2, probably due to an erroneous description of a single m _J state. On the other hand, SA-CAS­(5,7)­SCF/RASSI-SO calculations on Ln10 reproduce fairly well the CPL of the ⁴G_5/2 → ⁶H_7/2 band and highlight the fact that the transitions from m _J = 0′, 1′ on the low-energy side of the band do not lead to mutual cancellation but sum constructively, leading to the high g _lum factor observed at 595 nm (∼0.18). In general, these calculations allow one to trace temperature-dependent CPL spectra by taking into account the Boltzmann populations of the m _J levels of the emitting ⁴G_5/2 state.

Very recently, Avarvari et al. prepared Eu^III tris­(hexafluoro­acetylacetonato) complexes bearing 2,6-bis­(pyrazol-1-yl)­pyridine (bpp) camphor ligands, namely, bis­[(4S,7R)-camphor-2,2-bpp] (Ln12, Chart ) or its regioisomer bis­[(4S,7R)­camphor-1,2-bpp] (Ln13). Interestingly, these two complexes show almost mirror-like CPL spectra for the same absolute configuration of the camphor moiety. The CPL spectra were computed using the same procedure described above at the SA-CAS­(6,7)­SCF/RASSI-SO level of theory. The main features of the CPL spectrum of Ln12 were well reproduced, with the exception of the 0′ → 1 and 0′ → 3 transitions of the ⁵D₀ → ⁷F₂ manifold. The agreement is slightly less satisfying for higher-energy manifolds (⁵D₀ → ⁷F₃, ⁵D₀ → ⁷F₄), where multiple transitions overlap. As expected, the calculations revealed that the ⁵D₀ → ⁷F₁ transition is dominated by the magnetic dipole, while in ⁵D₀ → ⁷F₂, the EDTM and MDTM contributions are similar. Finally, given the suitability of such a computational procedure for this type of systems, the same method was also applied to compute the CPL spectrum of Ln13, for which the X-ray structure was not available. The issue was solved by allowing variations of the coordination geometry to maximize the match between the calculated and experimental CPL spectrum.

4. Key Concepts for the Rational Design of CPL-Active Materials

The aim of the present section is twofold. First, based on the survey presented in Section , we will recapitulate the efficiency of various calculation methodologies with the purpose of identifying the best practices to calculate CPL data in an accurate way. Section will try to answer the practical question as to which is the most convenient way of predicting CPL properties for specific or broad families of compounds. In Section , we will critically assess the current possibility of in silico approaches of aiding the research for more efficient CPL emitters by focusing on different structural or electronic properties which can be screened. From an empirical viewpoint, the available strategies for enhancing CPL properties were debated in several reviews or perspectives. ,,,,

4.1. Choice of Calculation Method

The very large majority of CPL calculations we found in the literature made use of TD-DFT. If we exclude open-shell metal complexes, in particular, lanthanide complexes, for which the use of a multireference calculation method is demanded, then the number of purely organic compounds for which CPL parameters have been evaluated with other methods than TD-DFT is very limited. A few cases concern CC2 calculations on helicenes , and CASSCF calculations on [2.2]­paracyclophanes, that is, systems for which TD-DFT, with the correct choice of functional, has been demonstrated to perform very well. In the above quoted examples, a direct comparison with TD-DFT results was not provided. In the widely discussed seminal paper by Mori and co-workers on multiple helicenes, the use of CC2 is well justified given the predictive nature of the study. For in silico screening of a limited set of potential CPL-active compounds, for which experimental data are not yet available, opting for a computationally demanding but accurate method is a rational choice. The literature also offers only a few examples of the direct comparison of CPL calculations run with TD-DFT and other methods. For small molecules such as ketones, we already noticed that the performance of TD-DFT is not worse than that for methods such as CC2, CCSD, CASSCF, and CASPT2 (Section and especially Figure ), although at least for B3LYP an error cancellation was invoked. ,, For the two biaryls B14a and B15a (Section , Chart ), CC2 outperformed CAM-B3LYP in the prediction of g _lum intensity. To the best of our knowledge, axially chiral BODIPY dimer BX2 (Section , Chart ) is the only case where TD-DFT failed to predict the correct CPL sign with any functional employed (M06-2X, CAM-B3LYP, and ωB97X-D), while the SCS-CC2 level of theory recovered the correct sign. This is, however, a rather peculiar example of an intrinsically weak and nonrobust CPL band.

Focusing on the choice of the density functional for TD-DFT calculations, CPL calculations align with the general principle that identifying the best-performing functional for each specific compound or class of compounds is more effective than searching for a single universally applicable functional. A thorough benchmark of density functionals for CPL calculations of an extended data set has never been produced, by analogy to the many existing ones for vertical transition energies, ,, oscillator strengths, , transition dipole moment direction, excited-state geometries, ,, and so on. As a matter of fact, while several reviews or best-practice indications on ECD calculations are available, , especially for the structural elucidation of natural products, ,,, benchmarks of TD-DFT and other electronic structure methods for rotational strengths run on extended molecular data sets are also rare, mostly focused on model molecules rather than real-life molecules, and also outdated. , In our literature survey, we encountered a few instances where different density functionals were benchmarked against the prediction of CPL properties of a specific compound. They include a pair of carbo[7]­helicene/helicenoids (H15a and H21, Chart , Section ); oxa­[n]­helicenes H43–H45 (n = 5, 7, 9, Chart , Section ); BODIPY derivative BX16 (Chart , Section ); macrocycle MC1b (Chart , Section ); a pyrene-appended catenane (P11, Chart , Section ); and fullerene derivative N15 (Chart , Section ). As noticed above, many ketones, diketones, and α,β- or β,γ-unsatured ketones were tested for several different calculation methods, but density functionals were limited to B3LYP and CAM-B3LYP (Figure , Section ). ,,,, Bella and co-workers ran the most extensive benchmark of which we are aware, by testing 10 different functionals on 14 chiral nanographenes (N1, N2a, and N3–N14, Chart , Section ). Apart from this latter example, no broad comparison of different calculation methods is available to date in the literature as far as CPL properties are concerned.

The most frequently used DFT functionals for each family of compounds treated in Section are listed in Table , together with the best-performing functional, if apparent from the literature. Quantitative data for some families of compounds, for which relatively large data sets are available, have been provided in Section . These include ketones and diketones (Section , Table ), helicenes and helicenoids (Section , Table and Figure ), biaryls (Section , Table and Figure ), and [2.2]­paracyclophane-OPE conjugates (Section , Table ). All available data on signed RE for calculated versus experimental g _lum values and g _lum/g _abs ratios, also comprising the just mentioned families, are collected in Figure .

