Challenges and Advances in Elucidating the Electronic Circular Dichroism of Crystalline Materials through First-Principles Calculations

Abstract

The electronic circular dichroism (ECD) of crystalline materials is governed by physical effects that are absent in isolated-molecule theory. Bloch-state delocalization makes position-operator matrix elements ill-defined and requires Berry-connection formulations of the periodic optical-activity response, while excitonic effects redistribute oscillator strength across the band gap. This Perspective critically surveys the hierarchy of first-principles methods that have been established to address these challenges. Molecular-cluster time-dependent density functional theory (TDDFT) provides chemical intuition and mechanistic insight but cannot represent the complete periodic response and can underestimate long-range crystal field effects. Periodic independent-particle calculations incorporate the crystalline environment and band-geometric response, whereas self-consistent coupled-perturbed approaches additionally include crystal-orbital relaxation. Adaptive Brillouin-zone sampling can accelerate convergence for localized one-particle contributions, although nonuniform meshes complicate momentum conservation in the correlated treatments. GW-BSE provides a rigorous framework for excitonic optical activity, whereas maximally localized Wannier function interpolation efficiently converges the DFT-level periodic quantities used to analyze the band-geometric response. A generally validated workflow that combines the BSE-level excitonic response with all geometric and interpolation ingredients has not yet been established. Finally, we discuss open challenges including fully relativistic treatments of chiral rare-earth halide perovskites and single-crystal anisotropy measurements.

1. Introduction

Chirality is a fundamental and widespread form of symmetry breaking in nature, with consequences ranging from the homochirality of biological macromolecules to the selective pharmacological activity of drug enantiomers and the emergent optical properties of chiral functional materials. − Electronic circular dichroism (ECD), which measures the difference in absorption between left- and right-circularly polarized light (CPL), provides a direct spectroscopic approach for investigating the electronic structure of chiral materials. Its spectral features encode the sign and magnitude of rotational strengths (R _λ), which reflect the relative orientation and magnitude of the electric- and magnetic-dipole transition moments of each electronic excitation. − In solution, ECD has become a standard tool for assigning absolute configuration and analyzing the conformations of chiral molecules, and the mature body of computational methodology based on time-dependent density functional theory (TDDFT) applied to isolated molecules now supports these applications reliably, delivering semiquantitative accuracy for a broad range of chiral organic chromophores. ,,

The crystalline solid state, however, presents a fundamentally richer and more challenging problem. Molecular crystals are periodic quantum-mechanical systems in which crystal packing, intermolecular interactions, and space-group constraints collectively shape the electronic structure and optical response in ways that have no molecular analogue. An achiral molecule may crystallize in one of the 65 Sohncke space groups, which lack improper symmetry operations, and acquire net ECD through packing-induced symmetry breaking. ,− Exciton coupling between neighboring chromophores produces Davydov-split Cotton effects whose sign pattern and peak separation encode intermolecular couplings that are inaccessible to isolated-molecule calculations. ,,− As illustrated in Figure , the ECD absorbance difference is ΔA = A _L – A _R = (ε_L – ε_R)l according to the Beer–Lambert law. It is governed by the rotational strength and produces a characteristic bisignate Cotton effect with a zero-crossing at wavelength λ₀. These collective, lattice-mediated phenomena mean that the spectrum of a molecular crystal cannot, in general, be reduced to the sum of its constituent molecular contributions. Applying a molecular TDDFT cluster approach to a crystal therefore encounters two fundamental obstacles: an artificial boundary created by truncating the periodic environment and the ill-defined position-operator matrix elements between extended Bloch states, which require a Berry-connection construction in reciprocal space. −

1.

Physical basis of ECD: the difference in absorption between left- and right-circularly polarized light by a chiral material and the associated bisignate Cotton effect.

This Perspective provides a critical assessment of the hierarchy of first-principles methods available for calculating the ECD of crystalline materials, from molecular cluster models through periodic independent-particle and self-consistent coupled-perturbed response to the full GW plus Bethe–Salpeter equation (GW-BSE) framework, alongside DFT-level Wannier interpolation, and introduces the geometric-phase picture that connects ECD spectra to the band geometry of the electronic band structure. As shown in Figure , a schematic overview of this hierarchy, organizing three levels of theory by computational complexity and the response effects included, is provided; this map provides an overview of Sections to , each of which discusses one model in detail. Section reviews the physical framework underlying the periodic optical-activity calculations, and Section concludes with open challenges and future directions for elucidating the ECD of crystalline materials based on first-principles calculations.

2.

