Evaluation of molecular photophysical and photochemical properties using linear response time-dependent density functional theory with classical embedding: Successes and challenges

Abstract

Time-dependent density functional theory (TDDFT) based approaches have been developed in recent years to model the excited-state properties and transition processes of the molecules in the gas-phase and in a condensed medium, such as in a solution and protein microenvironment or near semiconductor and metal surfaces. In the latter case, usually, classical embedding models have been adopted to account for the molecular environmental effects, leading to the multi-scale approaches of TDDFT/polarizable continuum model (PCM) and TDDFT/molecular mechanics (MM), where a molecular system of interest is designated as the quantum mechanical region and treated with TDDFT, while the environment is usually described using either a PCM or (non-polarizable or polarizable) MM force fields. In this Perspective, we briefly review these TDDFT-related multi-scale models with a specific emphasis on the implementation of analytical energy derivatives, such as the energy gradient and Hessian, the nonadiabatic coupling, the spin–orbit coupling, and the transition dipole moment as well as their nuclear derivatives for various radiative and radiativeless transition processes among electronic states. Three variations of the TDDFT method, the Tamm–Dancoff approximation to TDDFT, spin–flip DFT, and spin-adiabatic TDDFT, are discussed. Moreover, using a model system (pyridine–Ag₂₀ complex), we emphasize that caution is needed to properly account for system–environment interactions within the TDDFT/MM models. Specifically, one should appropriately damp the electrostatic embedding potential from MM atoms and carefully tune the van der Waals interaction potential between the system and the environment. We also highlight the lack of proper treatment of charge transfer between the quantum mechanics and MM regions as well as the need for accelerated TDDFT modelings and interpretability, which calls for new method developments.

I. INTRODUCTION

To study the systems at the atomic and molecular levels, one has to follow the principles of quantum mechanics (QM). Even though the first-principles electronic structure theory and the quantum dynamics methods have achieved great success in describing the ground state of molecular systems, computing excited states still faces many challenges because it implies not only to find the higher-energy solution of the electronic Schrödinger equation but also to solve many challenging conceptual and technical problems, such as the electron–nuclear coupling, the state mixing of the same spin and different spin multiplicities, or the entangled optical and dark processes, which usually require one to abandon the approximations such as the adiabatic Born–Oppenheimer (BO) and to calculate the accurate structure properties, such as potential energy surfaces, nuclear forces, harmonic and non-harmonic vibrational frequencies, nonadiabatic couplings (NAC), and spin–orbit couplings (SOC).¹ Furthermore, most of the photophysical and photochemical processes of technological and biological interest take place in the condensed phase or in the presence of an environment surrounding the molecular system such as solvents, protein scaffolds, and near semiconductor or metal surfaces. In this case, one has to account for the system–environment interaction for properly describing these processes.²

For composite systems, such as solute–solvent systems, fluorescent proteins, and even nanostructures that integrate molecules or semiconductors with metal nanoparticles (MNPs), a small region of the system plays a predominant role in the excited-state properties and processes of the whole system. Thus, the total system can usually be partitioned into an active region ranging from a single molecule to a few molecules and an embedding environment. The former is mainly responsible for the observed process or property, and the latter does not take a direct part in the process but acts as a perturbation affecting the electronic structure and the dynamics of the core active system. The active region is then treated by a high-level QM method, whereas the rest is described by a lower-level approach, such as molecular mechanical (MM) force fields and polarizable continnum models (PCMs). The integration of QM methods with classical descriptions within multiscale models provides a natural way to focus the computation on a small area in a larger environment without significantly increasing the computational cost.³ Currently, scientists have developed a family of multiscale theoretical models, such as QM/(MM),^4,5 QM/PCM,^6,7 QM/MM/PCM,⁸ and QM/electromagnetism (EM) (the hybrid QM and electromagnetic mechanics or electrodynamics method).^9–12