10. List of the Most Frequently Employed DFT Functionals Emerging from Our Survey for Each Family of Compounds.

Family	Document section(s)	Most used functional(s)	Best performing functional(s)
1. Ketones and diketones	3.1.1	CAM-B3LYP, B3LYP	CAM-B3LYP
2. Helicenes and helicenoids	3.2.1.1–3.2.1.2	B3LYP(-D3), CAM-B3LYP	B3LYP
3. Double and multiple helicenes	3.1.2.3	B3LYP(-D3), CAM-B3LYP	CAM-B3LYP
4. Biaryls	3.1.3	CAM-B3LYP, LC-PBE0*	CAM-B3LYP
5. Pyrene and perylene derivatives	3.1.4	CAM-B3LYP, PBE0	PBE0
6. [2.2]Paracyclophanes	3.1.5	B3LYP(-D3), CAM-B3LYP, M06-2X	M06-2X
7. Tetra-coordinated boron compounds	3.1.6	B3LYP, mPW1PW91	(No clear trend)
8. π-Conjugated oligomers	3.2.1	MN15	MN15
9. Macrocycles	3.2.2	B3LYP(-D3)	PBE0, B3LYP(-D3), CAM-B3LYP
10. Nanographenes	3.2.3	B3LYP	B3LYP
11. Closed-shell metal complexes	3.3.1–3.3.2	BHLYP	(No clear trend)

^aData only for singlet emissions included.

79.

Summary of literature data on TD-DFT CPL calculations on single (nonaggregated) organic molecules and closed-shell metal complexes. (a) Box plots of signed relative error between calculated and experimental g _lum for different functionals and any basis set; the boxes indicate a 25–75% interval, and the whiskers represent the range (×1.5) of the data outside the IQR. (b) Same as (a) for the g _lum/g _abs ratio. Numbers in italics indicate the number of entries.

The present statistical analysis includes data for S₁–S₀ transitions of single (nonaggregated) organic molecules and closed-shell metal complexes. It does not comprise supramolecular systems (Section ), triplet emitters (Section ), open-shell transition-metal complexes (Sections and ) where emission occurs from d–d transitions, and lanthanide complexes (Section ). Also excluded are the few calculations resulting in wrong predictions of the CPL sign as well as systems endowed with multiple excited-state conformers. The whole data set is composed of 297 entries for g _lum values and 112 entries for g _lum/g _abs ratios. The subsets with <5 entries per functional were excluded, whereas related functionals were grouped together: B3LYP-D3 with B3LYP-D3BJ; PBE0 with PBE0-D3 and PBE0-1/3. Among the functionals with a large number of entries (>10), B3LYP, CAM-B3LYP, and M06-2X appear to be the best-performing ones, followed by the PBE family. The good performance of MN15 is possibly biased by the fact that 21 of 26 entries refer to a homogeneous class of compounds, namely, [2.2]­paracyclophane-OPE conjugates. One of the surprises from our analysis is the bad performance observed for dispersion-corrected B3LYP (B3LYP-D3 and B3LYP-D3BJ). The use of a dispersion correction such as Grimme’s D3 type, with either the original zero-damping or the Becke-Johnson damping function (BJ), , is beneficial for the correct description of long-range electron correlation (London dispersion interaction) and the proper treatment of noncovalent interactions. However, since such a correction does not affect the linear response (vertical excitation energies, oscillator and rotational strengths), its impact must be sought in the prediction of the ground-state structures and only partially that of the excited-state structures: this is because the excited-state energy gradients in TD-DFT are usually computed as the sum of a ground state part and that explicitly due to excitation. The latter is not affected by the D3 protocol, which does not explicitly depend on density and therefore does not contribute to Casida’s equations and to the Z-vector term. Moreover, dispersion corrections are parametrized for the ground state. It is worth noticing that most of the data summarized in Figure were obtained by adopting the same functional for both S₁ geometry optimization and the S₁–S₀ transition. For small organic molecules, the TD-DFT geometry benchmark by Brémond et al. observed in fact that the inclusion of either the D3 or D3BJ dispersion correction did not affect the average signed and absolute errors of standard B3LYP with respect to the reference (CC3 and CASPT2). This study also points out that for small compounds with small reorganization density following an excitation (local valence singlets), B3LYP is among the best-performing exchange-correlation functionals (XCF), along with B97-D and B97-D3, which are absent from our survey as they have never been used for CPL calculations. Taken together with decent accuracy in the prediction of vertical transition energies for this specific type of excitation, , this may explain why standard B3LYP stands out as one of the best-performing functionals for CPL calculations, despite its well-known drawbacks, which have earned it an infamous reputation. Of course, this is not a general conclusion: depending on the character of the transition and the dimensions of the systems, accurate simulations need range-separated functionals like CAM-B3LYP and hybrid meta-GGA including a large fraction of exact exchange. , Turning back to dispersion corrections, a context where they might play a decisive role is in the prediction of excimer geometry. However, CPL calculations of excimer systems directly assessing the impact of D3­(BJ) corrections are scarce, and the results are not conclusive.

Summing up the dependence of the accuracy of calculated CPL data on the functional, in line with the recommended best practice for ECD calculations, we advise screening a minimum number of different functionals for each specific problem, including at least (a) a hybrid XCF such as B3LYP, PBE0, or PBE0-1/3; (b) a long-range-corrected XCF such as CAM-B3LYP; and (c) a functional from the Minnesota family such as M06-2X or MN15. Dispersion corrections should also be tested but checked against the corresponding noncorrected functional.

The number and type of basis sets employed for CPL calculations appear to be more spread than the functionals (Figure ). In general, Pople’s basis sets were the most used ones, followed by Ahlrichs’ and Dunning’s. The choice of a finite basis set unavoidably introduces some error in any calculation, yet DFT has an intrinsically weaker basis-set dependence compared to that of correlated wave function-theory-based methods. Recent literature assessing basis set performance in DFT has advocated for strict limitations on basis set selection. , However, these recommendations have since been challenged or reconsidered. , We notice that the simplest double-ζ basis sets with a minimal set of polarization functions, namely, 6-31G­(d), 6-31G­(d,p), [def2-]­SV­(P), and [def2-]SVP, whose use should be discouraged, cover 58% of all data; the percentage rises to 71% if one includes the 6-311G variants (without diffuse functions) which perform at approximately double-ζ quality. Especially infrequent is the use of diffuse functions or basis sets like def2-TZVP with flexible polarization functions.

80.

(a) Distribution of basis sets employed in TD-DFT CPL calculations on single (nonaggregated) organic molecules and closed-shell metal complexes. (b) Histogram plot of the ARE on calculated g _lum associated with different basis sets and three functionals. Numbers in italics indicate the number of entries.

To evaluate the impact of the basis set on calculated g _lum values, the dependence on the functional needs to be taken into account. Thus, in Figure b we collected the AREs for the three most frequently employed functionals (see above) and various basis sets. No clear trend emerges, apart from the expected poor performance of double-ζ basis sets and of 6-311G variants. In these cases, the addition of one set of extra polarization functions and of diffuse functions appears beneficial. Other aspects emerging from the collected data are difficult to rationalize, for example, the relatively bad performance of aug-cc-pVDZ in combination with CAM-B3LYP, especially if one considers that most data in this subset refer to small molecules (ketones and diketones, Section ). To summarize, our recommendations for the choice of the basis set for CPL calculations follow the general ones for best-practice DFT protocols: (a) use a relatively large basis set, including diffuse functions or flexible polarization functions, and (b) check for basis set convergence by varying the cardinal number (e.g., testing def2-SVP and def2-TZVP).