Hierarchy of three computational approaches for calculating the ECD of crystalline materials, organized by increasing computational complexity and accuracy: cluster models; periodic response at independent-particle (IPA) and self-consistent coupled-perturbed (SC–CP) levels; and GW-BSE.

2. Results and Discussion

2.1. Physical Framework: Bloch States, Berry Phases, and the Brillouin-Zone Integral

The fundamental contrast between the molecular and crystalline approaches to ECD calculation is summarized in Figure . On the left (the ECD calculation of a single molecule), an isolated molecule in a finite box has a wave function |Ψ­(r)|² that decays rapidly with distance from the molecular center, with amplitude dropping to zero well within a few angstroms. This physical localization is what makes isolated-molecule calculations tractable and gives them good physical intuition. On the right (the ECD calculation of a crystalline material), the same quantity for the Bloch state is uniform across all unit cells of the infinite periodic lattice, with the same amplitude at Cell 0, Cell 1, ···, Cell 9 and beyond. This symmetry-constrained delocalization is not a numerical artifact but a fundamental consequence of translational symmetry, and it is the reason the standard molecular formalism fails, as the position operator cannot be meaningfully applied to such a nondecaying wave function. For Ψ _nk ( r ) = e^{ik · r}u_{n k} ( r ), where u_nk ( r ) is the cell-periodic part satisfying u_nk ( r + R ) = u_nk ( r ) for all lattice vectors R and k is the crystal momentum defined over the first Brillouin zone (BZ), translational symmetry makes k a good quantum number, labeling the irreducible representations of the translational symmetry group. This is the defining structural difference between the crystalline material and an isolated molecule or cluster: whereas the molecular eigenstates are discrete and labeled by a single set of quantum numbers, the crystalline eigenstates form continuous bands parametrized by a crystal momentum that takes values across the compact reciprocal-space manifold. This contrast is directly summarized in Figure , where the periodic potential (V( r )) with lattice constant ( a ) gives rise to Bloch states that decompose as the product of a slowly varying plane-wave envelope (e ^{i k · r} ) and a rapidly oscillating cell-periodic part (u_nk ( r )). All the optical matrix elements must therefore be evaluated as integrals over this manifold rather than at a single point. The rotational strength that governs the ECD signal becomes the k -dependent quantity (R _nk ), and the ECD response is obtained from the Brillouin-zone-integrated optical-activity tensor, written schematically as eq , ,

3.

Schematic comparison of electronic structures between isolated molecules/clusters (left) and crystalline materials (right).

4.

Physical framework for calculating ECD in crystalline materials: real-space periodic potential and Bloch-state decomposition (left), and k-space geometric information within a periodic response calculation (right). The graphic is qualitative; eq gives the quantitative Berry-curvature definition used here. The ECD curve is schematic and represents the complete optical-activity tensor after Brillouin-zone integration; individual Berry-curvature regions do not map one-to-one onto spectral features.

$${\gamma }_{ijl}(\omega )={\int }_{\mathrm{BZ}}\frac{d^{3}k}{{\left(2\pi \right)}^3}{\mathcal{K}}_{ijl}(k,\omega )$$ 1

Here, γ _ijl (ω) is the optical-activity tensor, and K _ijl (k, ω) denotes its full reciprocal-space response kernel. The indices i, j, and l label Cartesian components. At the independent-particle level, the kernel includes band-pair transition matrix elements, occupation differences, and denominators or line-shape functions involving the energy difference ℏω – (E _mk – E _nk ). Eq summarizes the Brillouin-zone integration, rather than asserting a universal scalar rotational-strength prefactor; the measured absorbance difference additionally depends on propagation geometry and sample thickness. Evaluating the complete response in a gauge-consistent and origin-independent manner is the central computational challenge. ,