The QM methods for describing the molecular electronic systems can be divided into three categories: semi-empirical quantum chemistry methods, ab initio wavefunction-based methods, and density functional theory (DFT) methods.¹³ The post-Hartree–Fock (HF) methods have been very successful in describing the ground and excited-state electronic states of small molecules, especially when the molecules are near the equilibrium configuration. The semi-empirical methods are much faster than the ab initio ones. However, they require a couple of atom-specified parameters and are thus only suitable for parameterized systems. For a medium and even large sized molecular system, the most widely used method is DFT. The last two decades have witnessed an explosion of DFT-based calculations in the field of computational chemistry.^14–16 The modeling of ground state processes, such as chemical,¹⁷ surface,¹⁸ and enzyme reactions,¹⁹ is mostly carried out using Kohn–Sham DFT (KS-DFT)^20,21 calculations. Meanwhile, excited-state processes (such as UV-vis absorption,^22–25 fluorescence,^26,27 phosphorescence,^28–30 chemiluminescence,³¹ bioluminescence,^32–34 and energy transfer processes in natural and artificial photosynthetic systems and organic and dye-sensitized solar cells^35–40) have been modeled extensively using time-dependent DFT (TDDFT),^41,42 which is the excited-state counterpart of the KS-DFT.

The vast popularity of KS-DFT and TDDFT calculations arises from three factors. The first is their cost-effectiveness. They are simply orders of magnitudes cheaper than highly reliable quantum chemistry methods, such as coupled-cluster methods,^43,44 equation-of-motion coupled-cluster methods,^45–49 and multi-configurational and multi-reference wavefunction methods.^50–52 Moreover, the high-order energy derivatives, transition couplings, or molecular properties from DFT and TDDFT are relatively simple to be implemented and cost-efficient, which are essential for studying the rich chemical and physical events we are interested in, e.g., the evaluation of spectroscopic intensities and rates and quantum yields of excited-state processes. The second is their reasonable accuracy if a suitable exchange–correlation (XC) functional is employed for the particular problem at hand. For example, a valence electronic excitation might be best described with a conventional hybrid functional (such as B3LYP and PBE0), whereas a charge-transfer or Rydberg excitation necessitates the use of range-separated hybrid functionals.^53,54 Besides the above-mentioned two factors, the DFT/TDDFT methods have also been implemented in many efficient and user-friendly commercial or noncommercial software packages that were developed and made accessible to a wide community of researchers. This wide availability of DFT/TDDFT methods in modern computational chemistry programs combined with versatile and user-friendly visualization tools makes them accessible to researchers with no sophisticated background in theoretical chemistry.

Two versions of the TDDFT implementation have been familiar to the quantum chemistry community. One is to numerically solve the time-dependent Kohn–Sham equations directly in the real-time domain (RT-TDDFT).⁵⁵ The RT-TDDFT approach has been implemented in the real space, in plane wave bases, and in atomic orbital basis sets and has been used to calculate the linear and nonlinear spectra and describe the photochemistry by combining the Ehrenfest or surface hopping nuclear dynamics. Many review articles have summarized the recent progress of RT-TDDFT approaches (e.g., Refs. ^56–60 and references therein). The other is the implementation of TDDFT at the linear response level (LR-TDDFT),^61–66 which looks like the standard implementation of LR-time-dependent Hartree–Fock (TDHF) already available in most quantum chemistry codes. A set of static Casida’s equations was derived and solved to obtain the molecular excitation energies and transition vectors as well as the oscillator strengths. Within the framework of LR-TDDFT, the nuclear gradient^67–69 and Hessian^70,71 of excited-state energies and the nonadiabatic coupling between electronic states^72–74 have all been implemented. Consequently, this LR-TDDFT scheme has become the most widely used one to extract the energies of low-lying excited states, optimize excited-state geometries, calculate Stokes shifts, and explore the excited-state potential energy surfaces. Here, we only focus on reviewing approaches based on LR-TDDFT with a specific emphasis on their applications to the calculation of electronic structure quantities required by describing molecular excited-state properties and processes.

Most earlier TDDFT calculations were focused on gas-phase excited states. However, as mentioned earlier, there has been an ever-growing interest to extend these calculations to composite systems for the modeling of excited-state processes of a single or a few molecules embedded in a condensed-phase medium, such as solvents,^75–77 macromolecules,^78–80 and metal nanoparticles.^10,81–84 For these system–environment complexes, multiscale modeling approaches have been proposed with different levels of theories for the different regions. There are generally two categories of embedding methods for TDDFT calculations within the environment: density embedding and classical embedding. Within the category of density-embedding schemes, recent efforts include the projection-based embedding theory,^85,86 frozen-density embedding,⁸⁷ and polarized many-body expansion scheme⁸⁸. When the excited states of molecular aggregates are considered, the subsystem TDDFT methods have shown accuracy and efficiency for large systems⁸⁹ and inter-region charge-transfer excitations.^90,91 On the other hand, the hybrid QM/MM method has been more commonly used in TDDFT calculations with the aforementioned classical embedding, such as QM/MM, QM/PCM, QM/MM/PCM, and QM/EM. In the following, we mainly focus on TDDFT calculations with a classical embedding description of the environment using either PCM^6,7 or polarizable MM force fields (MMpol).⁹²