Turning to the role of solvent models, we must first observe that the interplay between solvent-specific polarization approaches and a TD-DFT description for excitation energies and oscillator strength is very delicate. Guido and co-workers provided an analysis including local, charge-transfer, and quadrupolar excitations and pointed out that the choice of both the DFT functional and the solvent polarization scheme has to be consistent with the nature of the studied electronic excitation. However, a conclusive analysis of the impact of solvent models on CPL calculations is still missing. Several reports described in Section made use of a polarizable continuum solvent model (PCM, Section ), most frequently in its integral equation formalism (IEF-PCM). , Other less common models are the PCM variant known as the universal solvation model for density (SMD), and the less refined conductor-like polarizable continuum models (C-PCM and COSMO). In all cases, except those described below, the linear response polarization (LR-PCM, Section ) formalism was adopted; however, it is often impossible to discern whether equilibrium or nonequilibrium calculations (Section ) were employed to estimate CPL parameters. Furthermore, no comparison was made between CPL parameters obtained in vacuo or with PCM, making the impact of LR-PCM methods impossible to assess.

Only a few reports mentioned in Section made use of a state-specific (SS) solvation approach (the external iteration scheme, see Section ). Crassous and co-workers evaluated the vertical emission energies of two carbo[6]­helicene derivatives with ICT behavior (H8a–b, Chart , Section ) by running single-point TD-DFT external iteration-PCM calculations on the S₁ state optimized at the TD-DFT LR-PCM level. The calculated emission energies for three different solvents reproduced the experimental trend much better than the LR-PCM approach alone. Also, g _lum values were in better agreement with the experiment yet observed values were still underestimated. In the simulation of CPL parameters for 1,1′-binaphthyl derivatives B16 and B17 (Chart ), Rapp and co-workers noticed a strong solvent influence on calculated g _lum values by comparing results obtained in vacuo and with the SS polarization method. In our study of carbo[7]­helicenoids H21, H22a, and H23a (Chart , Section ), we compared the results of TD-DFT calculations in vacuo with the LR-PCM and the state-specific VEM-UD model (Section ). In this case, LR-PCM appeared to overcorrect gas-phase values, reducing the calculated |g _lum| values excessively, whereas a better match with the gas phase was obtained with the VEM-UD model. The value measured later for H23a lies between LR-PCM and VEM-UD calculated values. Finally, we recall Barone and co-workers’ integrated QM/MD approach for CPL calculations as a unique example of explicit solvent treatment for CPL calculations, so far applied to ketones. In conclusion, the impact of solvation on the calculation of CPL properties remains hard to generalize based on the existing literature. While LR-polarization response models are widely accepted, state-of-the-art approaches such as the SS-polarization response and inclusion of explicit solvent molecules remain underutilized, offering opportunities for further advancement in the field.

Finally, vibronic calculations are nowadays feasible even for relatively large molecules. This is witnessed by several vibronic CPL calculations run on systems spanning from small molecules to nanographenes, as described in Section , most often in combination with TD-DFT and using Santoro and co-workers’ FCclasses code with different models (Section ). Supramolecular systems can be treated within the Frenkel exciton theory developed by Spano and co-workers (Section ), but its applications are so far limited. From the perspective of guiding the rational design of CPL emitters through in silico approaches, vibronic emission/CPL calculations appear especially useful in three different contexts:

• 1)a realistic prediction of the emission color, by simulating the whole emission envelope without recurring to an empirical bandwidth;
• 2)the evaluation or rationalization of wavelength-dependent dissymmetry values;
• 3)the justification of apparent sign inversion between ECD and CPL spectra due to Herzberg–Teller contributions.

All of them are supported by only a few literature examples, highlighting again a field with the potential for further development.

4.2. Simulation-Guided Design of CPL Emitters

The wide literature surveyed in Section fully demonstrates the current capability of different computational approaches to reproduce and rationalize CPL activity of almost any kind of nonaggregated molecular system, with the possible exclusion of lanthanide complexes, which represent a context still open to further computational development. The aim of the present section is to highlight the potential applications of QM calculations to predict, rather than interpret, CPL activity of chiral species. It is clear that only a highly accurate theoretical or computational approach, with full predictive capability, can effectively aid researchers in developing novel and efficient CPL emitters, eliminating the need for a time-consuming and costly trial-and-error synthetic process. We already discussed in detail several papers where a preliminary theoretical investigation facilitated the rational design of compounds with enhanced CPL activity. ,,,, They were based on tweaking different molecular and/or electronic properties, which serve as the basis for the division of the following sections.

4.2.1. Molecular Symmetry

Perhaps the molecular property which has been mostly debated in the literature as the single way to optimize CPL emission is molecular symmetry. The reasoning is based on the fact that some symmetry point groups impose geometrical restraints on the reciprocal orientation of electric and magnetic transition dipoles. For example, in the D ₂ point group, any allowed transition belongs to one of three irreps (irreducible representations) B₁, B₂, or B₃, which means that μ ₀₁ and m ₁₀ are necessarily aligned along the z, y, or x axis, respectively. In turn, this maximizes the geometrical term in the expression of g _lum (eq ), where cos θ _μm = ±1. This concept was especially exploited for the design of double helicenes, , multiple 1,1′-binaphtyl derivatives, propeller-shaped excimer emitters, , and cylindrically shaped macrocycles (detailed discussion in Sections and , and Scheme ).

Acting only on the molecular symmetry is, however, hardly a way to boost CPL efficiency over the limit imposed by the nature of the emitter in terms of its electronic properties. The already discussed case of double carbo[6]­helicenes H70 and H71 (Section , Chart ) reported by Mori and co-workers illustrates well the situation. With respect to the parent C ₂-symmetric carbo[6]­helicene (H1, Chart ), D ₂-symmetric double helicenes H70 and H71 were predicted to display enhanced dissymmetry values. This was indeed the case, although the final absolute g _lum values were not exceptional (2.5 × 10^–3 and 2.1 × 10^–3 for H70 and H71, respectively, with respect to 0.9 × 10^–3 for H1). This is due in part to the nature of the carbo[6]­helicene skeleton, whose S₁ state is dark.

In general, regardless of a highly symmetrical structure (in the ground state), the S₁ excited state may relax in a less symmetrical structure, thus thwarting the efforts made to optimize g _lum. For instance, H71 led to calculated θ _μm = 0° for the S₀–S₁ absorption from the D ₂-symmetric S₀ geometry and θ _μm = 130° for the S₁–S₀ emission from the actual, desymmetrized, S₁ geometry. The same phenomenon was noticed for several other compounds in our survey, including figure-of-eight biaryls (B34 and B35a–c, Section , Chart ), 1,1′-binaphthyl cyclic tetramers B37a–d (Section , Chart ), boron compounds (BX15a–f and BX21, Section , Charts and ), , several [2.2]­paracyclophane-OPE conjugates (Section , Charts and ), ,,, macrocyles made up of carbo[5]­helicene units linked by aromatic spacers (MC21–MC23, Section , Chart ), spirobi­(fluorene) S3 (Section , Chart ), and oligo­(spirofluorene)­s MC19a–b (Section , Chart ). The variability of structures for which this phenomenon occurs demonstrates its general and rather elusive nature, which is hardly to be anticipated by chemical reasoning only; on the contrary, it can be predicted by computational means before embarking on time-expensive chemical synthesis and enantioseparation. Further complicating the situation is the fact that a less symmetric structure in the ES may correspond to more intense transition dipole moments and vice versa. This is exemplified by BINOL-[5]­helicene dyads H68 and H69 (Chart and Figure , Section ), where C ₂-symmetrical structures, with an ideal angle of θ _μm = 0° for the S₁–S₀ transition, feature weak EDTM and MDTM, whereas C ₁-symmetrical structures, with θ _μm approaching 90°, feature much stronger EDTM and MDTM. Molecular rigidity offers a logical means of circumventing excited-state desymmetrization, as demonstrated by pyrenophanes (P20 and P21a–c, Chart , Section ) and double π-helical PDI derivatives (P22, P24, and P26, Chart , Section ). ,