The electric-dipole transition matrix element, conventionally written as ⟨Ψ _m | r ^|Ψ _n ⟩, is ill-defined for Bloch states because r ^ does not have well-defined matrix elements between states that extend across an infinite crystal. The resolution, developed within the modern theory of polarization, is to replace the position operator with the k -space derivative ⟨u_{n k} |i∇ _k |u_mk ⟩. , This replacement is gauge-dependent, meaning that it transforms nontrivially under the phase freedom of the Bloch states, and it must be handled with care to produce physical, origin-independent observables. The mathematical framework that achieves this is the theory of Berry phases: the physical quantity that replaces the position matrix element is the Berry connection A_nm ( k ) = i⟨u_{n k} |∇ _k |u_{m k} ⟩, which measures the geometric phase accumulated by the Bloch states as k is adiabatically transported along a path in the BZ. , The right panels of Figure visualize the k-resolved Berry curvature and a schematic ECD spectrum. Although regions of large Berry curvature can identify k-space regions relevant to the band-dispersion contribution, their signs and magnitudes do not map one-to-one onto individual ECD spectral features; the observable response follows from Brillouin-zone integration of the complete optical-activity tensor. The electric-dipole interaction may be represented in the length gauge (LG), where the transition moment involves the position operator r, or in the velocity gauge (VG), where it is expressed through momentum or velocity matrix elements. For exact eigenstates, the two representations are connected through the commutator of r with H; with approximate wave functions or finite basis sets, this equivalence is not exact. Under periodic boundary conditions, r is not a well-defined operator between extended Bloch states, whereas momentum and velocity matrix elements remain well-defined. A periodic LG treatment must therefore replace position matrix elements by a Berry-connection construction. ,, Because the position operator is not well-defined between extended Bloch states, a decomposition of crystalline optical activity into magnetic-dipole, electric-quadrupole, and band-dispersion terms must be interpreted with care. , The Berry-curvature-dependent contribution is not an ad hoc correction appended to otherwise molecular magnetic-dipole and electric-quadrupole terms. Rather, when periodic position matrix elements are represented by the Berry connection, the associated geometric contribution is incorporated into a translationally invariant periodic formulation of the conventional multipole response. , The orbital magnetic-dipole operator is proportional to r × p and therefore retains the periodic position-operator problem; the electric-quadrupole operator contains the same type of position dependence. In LG, both the electric-dipole factor and the magnetic-dipole/electric-quadrupole factor contain problematic position dependence. In VG, the electric-dipole factor is expressed through momentum or velocity, but the magnetic-dipole and electric-quadrupole factors still require a periodic reformulation.

Within the periodic formulation of Wang and Yan, the band-dispersion contribution enters together with the magnetic dipole and electric quadrupole terms in the full optical-activity tensor. It depends on transition energies, velocity-related quantities, and electric-dipole transition matrices across the Brillouin zone; it cannot in general be reduced to a scalar Berry curvature expression alone. The k-space geometry can be informative, but it does not establish a one-to-one correspondence between individual Berry curvature hotspots and ECD peak signs. A finite molecule or cluster does not explicitly represent this periodic band-dispersion contribution because it has no Brillouin zone or crystal-momentum band dispersion.

The Berry curvature is defined as eq , ,

$${\mathrm{\Omega }}_{n,z}(\mathit{k})=-2{\hslash }^2\mathrm{Im}{\sum }_{m\ne n}\frac{⟨u_{nk}|{\mathrm{v̂}}_x|u_{mk}⟩⟨u_{mk}|{\mathrm{v̂}}_y|u_{nk}⟩}{{({\epsilon }_m-{\epsilon }_n)}^2}$$ 2

where v̂ _a (k) = (1/ℏ)∂Ĥ _k /∂k _a (a = x, y) is the Cartesian velocity operator. The derivative acts on the cell-periodic Bloch Hamiltonian Ĥ _k = e ^–ik·r Ĥ₀ e ^ik·r , which satisfies Ĥ _k u _nk = E _nk u _nk for the unperturbed crystal Hamiltonian Ĥ₀; it is distinct from the external light–matter perturbation used to drive the response. The band Berry connection is A _n (k) = i⟨u _nk |∇ _k u _nk ⟩, and the corresponding Berry curvature is Ω _n (k) = ∇ _k × A _n (k). , In eq , ε _n denotes E _nk at fixed k. Its Brillouin-zone integral is related to the Chern number. This geometric formulation provides a translationally invariant representation of position-dependent multipole response in periodic systems and must be used as part of the complete gauge-consistent optical-activity tensor rather than as an independently appended correction. ,,,,

2.2. Cluster Models: Physical Intuition and Intrinsic Limitations

The molecular cluster approach treats a finite fragment of the crystal, typically one to a few dozen molecules extracted from the experimental structure determined by X-ray or neutron diffraction, using the full toolkit of modern molecular quantum chemistry. Molecular response methods include TDDFT and coupled-cluster approaches, whose accuracy depends on the electronic structure, basis set, and treatment of the environment. , The rotational strength R _λ of excitation λ is evaluated using the standard Rosenfeld expression (eq ),

$$R_{\lambda }=\mathrm{Im}⟨0|\mathrm{\mu ̂}|\lambda ⟩·⟨\lambda |\mathrm{m̂}|0⟩$$ 3

where μ̂ = −er̂ is the electric-dipole operator and m̂ = −(e/2m _e)L̂ is the magnetic-dipole operator. The imaginary part of the scalar product of these complex transition moments gives the rotational strength. In the velocity gauge, the full optical-activity tensor is origin independent even for a finite basis set. In the conventional length gauge, however, a finite basis introduces origin dependence, which can be removed using gauge-including atomic orbitals (GIAOs, or London orbitals). The origin-independent length-gauge formulation, LG­(OI), provides an alternative that avoids GIAOs and can be applied to both molecular and periodic calculations. , For isolated molecules in solution, these methods provide quantitative accuracy for small- to medium-sized organic chromophores and are widely used for absolute-configuration assignment.