While tremendous progress has been made in TDDFT/PCM and TDDFT/MM methodologies, the application of these methods to model excited-state properties and processes in a condensed-phase medium still faces three great challenges (in addition to the choice of suitable DFT XC functionals):

• •System–environment interactions. For each system–environment complex, there lacks a standard protocol for tuning the QM/MM electrostatic and van der Waals interactions between the system and its environment, which is critical to avoid over-polarization and geometry distortions. In addition, missing are approaches for describing full charge transfer excitations that move system electrons into the environment or partial charge transfer excitations where the participating system orbitals extend into the environment.
• •Computational cost. Standard single-point TDDFT excitation energy calculations and geometry optimizations are currently limited to systems with up to a couple of 100 atoms. To theoretically study a variety of molecular excited-state properties and processes, one has to calculate the derivatives of excitation energies with respect to perturbation parameters, such as the nuclear coordinates and external fields, the NACs, and the nuclear derivatives of the transition dipole or SOC matrix element. These quantities require even much higher computational cost than the excitation energy calculation. Another bottleneck in TDDFT/MMpol calculations arises from the need to solve for polarizable charges or dipoles, which can become rather expensive with a large polarizable MM region.^93,94
• •Interpretability. While numerous TDDFT calculations are routinely performed on new dye molecules and bioluminescence probes, few analysis tools beyond electron (difference and transition) densities, atomic partial charges, and excited-state and transition dipole moments are available for helping to directly interpret the observed photochemical properties.⁹⁵ New analysis procedures and design principles only started to emerge recently for predicting the substituent, solvent, and macromolecular effects on the measured spectra and quantum yield.^96–100

This Perspective is organized as follows. In Sec. II, we briefly introduce a variety of electronic structure quantities required by the evaluation of molecular properties, which are closely related to the calculations of the derivatives of the energy or wavefunction with respect to the nuclear coordinates or other perturbation parameters. The current development of hybrid TDDFT with MM and PCM is summarized in Sec. III, which is followed by a discussion on the choice of TDDFT methods in Sec. IV. We then elaborate in Sec. V on the three challenges listed above for the TDDFT modeling of excited-state processes in complex environments. A summary statement will be given in Sec. VI.

II. A VARIETY OF ELECTRONIC STRUCTURE QUANTITIES FOR EVALUATION OF MOLECULAR PHOTOPHYSICAL AND PHOTOCHEMICAL PROPERTIES

A. Quantities for describing properties of a specific electronic excited state

Molecular properties characterize molecules and their behaviors, and they represent the link between experimental observable quantities and theoretical calculations. There are a wide variety of molecular properties, which are related to numerous physical phenomena. At first, we address a variety of properties of molecules at a given electronic state, which are related with the calculations of energy derivatives with respect to the perturbation parameters.^101,102 For example, the forces and the force constants are given as the first and second derivatives of the molecular energy with respect to the nuclear coordinates, respectively,^67,103 and the infrared intensities can be computed as the cross second derivatives of the molecular energy with respect to the components of a static electric field and nuclear coordinates.^104–106 As such, formulating energy derivatives with respect to the external perturbation variables yields a wealth of time-independent molecular properties.

1. The excited-state energy gradient and Hessian within TDDFT

The energy gradient and hessian with respect to nuclear coordinates have been widely used to explore molecular potential energy surfaces and to calculate vibrational spectra and thermochemical parameters. For a given molecular geometry, the energy of the Ith excited state can be expressed as the sum of the ground-state energy and the excitation energy,

$$E_I=E_g+{\omega }_I,$$ (1)

where E_g is the ground-state energy and ω_I is the TDDFT excitation energy of the Ith excited state. Then, the energy derivatives of a given excited state I can be evaluated as the sum of the ground-state energy derivatives and the derivatives of the corresponding TDDFT excitation energy.