4.2.2. Molecular Structure: Axial and Helical Chirality

The dependence of any molecular property on the molecular structure is tautology. In this section, we focus on a specific way of changing the molecular structure, that is, modifying the amount of twist in compounds endowed with helical chirality or the torsional angle of compounds endowed with axial chirality. One of the advantages of QM calculations is that they can be run on proper molecular models, which capture the essential structural (chirality axis, twisted π-units) and electronic (chromophore or fluorophore) properties without sacrificing accuracy. This is certainly the case for many twisted aromatics and biaryl systems, where the molecular portion not conjugated to the aromatic core (e.g., a saturated tether) can be neglected in the calculations. Once a certain molecular arrangement is found to exhibit the expected properties, the proper structural elements (tethers) may be designed to impart the desired structure.

The CPL properties of several biaryl compounds, endowed with axial chirality, were explored through torsional scans of S₁ geometry, often run on simplified molecular models. It is not a surprise that all reports highlighted a pronounced dependence on (absolute) conformation. Taking 1,1′-binaphthyls as an example (Section ), the internaphthyl dihedral angle φ may be easily modulated by several means (substituents, open chain vs chain-bridged geometry, and tether length), and eventually it dictates the reciprocal arrangement between the aromatic rings and hence the chiroptical response in terms of sign, shape, and intensity. , In particular, the impact of open chain vs chain-bridged geometry has been summarized in Section (Figure ). Alternatively, the 1,1′-binaphthyl moiety serves as the source of chirality with predictable geometry to afford CPL-active scaffolds (Section ), including some exhibiting specific properties like ICT, ,, TICT, (MR-)­TADF, ,, and excimer CPL emission. ,

Turning to helical chirality, several studies explored the impact of helical length on photophysical and chiroptical properties. In Section , we described various literature examples devoted to the comparison of carbo­[n]­helicenes and hetero­[n]­helicenes with increasing size. − A noteworthy example was provided by helical nanographenes N23a–d (Chart , Section ), whose g _lum increases nonlinearly with the system size (Figure ).

An impactful way of modifying the degree of helicity of a helicene or helicenoid system, which also affects frontier orbitals to some extent (Section ), is by replacing a heteroatom with a congener. Brédas, Coropceanu and co-workers recently evaluated the impact of chalcogen substitution of different series of heterohelicenes (from H95a–c to H99a–c, Chart ). These are hetero[6]­helicenoids or hetero[8]­helicenoids with B,N or B,X motifs (X = O, S, Se) which impart SRCT character to the main electronic transition due to the alternate localization of the HOMO and LUMO. For the three series H95a–c, H96a–c, and H99a–c, the degree of helical twists increases upon increasing the X atom size, and in turn, this was transmitted to larger R ₁₀ and g _lum values. For series H98a–c, the degree of helical twist was consistent, as there were the R ₁₀ and g _lum values, whereas series H97a–c behaved more irregularly. Calculated g _lum values varied between ∼10^–3 and 10^–2. The calculations were run using the DLPNO-CCSD wave function method (Section ) in combination with the def2-SVP basis set.

As for intrinsically twisted π-systems, in Section , we described a paper by Bedi, Gidron, and co-workers on several tethered twistacenes (S10–S13, Chart ). The compounds are endowed with both helical and axial chirality and lend themselves to a systematic study of the impact of the twistacene structure on the chiroptical properties. By varying the twist of the anthracene core, also in concomitance with the anthracene-phenyl twist, the authors demonstrated that an increased twist correlates with an increase in rotational strengths (in both absorption and emission). The increase involves all key quantities, that is, |μ ₀₁ |, |m ₁₀ |, and cos θ _μm . Further discussion of the role of interchromophoric twist can be found in Sections and .

4.2.3. Molecular Structure Flexibility

Molecular flexibility affects CPL performance in a number of ways, first of all by favoring nonradiative decay processes which decrease QY and B _CPL and may hamper CPL measurements. On the other hand, some degree of flexibility is often beneficial for observing processes like TADF. A few examples from our survey illustrated several other specific effects on CPL performance. A trivial one is the presence of multiple excited-state structures which emit oppositely signed CPL, as is well known for some ketones (Section , Figure ) but also for other compounds such as mellein (S9a, Chart , Section ). Another obvious situation is the formation of an intramolecular excimer, which may be more or less hampered by the molecular skeleton. For example, in the pyrene- and perylene-substituted quaternaphthyl derivatives P4 (Chart , Section ) reported by Takaishi et al., , the molecular flexibility ensured by the quaternaphthyl skeleton was key to achieving excimer emission and CPL. An even more explicit example was offered by double helicates MC10a and MC10b (Chart , Section ). These two compounds differ in the flexible versus rigid scaffold: the more flexible MC10a undergoes intramolecular excimer formation with near-IR emission and g _lum = 1.1 × 10^–2 while the rigid analog MC10b has nonexcimeric emission in the visible range with g _lum = 7.1 × 10^–4.

Molecular flexibility is also strictly associated with the possibility of excited-state desymmetrization discussed in Section . One specific example was offered by [2.2]­paracyclophane-OPE conjugates O13 and O14a–b (Chart , Section ), where two phenyl rings attached to the opposite sides of [2.2]­paracyclophane were “stapled” together by means of an ethynyl moiety. The staple was demonstrated to suppress excessive relaxation in the S₁ state, which preserves the same geometry and symmetry as for the S₀ state. The same finding applies to D ₂-symmetric figure-of-eight macrocycles (MC21, Chart , Section ) and D ₃-symmetric triply twisted macrocycles (MC22a–c) investigated by Wu and co-workers. Here, the smaller and more rigid dimeric compounds MC21c and MC21d retained D ₂ symmetry in both the S₀ and S₁ states, while the more flexible trimeric compounds MC22a and MC22b lose their D ₃ symmetry in the S₁ states.

In addition to vertical calculations on S₀ and S₁ PES minima, the impact of molecular deformations on CPL properties may be studied through several means, ranging from simplified approaches, where low-frequency deformations are applied to S₁ geometry and vertical calculations are run thereof (see an example in Section ), to vibronic calculations (Section ) and the use of the nuclear ensemble approach (Section ). A further discussion of the role of the conformational disorder generated by molecular vibrations can be found in Section .