When applied to molecular crystals, the cluster approach encounters limitations associated with finite size and the treatment of the environment. Intermolecular couplings can modify transition energies and chiroptical intensities; therefore, the choice of cluster and its embedding must be tested. Forson et al. compared optical rotation calculated with density functional theory and periodic boundary conditions (DFT-PBC) against unit-cell molecular-cluster calculations for five chiral molecular crystals. Their results show that the treatment of periodic intermolecular interactions can be as important as the choice of functional and basis set; the periodic calculations do not uniformly give the closest numerical agreement with experiment. For aspartic acid and glutamic acid, the calculated optical rotation had the same qualitative trend as the reported measurements but was about 2 orders of magnitude smaller, motivating a reassessment of the experimental data.

The second layer of difficulty concerns the crystal’s electric field. Molecules within a crystal experience a periodic electrostatic potential from their neighbors that modifies transition energies through electrostatic shifts and polarization effects, renormalizes oscillator strengths, and alters rotational strengths in ways that cannot be captured by gas-phase calculations or simple macroscopic dielectric screening models. The correct treatment requires a self-consistent periodic calculation that converges under the influence of the full infinite crystal potential. The Balduf-Caricato formalism tackled this problem by deriving origin-invariant optical rotation tensors for chiral crystals within the DFT-PBC framework. They showed that the full origin-invariant tensor, which correctly combines the electric dipole–magnetic dipole and electric dipole–electric quadrupole polarizability contributions, gives results in significantly better agreement with experiment than either contribution alone, and that the choice of density functional is the primary determinant of the optical rotation magnitude, with the basis set and gauge playing secondary roles.

The third and most fundamental difficulty is that the cluster model has no natural way to treat the band-dispersion contribution to optical activity. This contribution, identified by Wang and Yan, arises from the k-dependence of transition quantities across the BZ and has no direct analogue in a finite cluster calculation, which lacks both a Brillouin zone and crystal-momentum band dispersion. Wang and Yan showed explicitly for elemental tellurium and chiral carbon nanotubes that the band-dispersion contribution is quantitatively significant and must be included for agreement with experiment. A cluster calculation cannot capture this contribution because it lacks the concept of crystal momentum entirely. This is not merely a practical limitation but a fundamental structural limitation of the finite-system approach when applied to periodic materials.

Despite these limitations, cluster models retain genuine utility for certain well-defined systems. For molecular crystals with tightly bound, localized excitons where intermolecular electronic overlap is negligible and the crystal field perturbation is small, a nearest-neighbor dimer or small oligomer cluster can reproduce the essential bisignate Cotton effect at a modest computational cost, providing a useful first assignment of the spectrum. Cluster calculations also serve as indispensable benchmarks for validating periodic implementations since the convergence of cluster results to the periodic limit provides a stringent test of both codes. The chemical interpretability of cluster results, particularly the ability to decompose the total ECD signal into contributions from individual chromophore pairs using excitonic coupling models, provides physical insight that is less directly accessible in the periodic framework. These complementary strengths mean that cluster and periodic approaches are best viewed as tools that support rather than replace one another.

Despite the intrinsic limitations discussed above, cluster-based TDDFT calculations remain the most widely deployed approach for interpreting experimental ECD spectra of chiral organic and hybrid crystalline materials in practice. In chiral perylene diimide (PDI) crystals, for instance, Li et al. employed molecular cluster TDDFT to rationalize how ortho-substitution of a cyano group tunes the angle between the electric and magnetic transition dipole moments, thereby amplifying the absorption asymmetry factor g _abs by a factor of 3 relative to the unsubstituted analog. Similarly, Gao et al. employed cluster DFT and TDDFT calculations to reveal the energy origin of chirality reversal between monomeric and crystalline forms of tetrachlorinated diperylene diimides, connecting the computed rotational strength of individual helicene conformers to the experimentally observed sign flip in the solid-state ECD spectrum. In chiral perovskites, Fortino et al. applied DFT and TDDFT to model the chiroptical response of (R/S-MBA)₂PbI₄ and (R/S-MBA)₂SnI₄ clusters, identifying metal–ligand electronic overlap as the primary driver of chirality transfer from the organic cation to the inorganic sublattice, while Alehyane et al. combined ab initio molecular dynamics with TDDFT on cluster geometries to dissect the separate contributions of the chiral ligand and the metal–halide framework to the ECD signal of S-(3Br-MBA)₂PbI₄. These examples illustrate a common pattern: the cluster approach succeeds when chiroptical activity is dominated by the local electronic structure of a single chromophore or a tightly coupled chromophore pair, and when the primary goal is mechanistic assignment rather than quantitative prediction of absolute intensities. In this regime, cluster TDDFT provides chemically interpretable access to transition dipole orientations, rotational strength decompositions, and structure–property relationships that would be considerably more difficult to extract from a fully periodic calculation. Representative examples of this approach across chiral organic and hybrid crystalline materials are collected in Figure .