In the framework of LR-TDDFT, the excitation energy is defined as the pole of the LR function to an external perturbation. The excitation energies and the corresponding transition vectors can be obtained from an usual non-Hermitian eigenvalue equation,⁴²

$$\left(\begin{array}{cc}\hfill \mathbf{A}\hfill & \hfill \mathbf{B}\hfill \\ \hfill {\mathbf{B}}^{†}\hfill & \hfill {\mathbf{A}}^{†}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \mathbf{X}\hfill \\ \hfill \mathbf{Y}\hfill \end{array}\right)=\omega \left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right)\left(\begin{array}{c}\hfill \mathbf{X}\hfill \\ \hfill \mathbf{Y}\hfill \end{array}\right).$$ (2)

The matrix representations of A and B in the molecular orbital (MO) basis are written as

$$A_{ia\sigma ,jb{\sigma }^{\prime }}={\delta }_{ab}{\delta }_{ij}{\delta }_{\sigma {\sigma }^{\prime }}({\epsilon }_{a\sigma }-{\epsilon }_{i{\sigma }^{\prime }})+\frac{\partial F_{ia\sigma }}{\partial P_{jb{\sigma }^{\prime }}},$$ (3)

$$B_{ai\sigma ,bj{\sigma }^{\prime }}=\frac{\partial F_{ai\sigma }}{\partial P_{jb{\sigma }^{\prime }}}.$$ (4)

Here, ɛ_pσ is the pth σ (σ = α or β) spin Kohn–Sham MO energy. The indices i, j, k, …; a, b, c, …; and p, q, r, … denote occupied, virtual, and generic MOs, respectively. The transition vectors |X⟩ and |Y⟩ are defined on the same Hilbert spaces of occ ⨂ vir, which satisfy the following biorthonormal condition:

$$⟨{\mathbf{X}}_I,{\mathbf{Y}}_I|{\mathbf{X}}_J,-{\mathbf{Y}}_J⟩={\delta }_{IJ}.$$ (5)

After these transition vectors are solved, the excitation energy of the Ith TDDFT excited state can be expressed as

$$\begin{array}{ll}\hfill {\omega }_I& =\frac{1}{2}{({\mathbf{X}}_I+{\mathbf{Y}}_I)}^{†}\cdot (\mathbf{A}+\mathbf{B})\cdot ({\mathbf{X}}_I+{\mathbf{Y}}_I)\hfill \\ \hfill & +\frac{1}{2}{({\mathbf{X}}_I-{\mathbf{Y}}_I)}^{†}\cdot (\mathbf{A}-\mathbf{B})\cdot ({\mathbf{X}}_I-{\mathbf{Y}}_I).\hfill \end{array}$$ (6)

Direct differentiation of ω_I in Eq. (6) with respect to the nuclear coordinates (R) requires one to calculate first-order nuclear derivatives of molecular orbital coefficients (C) and transition vectors (X and Y). In order to avoid these calculations, one would define an auxiliary energy functional as⁶⁸

$$\begin{array}{ll}\hfill {\mathcal{L}}_e[\mathbf{X},{\omega }_I,\mathbf{C},\mathbf{Z},\mathbf{W}]& =E_g+{\omega }_I-{\lambda }_I(\mathrm{Tr}({\mathbf{X}}_I^2-{\mathbf{Y}}_I^2)-1)\hfill \\ \hfill & +\sum _{ai\sigma }{Z}_{ai\sigma }{F}_{ai\sigma }+\sum _{pq\sigma }{W}_{pq\sigma }(S_{qp\sigma }-{\delta }_{pq}).\hfill \end{array}$$ (7)

With the construction of the Lagrangian ${\mathcal{L}}_e$, one has

$$\frac{d{E}_I}{d{R}_i}=\frac{\partial {\mathcal{L}}_e}{\partial R_i}$$ (8)

since the ${\mathcal{L}}_e$ is stationary with respect to all the variational parameters, including C, X, and Y. The Lagrange multipliers Z and W are determined by the condition

$$\frac{\partial {\mathcal{L}}_e}{\partial C_{\mu p\sigma }}=0,$$ (9)

and the equation derived thereof is called the z-vector equation.¹⁰⁷ Once the energy gradient of an excited state is formulated, one can straightforwardly differentiate the energy gradient to derive the excited-state Hessian. The calculation of the excited-state Hessian requires one to calculate the first-order nuclear derivatives of molecular orbital coefficients (C^R) and transition vectors (X^R and Y^R) explicitly, which can be obtained by solving the coupled perturbed self-consistent field (CPSCF) equations¹⁰³ and CP-TDDFT equations,⁷¹ respectively. A detailed formulation of the energy Hessian of TDDFT excited states can be found in Ref. 70 [with the Tamm–Dancoff approximation (TDA)] and Ref. 71 (full TDDFT). The reader is referred to Refs. ^69–71 and ^108–116 for additional details.