4.2.4. Manipulation of Frontier Molecular Orbitals

Tweaking the relative energy, shape, and localization of FMOs is an obvious way to change and control the nature of S₀–S₁ and S₁–S₀ transitions and hence the associated dipolar and rotational strengths. Of the many existing possibilities to change the MO manifold, we consider here the structural modifications that aim at improving the CPL activity of a certain chiral skeleton. The most obvious of such modifications is the introduction of substituents on aromatic rings. Qualitative MO theory appropriately describes the effects of electron-donating and electron-withdrawing groups (EDG and EWG) on the FMO levels, and such effects are very easily predicted by calculations. This approach was applied to transform carbo­[n]­helicenes, with generally low quantum yields and g _lum values, into several efficient derivatives (discussion in Section ). A common strategy consists of the introduction of both EDG and EWG substituents, which promote ICT transitions with g _lum > 10^–2. ,, In some situations, the direct conversion of an EDG substituent to an EWG one may also be accomplished, affecting FMO levels and transition dipoles; one example is phosphine oxidation to phosphine oxide (H14a–b, Chart ).

Another possibility consists of tweaking the energies of quasi-degenerate FMOs so that the relative energy of a dark S₁ and a bright S₂ state may be interchanged, turning a poorly emissive species into a CPL-active one. Again, the rich chemistry of helicene and helicenoids offers various examples of this approach, based on substituent effects and/or helicene length (see Section and especially Figures and ). ,

A computation screening study aimed at the design of MR-TADF CPL-emitters (H100a _R -j _R, Chart ), based on the optimization of EDG and EWG peripheral substituents of quinolino­[3,2,1-de]­acridine-5,9-dione (QAO), was reported by Li, Chen, and co-workers. First, the HOMO/LUMO levels and λ_max/λ_em of model H100a _H were benchmarked against different functionals with variable HF%, leading to the choice of PBE0 for geometry optimizations and M062X for transition properties. FMO plots of all compounds (with R = H) demonstrated that for the molecules with electron-rich Ar groups (H100a _H –e _H) the HOMO is mainly distributed on the Ar and the adjacent benzene rings while the LUMO lies on the QAO core. Conversely, for the molecules with electron-poor Ar groups (H100f _H –j _H), the HOMO lies on the QAO core, while the localization of LUMO is more variable. HOMO–LUMO gaps vary accordingly, from a minimum of 2.89 eV for H100e _H to a maximum of 3.95 eV for H100i _H. NTO analysis of S₁–S₀ transitions further highlighted the substituent role, evidencing in some cases a SRCT character. Then, torsional energy scans around angle φ₁ (Figure a) were run on H100a _H to investigate the impact of the relative orientation between QAO and the carbazole moiety. Optimal dissymmetry values, both in absorption and emission, were detected for φ₁ ≈ 60–80°. This led to considering a series of substituents R on the meta position of the appended phenyl unit (Chart ) as a means to regulate θ _μm and hence g _lum (Figure b). In this way, the g _lum value could be theoretically improved by 1 order of magnitude with respect to the parent compounds with R = H, surpassing in most cases 10^–2, with a maximum g _lum = 6 × 10^–2 predicted for H100d _C(CN)3. The best compromise between g _lum and QY suggested H100b _CF3 as the best-performing CPL emitter. Finally, TADF properties were evaluated by means of SOC calculations. The study beautifully demonstrates that optimized substituents, in terms of electronic properties and steric hindrance, may effectively enhance the chiroptical performance of QAO-based CP-MR TADF molecules. A total of 50 compounds were considered, clearly demonstrating the significant added value of in silico predictions.

81.

(a) Calculated g _abs, g _lum, and θ _μm for H100a _H as a function of dihedral angle φ₁. (b) Calculated φ₁, θ _μm and g _lum for compounds H100a _R -j _R with different R groups (shown on the x axis). Reprinted with permission from ref . Copyright 2023 The Royal Society of Chemistry.

It is obvious that for a medium-sized SOM like carbo[6]­helicene (H1, Chart ) the number and type of substituents which can be introduced are practically infinite. Even by restricting to a specific substitution such as halogenation, the number of possible monohalogenated, dihalogenated, and so on carbo[6]­helicene derivatives increases exponentially with the number of substituents; for example, it amounts to ∼2.3 × 10⁵ for 5 halogen atoms. Such a situation hampers any systematic treatment but may be tackled by means of ML techniques. Uceda et al. demonstrated the effectiveness of such an approach for predicting R _0n values (ECD intensities) of multiple halogenated carbo[6]­helicenes derivatives. Apart from designing the most efficient compound (in this case, 2,3,14,15-tetrabromo[6]­helicene reached the highest R _0n value, ∼20% higher than the parent H1), ML approaches might be useful to highlight those positions on a certain chiral skeleton which are expected to mostly affect, positively or negatively, its chiroptical properties. To the best of our knowledge, ML techniques have not been applied so far to analyze and predict CPL quantities.

A more complicated, from the synthetic viewpoint, but often more effective process is the substitution of ring carbons by heteroatoms, that is, of carbon-only aromatic rings by heteroaromatic rings. We have noticed in Section how the evolution of pure carbo­[n]­helicenes into hetero­[n]­helicenes of various kinds is an effective way of improving photophysical and chiroptical properties. An explicit discussion of such an approach was provided by Hirose and co-workers and is summarized in Section . Switching from carbo­[n]­helicenes with n = 5, 7, and 9 to their analogs with terminal 2,1,3-thiadiazole rings (H35–H37, Chart ), the quasi-degeneracy between HOMO and HOMO–1 and between LUMO and LUMO+1 was lifted. Moreover, compounds H36 and H37 acquired HOMOs and LUMOs of different symmetry (belonging to A or B irrep in the C ₂ point group; see Figure , Section ), which is beneficial for a maximization of the magnetic dipole transition moment in both absorption and emission and hence of dissymmetry values. A second example was provided by the work of Staubitz and colleagues on the incorporation of B–N groups into carbo[5]­helicene and carbo[6]­helicene scaffolds (H48a–b and H49, Chart ). Conversely, 1,4-substitution of carbon atoms on a helicene skeleton by B and N may boost the RISC process from T₁ to S₁ states and impart MR-TADF behavior to helicene derivatives (Section ), which opens the way for CP-OLED applications. ,

Building on the same concept of maximizing |m ₁₀ | by manipulating the MO manifold through carbon-to-heteroatom substitution, Zhang, Long, and co-workers proposed a possible evolution of cylindrically shaped macrocycles MC1b and MC3 into their B,N-embedded analogs MC4 and MC5 (Section , Chart ). The proposed replacement (for the moment, only in silico) of C atoms with B and N atoms in specific positions would dislocate HOMO and LUMO from each other (Figure ), creating the premise for SRCT transitions endowed with a large magnetic transition dipole.

4.2.5. Transition Densities

Several papers surveyed in Section reported, among calculated quantities, transition density plots as a straightforward way to inspect the character of S₀–S₁ and S₁–S₀ transitions (Section ). The most insightful analyses came from transition dipole density plots, which allow one to visualize the distribution of electronic transition density in terms of electric and magnetic dipole transition moments and thus to decompose EDTM and MDTM into (x, y, z) components and atom-centered contributions. ,,,, Such a piece of information is then a starting point to explore possible substitution strategies or skeleton modifications.