5.

Cluster-based TDDFT calculations of ECD in chiral organic and hybrid crystalline materials. (a) Electric (μ, red) and magnetic (m, blue) transition dipole moments in (S)-4PH-PDI-Ph and (R)-4CN-PDI-Ph crystals, showing how cyano ortho-substitution reduces the μ–m angle from 99° to 25°. Reproduced with permission from ref . Copyright 2024 Wiley-VCH GmbH. (b) Left: quantum mechanics/molecular mechanics (QM/MM) simulated ECD spectra of (SSSS)-MM and (RRRR)-PP crystal aggregates. Right: Transition dipole moments for (SSSS)-4CldiPDI in the crystal state. Reproduced from ref with permission from the Royal Society of Chemistry. (c) Periodic boundary condition model of (S-3Br-MBA)₂PbI₄ with the QM unit cell (red octahedra and black organic ligands) embedded in a periodic environment. Reproduced with permission from ref . Copyright 2025 Wiley-VCH GmbH. (d) The ECD spectra of R- and S-MBA₂PbI₄ from ab initio molecular dynamics (MD) ensemble calculations, and the experimental ECD spectra are also shown. Reproduced from ref with permission from the Royal Society of Chemistry.

2.3. Periodic Independent-Particle and Coupled-Perturbed Response Approaches

Periodic response calculations solve the ground-state Kohn–Sham problem under Bloch periodic boundary conditions and evaluate the optical response in reciprocal space. Plane-wave implementations such as VASP and QUANTUM ESPRESSO provide established ground-state solvers. , A common low-cost baseline is the independent-particle approximation (IPA), which neglects electron–hole interactions and treats the response as a superposition of single-particle Kohn–Sham transitions. , The IPA is computationally efficient and provides a transparent starting point for systems in which excitonic effects are modest. It is distinct from self-consistent coupled-perturbed periodic response, which includes the relaxation of the crystalline orbitals induced by the external perturbation.

Although electric-dipole momentum or velocity matrix elements are well-defined under periodic boundary conditions, the magnetic-dipole transition moment is not automatically well behaved because the orbital magnetic-dipole operator contains r × p. The electric-quadrupole operator likewise retains position dependence. Translationally invariant periodic formulations therefore express these quantities through Berry connections and velocity-based sum-over-states relations. For example, the electric-quadrupole contribution derived by Pozo Ocaña et al. can be written using velocity matrix elements and energy denominators as

$${\mathrm{q̅}}_{ln}^{bc}=-\frac{e}{2}{\sum }_{p\ne l,n}[\frac{v_{lp}^b{v}_{pn}^c}{{\omega }_{pl}{\omega }_{pn}}+(b\leftrightarrow c)]$$ 4

This expression is origin independent and reduces to the standard molecular formalism in the isolated-system limit, providing a rigorous connection between molecular and periodic formalisms. The complete optical-activity tensor obtained from a consistent combination of magnetic dipole, electric quadrupole, and band-dispersion contributions is gauge invariant. The individual terms are useful for analysis, but they should not be interpreted as independently appended or separately observable contributions outside the complete periodic formulation. ,,

A practical challenge in IPA calculations is achieving numerical convergence with respect to both the number of empty bands and the Brillouin-zone sampling density. Explicit empty-state summations can be avoided by evaluating Bloch-wavevector derivatives directly through finite differences. Brillouin-zone convergence is more subtle because the k-dependence of the response is often highly nonuniform and concentrated near band degeneracies and avoided crossings. For trigonal tellurium, Wang and Yan showed that an adaptive mesh targeting such hotspots achieved convergence with 1174 irreducible k-points, compared with 46,782 points on a uniform mesh. Adaptive refinement can therefore reduce the cost of localized one-particle response. This advantage is not universal: nonuniform meshes require additional care in enforcing momentum conservation in one- and two-electron integrals, which is essential for interband electron correlation and electron–hole coupling. Uniform or commensurate meshes, potentially combined with interpolation, are convenient options for correlated calculations.