2. Electric dipole moments, polarizabilities, and infrared/Raman intensities

The molecular dipole moment and polarizability of a given electronic state are defined as

$${\mu }_x=-\frac{\partial E}{\partial F_x},$$ (10)

$${\alpha }_{xy}=\frac{\partial {\mu }_x}{\partial F_y}=-\frac{{\partial }^{2}E}{\partial F_x\partial F_y},$$ (11)

where F_x/F_y are the x/y components of the perturbing electric field. The expectation value of the electric dipole moment of a specific electronic state I is usually calculated as

$${\overline{\mu }}_x^I=⟨{\mathrm{\Psi }}_I|{\widehat{\mu }}_x|{\mathrm{\Psi }}_I⟩=\mathrm{Tr}({\mathbf{P}}_I{\mathit{\mu }}_x).$$ (12)

For the Ith excited state, its one-electron reduced density matrix, P_I, is composed of^{70,71,116,117}

$${\mathbf{P}}_I={\mathbf{P}}_0+\mathrm{\Delta }{\mathbf{P}}_I+{\mathbf{Z}}_I.$$ (13)

Here, P₀ is the reduced one-electron density matrix of the ground state; ΔP_I is the unrelaxed difference density matrix, $\mathrm{\Delta }{\mathbf{P}}_I=\frac{1}{2}{\mathbf{C}}_{\text{v}}({\mathbf{X}}_I+{\mathbf{Y}}_I){({\mathbf{X}}_I+{\mathbf{Y}}_I)}^{†}{\mathbf{C}}_{\text{v}}^{†}+\frac{1}{2}{\mathbf{C}}_{\text{v}}({\mathbf{X}}_I-{\mathbf{Y}}_I){({\mathbf{X}}_I-{\mathbf{Y}}_I)}^{†}{\mathbf{C}}_{\text{v}}^{†}-\frac{1}{2}{\mathbf{C}}_{\text{o}}{({\mathbf{X}}_I+{\mathbf{Y}}_I)}^{†}({\mathbf{X}}_I+{\mathbf{Y}}_I){\mathbf{C}}_{\text{o}}^{†}-\frac{1}{2}{\mathbf{C}}_{\text{o}}{({\mathbf{X}}_I-{\mathbf{Y}}_I)}^{†}({\mathbf{X}}_I-{\mathbf{Y}}_I){\mathbf{C}}_{\text{o}}^{†}$, where C_o and C_v are the occupied and virtual MO coefficient matrices, respectively; and Z_I corresponds to the z-vector, denoting the relaxed part of the excited-state density matrix.

Moreover, the Ith electronic state’s polarizability tensor can be calculated as

$${\overline{\alpha }}_{xy}^I=\mathrm{Tr}({\mathbf{P}}_I^{F_y}{\mathit{\mu }}_x),$$ (14)

where ${\mathbf{P}}_I^{F_y}$ denotes the derivative of the one-electron density matrix of a given electronic excited state with respect to the y component of an external field. This expression indicates that it is much easier and faster to calculate the polarizability than the nuclear Hessian because the former is merely related to the energy derivatives with respect to external electric fields so that the explicit derivatives of atomic orbitals in one- and two-electron integrals with respect to nuclear coordinates are not required.