A study focused on the role played by transition density in obtaining efficient CPL emitters was reported by Chen et al. They investigated four families of compounds with different types of chirality. Planar chirality was exemplified by a carbazole-substituted cyclophane derivative (C15, Chart ); central chirality was exemplified by a bis­(5,6-di­(9H-carbazol-9-yl)­phthalimide derivative of 1,2-cyclohexanediamine (S18, Chart ) and a spirosulfone (S19); helical chirality was exemplified by carbo[6]­helicene H2 (Chart ), hetero[6]­helicene H6, and monoplatina[6]­helicene M1 (Chart ); and axial chirality was exemplified by carbazole-functionalized 3,3′,4,4′-tetrahydro-BINOL derivatives B25a–c (Chart ) and 1,1′-biphenyl derivative B66a (Chart ). A common feature of most of the chosen compounds (except helicenes) is their D–A dyad nature. The analysis started by considering two isomeric biaryls, 4-(1H-pyrrol-2-yl)­benzonitrile and 4-(1H-pyrrol-1-yl)­benzonitrile, where the pyrrole acts as a donor (D) and benzonitrile acts as an acceptor (A). The S₁–S₀ transition density and rotational strengths R ₁₀ of the two models were evaluated as a function of the aryl–aryl torsional angle α (Figure ). The C2′–C4 junction in the former compound allows for full conjugation and delocalization of transition density over the D–A dyad, and R ₁₀ depends continuously on α. On the contrary, the N–C4 junction in the latter compound reduces delocalization because the pyrrole HOMO has a node on the N atom. Hence, the transition density is more localized, and R ₁₀ has a discontinuous and less pronounced dependence on α. The two models epitomize molecular systems of two kinds, both belonging to the D–A type: the first is endowed with large delocalization of the transition density, whose CPL activity can therefore be modulated by the twist angle between D and A moieties; the second is endowed with localized transition density whose CPL activity is less sensitive to structure. An analysis of the real molecules listed above confirmed the results obtained for the models. Moreover, by using a nuclear ensemble approach to simulate band-shape broadening (Section ), the authors disclosed that in the case of unrestricted D–A twisting, the conformational disorder generated by molecular vibrations can contribute to reducing the CPL signal. Based on the observed results, they concluded their survey by envisaging possible structural modifications to further improve the CPL performance of hetero[6]­helicene H6 and suggested that the replacement of the central N atom with C· (radical) would enhance the g _lum by 1 order of magnitude.

82.

Dependence of calculated R ₁₀ as a function of the dihedral angle α for (a) 4-(1H-pyrrol-2-yl)­benzonitrile and (b) 4-(1H-pyrrol-1-yl)­benzonitrile. In the upper right part of each panel, the S₁–S₀ transition density is shown for specific α values together with EDTM (red) and MDTM (blue). Adapted from ref under a CC-BY-NC 3.0 unported license.

From the latter discussion, it emerges clearly that factors such as interchromophoric twist, MO delocalization, and transition density are strictly interrelated and manipulating one of them implies acting on all of them simultaneously.

4.2.6. Macrocycle Size

In Section , we noticed that macrocyclic compounds such as molecular belts tend to feature the largest dissymmetries among single organic compounds. It is therefore obvious looking at this family of compounds to push dissymmetry values toward their theoretical limit of ±2. As for ECD, a recent theoretical study on several molecular belts started from considering MC1b (also known as [4]­CC_2,8, Chart ) with four chrysene units, exhibiting the record g _abs = 0.71. This latter value may be further pushed up by increasing the ring size (repeating units): a value of g _abs ≈ 2 was in fact predicted for [28]­CC_2,8 with 28 chrysene units. In CPL, however, two factors contrast a straightforward relationship between macrocycle size and dissymmetry values: (a) the possible degeneracy of the lowest-lying excited states, which hampers full control of the nature of the emissive S₁ state, and (b) the geometry relaxation of the S₁ state, which can break (lower) its symmetry by a pseudo-Jahn–Teller effect, affecting the reciprocal orientation between EDTM and MDTM. The latter issue, which has been already emphasized in Section , was further confirmed by a recent computational study by Brédas, Coropceanu, and co-workers on cyclic 1,4-phenylene-2,6-naphthylene oligomers. Moreover, other examples from our survey highlight a rather complex relationship between chiroptical properties and macrocycle size. For instance, triptycene-fused nanohoops MC7a–f (Section , Chart ) feature an inversely proportional relationship between dissymmetry values and ring size (Figure ), while for twisted cyclic oligomers P22–P31 (Section , Chart ) based on perylene diimide, the performance of the larger systems (tetramers) depended on the specific diastereomer considered (Figure ). ,

5. Perspectives and Conclusions

Despite the relatively recent history of ab initio CPL calculations, the methodologies have already reached full maturity in the case of single organic molecules (SOMs). Calculations provide a powerful framework to rationalize experimental results, and, in some cases, when the measurement is ambiguous and cannot be repeated, even to check their accuracy and to visualize the key observables involved in CPL activity. Despite seldom being done, it is perfectly feasible to use ab initio results as a predictive tool, thus targeting synthetic efforts toward the most suitable molecular candidates. In the context of SOMs, some discrepancy persists among different classes of compounds as far as the accuracy of calculations is concerned. For example, looking at the two broadest families covered in our survey, we noticed that the accuracy reached by TD-DFT calculations, in terms of agreement with experimental data, is significantly higher for helicenes and helicenoids than for biaryls. However, even for this latter family, the accuracy remains high enough to guarantee a meaningful prediction of CPL-related properties. A positive message from our survey comes from noticing the efficiency of ab initio calculations even for very large single-molecule systems like nanographenes. TD-DFT offers unrivaled computational efficiency but struggles with functional sensitivity and limitations in describing long-range charge transfer and double-excitation character, particularly in extended conjugated systems. Simulating CPL at higher levels of theory than TD-DFT without sacrificing scalability remains a significant challenge. High-accuracy methods like EOM-CCSD and CASPT2 provide reliable results but are computationally intensive, preventing full applicability to large systems. Fragment-based approaches and localized wave function methods, such as DLPNO-CCSD and spin-flip TD-DFT (SF-TD-DFT), show promise but are not yet widely applied to CPL simulations.

Moving beyond SOMs, the situation becomes more complicated for two main reasons: compounds containing heavy elements and unpaired electrons pose problems of electronic structure description; aggregate systems pose problems of structure uncertainty, disorder, and size.

Lanthanide complexes are the most useful CPL emitters as isolated species, but their computational description is very challenging. Although relativistic multireference methods offer in principle the right solution for the most relevant problems (orbital degeneracy, spin–orbit coupling, and relativistic effects), their application is not trivial at all. This is witnessed by the scarce number of examples found in the literature, which still manifest accuracy issues and heavy dependence on the choice of active space. Both metal complexes and conjugated systems at conical intersections exhibit strong static correlation, and as such they are either untreatable by DFT or very functional-dependent. However, the computational cost and methodological complexity of multireference methodsthe need to define an active space, avoiding intruder states and ensuring size consistencyrender them challenging to apply, especially for more complex systems. The development of tools for accurate calculations of CPL properties of lanthanide complexes with a broad application represents, in our opinion, the most important challenge in the field.