The fundamental limitation of the IPA is its neglect of electron–hole interactions. Electron–hole interactions can substantially redistribute optical spectral weight relative to independent-particle calculations. , The IPA places spectral weight at and above the independent-particle gap and misses its redistribution into below-gap excitonic states. For the chiral perovskites studied by Multunas et al., the IPA misses the first excitonic ECD feature and does not reproduce the measured ECD quantitatively across the full spectral range. The IPA nevertheless retains important band-geometric contributions and can provide a useful baseline for covalent and metallic chiral solids when excitonic effects are modest; its quantitative reliability must be established for each material rather than assumed from the band-dispersion term alone. ,

Beyond the unrelaxed IPA, the self-consistent coupled-perturbed (SC–CP) periodic response includes the self-consistent relaxation of crystalline orbitals induced by the applied perturbation. This relaxation should not be conflated with the specific screened electron–hole interaction kernel introduced in GW-BSE calculations; they are distinct response ingredients and approximations. Rérat and Kirtman implemented an SC–CP method in the CRYSTAL code using the velocity gauge and reported calculations for α-quartz, tartaric acid crystal, and chiral carbon nanotubes; their study also reported separate IPA results. Desmarais et al. subsequently examined the effect of modern functionals and showed that exchange–correlation treatment and orbital relaxation are important for quantitative optical rotation. Forson et al. likewise used a coupled-perturbed DFT-PBC response for five chiral molecular crystals, demonstrating the importance of intermolecular interactions and, in two cases, identifying likely problems in experimental assignments. These studies benchmark relaxed periodic response and should not be classified as IPA-only calculations. Figure therefore compares examples obtained with periodic independent-particle and coupled-perturbed approaches.

6.

Applications of periodic independent-particle and coupled-perturbed response calculations to chiral crystalline systems. (a–c), from ref : (a) k-resolved optical rotation of trigonal Te in the two-dimensional hexagonal Brillouin zone at ℏω = 0.3 eV, showing contributions concentrated near the H-point corners; (b) decomposition of the optical rotatory power of Te into magnetic-dipole (m), electric-quadrupole (q), and band-dispersion (v) terms for light propagating parallel and perpendicular to the optic axis; and (c) calculated ECD spectra of a chiral (6,4) carbon nanotube resolved by contribution for light propagating perpendicular and parallel to the tube axis. Reproduced with permission from ref . Copyright 2023 American Physical Society. (d), from ref : optical rotatory dispersion of benzil calculated by coupled-perturbed DFT-PBC using the modified velocity gauge (MVG) and LG­(OI), compared with experiment. Reproduced with permission from ref . Copyright 2024 Wiley Periodicals LLC.

2.4. GW-BSE and Wannier Interpolation

Taken together, the studies summarized above demonstrate that first-principles calculations of chiroptical response in crystalline solids are feasible at both independent-particle and self-consistent coupled-perturbed levels. Two principal challenges nevertheless remain. First, Brillouin-zone convergence can be demanding because the optical-activity integrand may be strongly localized in k-space. Second, strongly excitonic systems may require a treatment with a screened electron–hole interaction kernel beyond the IPA and DFT-response approximations considered here. This requirement is distinct from the self-consistent orbital relaxation included in SC–CP. The quadrupole contribution can also be quantitatively important, the full orientation dependence of the optical-activity tensor must be retained for anisotropic crystals, and spin–orbit coupling must be treated carefully in heavy-element systems. The remainder of this section addresses these challenges.

For organic molecular crystals and other systems where excitonic effects are large, accurate treatment requires going beyond the IPA and incorporating electron–hole interactions from first principles. The standard approach within many-body perturbation theory (MBPT) proceeds in two sequential steps that together constitute the GW-BSE framework. In the first step, the quasiparticle band structure is obtained within the GW approximation, in which the exchange-correlation self-energy Σ­( r , r ′;ω) is approximated as Σ = (i/2π)∫GWdω′, where G is the single-particle Green’s function and W = ε^–1 v _c is the screened Coulomb interaction. The GW self-energy accounts for the dynamical, nonlocal, and frequency-dependent character of the electron–electron interaction. GW yields quasiparticle energies, not the optical gap itself; comparison with optical absorption requires electron–hole interactions to be included at the BSE stage. ,,−