One may be further interested in the nuclear derivatives of the above two quantities (μ and $\overline{\mathit{\alpha }}$) since they are closely related to the calculations of infrared (IR) and Raman intensities, respectively.^118–120 In the harmonic approximation, the transition electric dipole moment, $⟨{\mu }_{fi}^I⟩$, and the polarizability, $⟨{\alpha }_{fi}^I⟩$, between the initial and final vibronic states, which enter the expressions of IR absorption and Raman scattering intensities, can be expanded in a power series of the molecular normal coordinates Q as

$$〈{\mu }_{fi}^I〉={\overline{\mu }}^I(0)〈{\mathrm{\Lambda }}_{If}|{\mathrm{\Lambda }}_{Ii}〉+\sum _k{\left(\frac{\partial {\overline{\mu }}^I}{\partial Q_k}\right)}_0〈{\mathrm{\Lambda }}_{If}|Q_k|{\mathrm{\Lambda }}_{Ii}〉+\cdots ,$$ (15)

$$〈{\alpha }_{fi}^I〉={\overline{\alpha }}^I(0)〈{\mathrm{\Lambda }}_{If}|{\mathrm{\Lambda }}_{Ii}〉+\sum _k{\left(\frac{\partial {\overline{\alpha }}^I}{\partial Q_k}\right)}_0〈{\mathrm{\Lambda }}_{If}|Q_k|{\mathrm{\Lambda }}_{Ii}〉+\cdots .$$ (16)

Under the dipole approximation, the IR intensity corresponding to the transition from the ith vibrational state associated with the electronic state I to the fth vibrational state within the same electronic state is proportional to ${\left(\frac{\partial {\overline{\mu }}^I}{\partial Q_k}\right)}_0^2$, and the Raman intensity is closely related to the calculation of nuclear derivatives of molecular polarizability.^118,121 Here, we assume that the pure-spin BO (psBO) wavefunctions of initial and final vibronic states can be written as products of an electronic wavefunction Ψ and a vibrational wavefunction Λ, such as ${}^{\sigma }\mathrm{\Phi }_n(q,Q)={}^{\sigma }\mathrm{\Psi }_N(q,Q){\mathrm{\Lambda }}_{Nn}(Q)$, with normal coordinates $Q_k={\sum }_{i=1}^{3N}{l}_{ik}{q}_i$. The q_i’s denote the mass-weighted Cartesian coordinates, and the transformation coefficients l_ik are chosen such that the potential energy does not depend on cross products.

B. Quantities for describing transitions between electronic states

Transition couplings between different electronic states play central roles in many physical and chemical processes. To characterize a photophysical process, such as optical absorption and radiative emission (fluorescence and phosphorescence), or radiativeless transition, such as internal conversion (IC), intersystem crossing (ISC), and decay through a conical intersection (CI), one usually has to account for simultaneous changes in the vibrational and electronic states. Within the perturbation theory framework, one will need to combine the electronic structure theories and the quantum dynamics methods. The former is adopted to obtain the electronic structure parameters, and the latter is used to describe the nuclear dynamics.

According to the Fermi’s golden rule,¹²² the transition rate (k) from the initial state i to a dense manifold of final states f can be defined as

$$k_i=\frac{2\pi }{\hslash }\sum _f{⟨{\mathrm{\Phi }}_f|{\widehat{H}}^{\prime }|{\mathrm{\Phi }}_i⟩}^2\delta \left(\hslash \omega -(E_f-E_i)\right),$$ (17)

where Φ_i and Φ_f are the wavefunctions of the initial and final vibronic states, respectively, and ω is the frequency of the external radiation. ${\widehat{H}}^{\prime }=\widehat{H}-{\widehat{H}}^{el}$ (the electronic Hamiltonian) denotes the perturbation Hamiltonian and can be written as

$${\widehat{H}}^{\prime }=-\vec{\mu }\cdot \vec{f}+{\widehat{T}}_{\text{N}}+{\widehat{H}}_{\mathrm{S}\mathrm{O}},$$ (18)

where the first, second, and third operators correspond to the matter–field interaction, the nuclear kinetic energy, and spin–orbit interaction, respectively. The radiative transition, such as electronic absorption or emission, is usually governed by the matrix element of $⟨{\mathrm{\Phi }}_f|\widehat{\mu }|{\mathrm{\Phi }}_i⟩$ while the nonradiative processes, such as IC and ISC or reversed ISC (RISC), are determined by the matrix elements of the latter two operators, respectively. For example, if we restrict ourselves up to second-order terms, the coupling matrix element of an ISC process can be expressed as