Large aggregate systems would greatly benefit from accurate predictions, as they find direct applications in electronic and photonic devices. The active layer of these devices is often in the form of a thin film, and emerging optoelectronic properties rely on processes occurring in the condensed phase, like charge and exciton transport, that in principle involve tens of molecules arranged in a tridimensional array. Therefore, a description of aggregate phases is compulsory, which is severely complicated by at least two questions:

• 1)The supramolecular structure of the aggregate (or aggregates, in the common case of heterogeneous systems) is often not known. Chiral structures in aggregate materials manifest themselves at various scales and hierarchical levels, from first-order supramolecular packing (dimer/oligomer, ∼1–10 nm), to fibrils/bundles of fibrils (∼10–100 nm), and then to fibers (>1 μm); the picture can be further complicated by the presence of grains or domains due to phase segregation. In several cases, the structural details cannot be inferred by experimental techniques alone. Purely computational approaches, such as molecular dynamics, may also fail to give definitive results, as many structures corresponding to several shallow energy minima could be found, related to aggregation pathway complexity. In particular, excited-state geometry optimization of large aggregates with variational methods faces many challenges, such as variational collapse, root flipping, and the scarcity of reliable gradients, which make it hard to locate true excited-state minima.
• 2)Even if the structure is known, modeling CPL in supramolecular and periodic systems presents intrinsic complexities. Although atomistic models of dimers and oligomers are increasingly utilized, extending simulations to “infinite” systems is still limited. Implementations of TD-DFT under PBC have progressed, yet achieving a gauge-invariant description of magnetic dipole transition moments remains elusive. Efficient algorithms for periodic excited-state geometries and the treatment of disorder and exciton–vibrational coupling are necessary for predictive simulations in solid-state emitters, such as OLEDs, but they remain to be realized.

Additionally, CPL calculations are lagging behind with respect to various interesting CPL-active systems. One example is offered by perovskites. If, on one hand, perovskite crystal structure is often precisely known, then the calculations of the chiroptical property necessitate a good description of the spin–orbit coupling and other relativistic effects. Given the high number of atomic orbitals to be considered, even TD-DFT calculations are less feasible. On the other hand, good results were obtained in the case of ECD calculations on chiral halide perovskites by using a parametrized tight-binding (TB) model. A second example is offered by carbon dots. Despite the current interest in CPL-active carbon dots, to date there are no reports of ab initio calculations of their chiroptical properties. For these systems, however, the main limitation to overcome is often the lack of a molecular model for the structure of the dot itself.

In addition to system-dependent issues, various general issues offer a future field of challenge on the theoretical and computational ground. Vibronic coupling plays a crucial role in replicating the intensity and line shape of experimental CPL spectra. While the Franck–Condon and Herzberg–Teller (FC/HT) expansions are standard, fully time-dependent approaches such as correlation function methods offer access to spectral band shapes, including thermal effects and broadening. However, nonadiabatic dynamics in CPL, especially involving excited-state interconversion or emission from higher singlet/triplet states (anti-Kasha behavior), remain underexplored.

The consistent evaluation of rotational strengths, especially in coupled-cluster and multireference methods, continues to suffer from gauge-origin dependence. Techniques like the use of GIAOs and response-theory corrections have been proposed, but a consensus approach, particularly in the solid state, has yet to be established.

Finally, accurate modeling of dynamic environments remains a frontier. Embedding schemes such as the PCM, QM/MMpol, and polarizable embedding (PE)-TD-DFT have improved the modeling of solvation effects. As a matter of fact, a very large majority of CPL calculations found during our survey were run in vacuo, despite the crucial impact that solvent may have on photophysics in general and CPL properties in particular. A limited number of calculations, less than 20% of the total, included a continuum solvent model in the linear response polarization formalism, while state-specific polarization approaches are still extremely rare. It must be recalled here that many compounds manifest solvent-dependent emission and CPL activity, often as a consequence of specific phenomena like TADF, ICT, and TICT behavior, excited-state equilibria including ESIPT, and so on. The use of a state-specific polarization approach, or of explicit solvation models, appears to be mandatory for a meaningful theoretical treatment of these phenomena. However, the inclusion of a state-specific polarization response and dispersion within excited states is still under development. The role of the solvent electron dynamic response in modulating CPL spectral shape remains a key area for future research.

Theoretical research in CPL continues to evolve, opening up several promising directions and emerging opportunities:

• Time-Resolved and Ultrafast CPL Spectroscopy: Ultrafast time-resolved CPL (TR-CPL) could represent a frontier technique capable of probing the dynamics of chiral excited states on femtosecond to nanosecond time scales. Only very recently did femtosecond TR-CPL experiments appear in the literature, , but theoretical studies are not yet available. From a theoretical perspective, a straightforward extension of time-resolved ECD methods − to CPL is not feasible because the latter is an emissive phenomenon: a simple time-dependent propagation starting from the excited state will rather provide time-resolved excited-state ECD. Therefore, a description of TR-CPL will require an extension of the excited-state quantum electrodynamics treatment. Work on this topic is currently ongoing by some of us. Even further, a confluence of nonadiabatic dynamics, excited-state QED, and trajectory-based simulations such as surface hopping or multiconfigurational Ehrenfest dynamics will offer the potential to unravel chiral photophysics in real time, including processes like singlet fission, ISC, and ultrafast structural relaxation in chiral systems.

• Machine Learning and Surrogate Models: The construction of surrogate models for CPL properties, trained on high-level reference data, could be an emerging route to accelerate predictions across chemical space and yield novel efficient materials. , Graph neural networks , could encode rotational properties and allow the prediction of full rotational strength tensors and emission dissymmetry factors.

• Chiral Plasmonic and Strong Coupling Regimes: Coupling chirality to nanophotonic environments such as plasmonic cavities or photonic crystals introduces opportunities for CPL enhancement, control, and manipulation. − Theoretical tools that include both the electronic structure of the emitter and the quantum nature of the electromagnetic field (i.e., quantum cavity electrodynamics) would be essential to simulate CPL in strong light–matter coupling regimes. −

• As stressed above, CPL in molecular aggregates emerges from the intricate interplay among exciton delocalization, vibrational dynamics, and chiral supramolecular structures. Accurately modeling these interactions necessitates a comprehensive quantum dynamical treatment of exciton–phonon–photon coupling, which remains a significant challenge. Recent advancements in computational methods, such as tensor network approaches and mixed quantum-classical techniques, could offer promising avenues to address this complexity.