In the second step, electron–hole interactions are incorporated by solving the Bethe-Salpeter equation (BSE), which takes the form of an eigenvalue problem for two-particle excitonic states eq ,

$${Ĥ}_{\mathrm{BSE}}|{\mathrm{\Psi }}_{\lambda }⟩=E_{\lambda }|{\mathrm{\Psi }}_{\lambda }⟩$$ 5

where Ĥ _BSE = H _diag + 2v_x – W acts on electron–hole pair amplitudes, with H _diag containing the quasiparticle energies, v_x the bare exchange interaction responsible for local-field effects, and W the screened direct Coulomb interaction responsible for excitonic binding. Solutions to the BSE yield excitonic eigenvalues E _λ and eigenvectors $|{\mathrm{\Psi }}_{\lambda }⟩={\sum }_{cv\mathit{k}}{A}_{cv\mathit{k}}^{\lambda }|c\mathit{k}⟩⟨vk|$ that mix electron–hole pairs at all k -points across the BZ, from which optical matrix elements and rotational strengths can be computed. The BSE captures both the redistribution of oscillator strength from independent-particle transitions to bound excitonic states and the renormalization of the optical gap by the electron–hole binding energy.

Wang and Yan developed a GW-BSE theory of optical activity and applied it to α-quartz. Their formulation treats dielectric spatial dispersion through two complementary forms, exciton envelope modulation (EEM) and sum-over-exciton-states (SOXS) expansion. The EEM form captures the low-frequency response, whereas the SOXS form is required to reproduce the full frequency dependence. Comparison with independent-particle and local-field treatments demonstrates the importance of excitonic many-body effects for the optical rotatory dispersion of α-quartz.

A major practical bottleneck for GW-BSE calculations of optical activity is the need for dense BZ sampling. Direct GW-BSE calculations on sufficiently fine k-grids are computationally inaccessible for most systems of interest. At the DFT level, Wannier interpolation, as implemented in the Wannier90 code, provides an efficient route to interpolate periodic single-particle quantities and optical matrix elements from a coarse k-grid to dense grids. ,, In this DFT/Wannier procedure, a unitary transformation converts Bloch states into maximally localized Wannier functions by minimizing the Marzari–Vanderbilt spread functional. The resulting real-space representation makes the single-particle Hamiltonian and relevant optical matrix elements short-ranged, so that they can be recovered on dense k-grids by Fourier transformation. , This established single-particle interpolation should not be conflated with an already validated, negligible-cost interpolation of all quantities entering the GW-BSE optical-activity kernel. Extending interpolation reliably to the complete excitonic response remains an important methodological challenge.

Taken together, the Wannier-interpolation strategy of Multunas et al., which accelerates Brillouin-zone convergence by one to two orders of magnitude, and the GW-BSE extension of Wang et al., which incorporates excitonic contributions to the optical-activity tensor through a first-order expansion in the photon wavevector, are individually validated and complementary components of predictive first-principles chiroptical spectroscopy of crystalline solids. They address distinct practical and physical challenges, respectively; however, a single generally applicable workflow that simultaneously combines excitonic and band-geometric effects with efficient interpolation has not yet been established. Figure illustrates the quantitative agreement with experiment achieved by these complementary methods: the Wannier-interpolated DFT reproduces the ECD of elemental Se and the optical rotation of Te (panels a, b), while the GW-BSE framework resolves the long-standing theory–experiment discrepancy for α-quartz (panels c, d).

7.

GW-BSE and Wannier interpolation calculations of chiroptical properties in crystalline systems. (a) Predicted ECD spectrum of trigonal Se compared with experiment, showing good agreement in both sign and relative magnitude after a scissor correction of the band gap. (b) Optical rotatory dispersion of trigonal Te obtained by Kramers–Kronig integration of the predicted ECD spectrum; inclusion of the electric-quadrupole (EQ) contribution is essential to reproduce even the correct sign of the experimental optical rotation. Reproduced with permission from ref . Copyright 2023 American Physical Society. (c) Optical rotatory dispersion of α-quartz calculated within the GW-BSE framework compared with experiment, demonstrating that many-body excitonic corrections are necessary to resolve the long-standing discrepancy between theory and experiment. (d) Energy-renormalized optical rotatory dispersion of α-quartz from the GW-BSE sum-over-exciton-states (SOXS) formulation, confirming quantitative agreement with experiment after accounting for quasiparticle energy renormalization. Reproduced with permission from ref . Copyright 2026 American Physical Society.

2.5. Outlook and Open Challenges

The past decade has witnessed a transformation in our ability to compute and understand ECD in crystalline materials, driven by the convergence of Wannier-function technology, many-body perturbation theory, and the geometric-phase formalism. Periodic single-particle and coupled-perturbed response, Wannier interpolation, and GW-BSE now provide rigorous and validated component methods for increasingly complex crystalline systems, while the connection to Berry curvature has opened a useful conceptual language for analyzing band-geometric aspects of these spectra. ,,, Nevertheless, a fully unified workflow merging excitonic and geometric contributions is not yet generally established, and several important challenges remain before the field can be considered truly mature.

2.5.1. Spin–Orbit Coupling and Heavy-Element Systems

Chiral perovskites containing heavy atoms such as Pb or Bi have attracted exceptional experimental attention in recent years due to their potential in chiral optoelectronic and spintronic applications. ,,, In these systems, spin–orbit coupling (SOC) is a leading-order effect on the electronic structure, splitting bands and strongly renormalizing optical gaps and transition intensities. Multunas et al. found that SOC influences the ECD of chiral perovskites primarily through modifications of the electronic energies rather than through direct spin matrix element contributions, which is encouraging, as it suggests that scalar-relativistic or spin-free calculations followed by a second-variation or two-step SOC treatment may be adequate for many systems, substantially reducing the computational cost. Nevertheless, chiral rare-earth halide perovskites have attracted growing attention owing to their highly predictable and tunable emission properties arising from 4f→4f transitions, yet the direct participation of 4f electrons in the low-energy optical response, combined with the intricate interplay between SOC, crystal field splitting, and interconfigurational mixing, poses severe challenges for a first-principles theoretical description, and their ab initio treatment remains an important open problem in the field. −

2.5.2. The Anisotropy Problem and Connection to Experiment

The ECD spectrum of a single crystal is intrinsically anisotropic: the measured absorbance difference depends strongly on the orientation of the crystal relative to the propagation direction and polarization of the incident light. This orientation dependence is encoded in the full optical-activity tensor γ _ijl (ω), a rank-3 pseudotensor that transforms under the point-group operations of the crystal class and can have a symmetry-restricted set of independent components. In practice, most ECD measurements on crystalline materials use powders or polycrystalline thin films, whose signals depend on the distribution of crystallite orientations. Orientational averaging can discard structural information contained in the anisotropic tensor components, and agreement between a scalar-averaged theoretical prediction and experiment is a weaker validation than a full tensor comparison. First-principles calculations naturally yield the complete tensor, enabling direct comparison with angle-resolved single-crystal ECD. Multunas et al. demonstrated that the unit-cell directions with the largest ECD can be identified computationally for two-dimensional chiral perovskites and that organic- and inorganic-sublattice contributions are spectrally and orientationally distinct. Developing standardized single-crystal ECD measurements and computational protocols for full-tensor comparison is therefore an important goal. Related polarization-resolved measurements by Wang et al. revealed linearly polarized photoluminescence from ultrasmall CsPbBr₃ nanoparticles under linearly polarized excitation, illustrating a polarization-dependent optical response distinct from circular dichroism.

2.5.3. Toward a Unified Excitonic-Geometric Framework

Perhaps the most important challenge is to unify the GW-BSE description of excitonic optical activity with the Berry-connection formulation of band-geometric contributions in a single coherent framework. , These approaches currently coexist in the literature but have not been fully merged. GW-BSE is appropriate for systems with large exciton binding energies, whereas independent-particle band-geometric formulations are most reliable when excitonic effects are weak. Between these limits lies a broad class of materialsincluding chiral perovskites, transition-metal dichalcogenides, and topological semimetalsin which excitonic and band-geometric contributions can both be important. The formulation of Wang et al., in which the GW-BSE dielectric response is expanded to first order in q and the optical-activity tensor contains both excitonic matrix elements and Berry-connection contributions, provides a route toward such a unified description. Combining this formulation with Wannier interpolation and translationally invariant periodic multipole theory could make the treatment computationally tractable across a wider range of chiral crystalline materials. ,

3. Conclusions

This Perspective has surveyed first-principles methods for calculating the electronic circular dichroism of crystalline materials, from molecular-cluster TDDFT through periodic independent-particle and self-consistent coupled-perturbed response to GW-BSE, alongside DFT-level Wannier interpolation. The central theoretical requirement is a gauge-consistent periodic treatment of position-dependent multipole matrix elements through Berry-connection and related translationally invariant formulations, together with an explicit treatment of electron–hole interactions when excitonic effects are strong. Band-geometric contributions, Brillouin-zone interpolation, orbital relaxation, and excitonic response are individually validated, complementary parts of the computational toolkit rather than universally separable levels of accuracy. A general framework that fully unifies excitonic and geometric contributions remains unresolved. Remaining challenges include relativistic treatments of chiral rare-earth halide perovskites, benchmarking the full optical-activity tensor against single-crystal anisotropy measurements, and combining theoretical calculations with targeted experiments.

The authors declare no competing financial interest.

Published as part of ACS Nanoscience Au special issue “2026 Rising Stars in Nanoscience”.