Glossary

Abbreviations

ACN

Acetonitrile

ACQ

Aggregation-Caused Quenching

Ad

Adiabatic

ADC

Algebraic Diagrammatic Construction

Ad-MD

Adiabatic Molecular Dynamics

AH

Adiabatic Hessian

AIE

Aggregation-Induced Emission

AIEgen

AIE Luminogens

ANO

Atomic Natural Orbital

ARE

Absolute Relative Error

ASC

Apparent Surface Charge

B _CPL

CPL Brightness

BINOL

1,1′-Binaphthyl-2,2′-diol

BJ

Becke–Johnson Damping

BODIPY

Boron-dipyrromethene

BOMD

Born–Oppenheimer Molecular Dynamics

CAS

Complete Active Space

CASPT2

Complete Active Space with Perturbative Corrections

CASSCF

Complete Active Space Self-Consistent Field

CC

Coupled-Cluster

CC3

Coupled-Cluster with Triples

CCSD

Coupled-Cluster with Single and Double Excitations

CCSDT

CCSD Plus Triple Excitations

CCSDTQ

CCSDT Plus Quadruple Excitations

CCSDTQP

CCSDT Plus Pentuple Excitations

CI

Configuration Interaction

CIS

Configuration Interaction with Single Excitations

CNLC

Chiral Nematic Liquid Crystals

COSMO

Conductor-like Screening Model

CPCM

Conductor Polarizable Continuum Model

CPL

Circularly Polarized Luminescence

CP-OLED

Circularly Polarized Organic Light-Emitting Diode

CPP

Circularly Polarized Phosphorescence

CREST

Conformer–Rotamer Ensemble Sampling Tool

CSF

Configuration State Functions

DCM

Dichloromethane

dd

Domain Decomposition

dd-COSMO

Domain Decomposition Conductor-like Screening Model

DFT

Density Functional Theory

DLPNO

Domain-Based Local Pair Natural Orbital

DMF

Dimethylformamide

DMSO

Dimethyl Sulfoxide

ECD

Electronic Circular Dichroism

EDTM

Electric Dipole Transition Moment

EOM

Equation of Motion

ES

Excited State

ESIPT

Excited-State Intramolecular Proton Transfer

EXAT

EXcitonic Analysis Tool

FC

Franck–Condon

FCHT

Franck–Condon/Herzberg–Teller

FDCD

Fluorescence-Detected Circular Dichroism

FEDA

Frenkel Exciton Decomposition Analysis

FMO

Frontier Molecular Orbitals

FQF

Fluctuating Charges and Fluctuating Dipoles Model

fwhm

Full-Width at Half-Maximum

g _abs

Absorption Dissymmetry Factor

GGA

Generalized Gradient Approximation

GIAO

Gauge-Including Atomic Orbitals

g _lum

Luminescence Dissymmetry Factor

GS

Ground State

HOMO

Highest Occupied Molecular Orbital

HT

Herzberg–Teller

HWHM

Half-Width at Half-Maximum

ICD

Induced CD

ICPL

Induced CPL

ICT

Intramolecular Charge Transfer

IEFPCM

Integral Equation Formalism Polarizable Continuum Model

ISR

Intermediate State Representation

LR

Linear Response

LR-CC

Linear Response Coupled Cluster

LSCM

Laser Scanning Confocal Microscopy

LUMO

Lowest Occupied Molecular Orbital

MAE

Mean Absolute Error

MDTM

Magnetic Dipole Transition Moment

ML

Machine Learning

MM

Molecular Mechanics

MO

Molecular Orbital

MR-TADF

Multiresonant Thermally Activated Delayed Fluorescence

NEVPT2

N-Electron Valence State Perturbation Theory

NHC

N-Heterocyclic Carbene

NIR

Near-Infrared

NTO

Natural Transition Orbital

OFET

Organic Field-Effect Transistor

ONIOM

N-Layered Integrated Molecular Orbital and Molecular Mechanics

PBC

Periodic Boundary Conditions

PCM

Polarizable Continuum Model

PEM

Photoelastic Modulator

PES

Potential Energy Surface

PMM

Perturbed Matrix Method

PTD

Perturbation-to-Density

PTE

Perturbation-to-Energy

PTED

Perturbation-to-Energy and Density

QAO

Quinolino­[3,2,1-de]­Acridine-5,9-Dione

QC

Quantum Center

QED

Quantum Electrodynamics

QM

Quantum Mechanics

QY (Φ_f)

Quantum Yield

RAS

Restricted Active Space

RASSI

Restricted Active Space State Interaction

RC

Relativistically Contracted

RE

Relative Error

RISC

Reverse Intersystem Crossing

RMS

Rotated Multistate

RT

Room Temperature

SA-CASSCF

State-Averaged Complete Active Space Self-Consistent Field

SARC

Segmented All-Electron Relativistically Contracted Basis Set

SCF

Self-Consistent Field

SMD

Universal Continuum Solvation Model

SOC

Spin–Orbit Coupling

SOM

Single Organic Molecules

SPINOL

1,1′-Spirobiindane-7,7′-diol

SRCT

Short-Range Charge Transfer

SS-CASSCF

State-Specific Complete Active Space Self-Consistent Field

SVD

Singular Value Decomposition

TADF

Thermally Activated Delayed Fluorescence

TDA

Tamm-Dancoff Approximation

TD-DFT

Time-Dependent DFT

TD-HF

Time-Dependent Hartree–Fock

THF

Tetrahydrofuran

TICT

Twisted Intramolecular Charge Transfer

TMS

Trimethylsilyl

TSC

Through-Space Conjugation

TS-ICT

Through-Space ICT

UV

Ultraviolet

VAPOL

2,2′-Diphenyl-3,3′-biphenanthr-4-ol

VEM

Vertical Excitation Method

VG

Vertical Gradient

VH

Vertical Hessian

XCF

Exchange-Correlations Functional

XDW

Extended Dynamically Weighted

XMS

Extended Multistate

ZORA

Zeroth-Order Regular Approximation

Λ_CPL

Circular Polarization Luminosity Index

Biographies

Ciro A. Guido is an associate professor of theoretical chemistry at the Department of Science and Technological Innovation, University of Eastern Piedmont, Italy, where he leads the LIMELab group. In 2011, he earned his Ph.D. in theoretical chemistry from the Scuola Normale Superiore di Pisa and then held research positions at several prominent institutions, including CECAM at EPFL, Chimie ParisTech, École Centrale Paris, Université de Nantes, the University of Pisa, and the University of Padua and served as an assistant professor at the University of Siena. His research lies at the intersection of quantum chemistry and photophysics, with a focus on the development and implementation of methods for the electronic structure of excited states, the theoretical description of dispersion and polarization effects in polarizable media and metal nanoparticles, the formulation of an Open Quantum Systems approach to describe the coupled solute–solvent electronic dynamics, and the theoretical design of electron energy loss spectroscopy sensitive to chirality and multipolar transitions. He currently leads research projects focused on optoelectronic materials and the photophysics of rhodopsins.

Francesco Zinna received his Ph.D. in chemistry and materials science at the University of Pisa in 2016, followed by postdoctoral research at the University of Geneva. In 2018, he came back to Pisa as an assistant professor, and in 2024, he was promoted to associate professor. In 2022, he was awarded the Ciamician Medal of the Italian Chemical Society for his contributions to the development of frontier chiroptical techniques. Since his Ph.D., he has worked in the field of chirality, with a specific focus on chiroptical spectroscopies and, in particular, circularly polarized luminescence and chiral materials. He is currently the PI of a FIS (Italian Science Fund) project focusing on the supramolecular structures and properties of chiral organic conjugated compounds.

Gennaro Pescitelli received his Ph.D. (2001) in chemistry from the University of Pisa and did postdoctoral research at Columbia University. He was appointed associate professor in organic chemistry at the University of Pisa in 2014 and full professor in 2023. He is the coauthor of over 260 publications, including several reviews on the principles and applications of chiroptical spectroscopies. His research is focused on spectroscopic and computational investigations of chiral organic molecules, especially natural products, metal compounds, organic crystals, organo-gelators, conjugated functional oligomers, and polymers. He is associate editor of the journal Chirality (Wiley).

The authors declare no competing financial interest.