$$\begin{array}{ll}\hfill H_{fi}^{\prime }& =〈{}^3\mathrm{\Phi }_f\mid {\widehat{H}}_{\text{SO}}\mid {}^1\mathrm{\Phi }_i〉\hfill \\ \hfill & +\sum _{m\ne i}[〈{}^3\mathrm{\Phi }_f\mid {\widehat{H}}_{\text{SO}}\mid {}^1\mathrm{\Phi }_m〉〈{}^1\mathrm{\Phi }_m\mid {\widehat{T}}_{\text{N}}\mid {}^1\mathrm{\Phi }_i〉/(E_i-E_m)]\hfill \\ \hfill & +\sum _{n\ne f}[〈{}^3\mathrm{\Phi }_f\mid {\widehat{T}}_{\text{N}}\mid {}^3\mathrm{\Phi }_n〉〈{}^3\mathrm{\Phi }_n\mid {\widehat{H}}_{\text{SO}}\mid {}^1\mathrm{\Phi }_i〉/(E_f-E_n)],\hfill \\ \hfill & =H_{\text{SO}}^{FI}{\mid }_{Q=0}〈{\mathrm{\Lambda }}_{Ff}(Q^{\prime })|{\mathrm{\Lambda }}_{Ii}(Q)〉\hfill \\ \hfill & +\sum _{k=1}^{3N-6}\left(\frac{\partial H_{\text{SO}}^{FI}}{\partial Q_k}{\mid }_{Q=0}〈{\mathrm{\Lambda }}_{Ff}(Q^{\prime })|Q_k|{\mathrm{\Lambda }}_{Ii}(Q)〉\right)\hfill \\ \hfill & -\sum _{k=1}^{3N-6}{\left[\sum _{M\ne I}\frac{H_{\text{SO}}^{FM}{d}_{MI,1}^k}{(E_I-E_M)}+\sum _{N\ne F}\frac{d_{FN,3}^k{H}_{\text{SO}}^{NI}}{(E_F-E_N)}\right]}_{Q=0}\hfill \\ \hfill & \times 〈{\mathrm{\Lambda }}_{Ff}(Q^{\prime })|\frac{\partial }{\partial Q_k}|{\mathrm{\Lambda }}_{Ii}(Q)〉.\hfill \end{array}$$ (19)

Here, the summations extend over the complete sets of psBO vibronic states of the given multiplicity, and the psBO wavefunctions of the initial and final vibronic states have been written as products of an electronic wavefunction Ψ and a vibrational wavefunction Λ. The relation

$$\begin{array}{ll}\hfill ⟨{}^{\sigma }\mathrm{\Phi }_e\mid {\widehat{T}}_{\text{N}}\mid {}^{\sigma }\mathrm{\Phi }_r⟩& \sim -{\hslash }^2\sum _{k=1}^{3N-6}⟨{}^{\sigma }\mathrm{\Psi }_E\mid {\nabla }_k\mid {}^{\sigma }\mathrm{\Psi }_R⟩\hfill \\ \hfill & \times ⟨{\mathrm{\Lambda }}_{Ee}(Q^{\prime })|{\nabla }_k{\mathrm{\Lambda }}_{Rr}(Q)⟩\hfill \end{array}$$ (20)

has been applied, and

$$d_{ER,\sigma }^k=⟨{}^{\sigma }\mathrm{\Psi }_E\mid \partial /\partial Q_k{}^{\sigma }\mathrm{\Psi }_R⟩$$ (21)

is defined as the NAC vector between two excited states of the given spin multiplicity σ. The spin–orbit coupling matrix element (SOCME) has been assumed to be normal mode-dependent, and it reads

$$H_{\text{SO}}^{FI}=⟨{\mathrm{\Psi }}_F|{\widehat{H}}_{\text{SO}}|{\mathrm{\Psi }}_I⟩=H_{\text{SO}}^{FI}(Q=0)+\frac{\partial H_{\text{SO}}^{FI}}{\partial Q}{|}_{Q=0}Q+\cdots ,$$ (22)

which thus includes the Franck–Condon (FC) and Herzberg–Teller (HT) contributions. Although Eq. (19) specifically refers to a S → T crossing, the corresponding expression for the RISC is readily written down.

The transition dipole moment between pure S₀ and T₁ states is zero. In the case of phosphorescence, one has to resort to perturbation theory by considering the SOC perturbation to give rise to admixtures of states with opposite spin.

It is clear that to quantitatively describe the radiative and radiationless processes, such as the molecular vibronic spectra¹²³ and the transition rates, within the perturbation theory framework, one has to know the geometric and electronic structure parameters at first. More specifically, the energy gradients of ground and excited states are required to reach the minimum on each potential energy surface. Meanwhile, Hessians of the ground- and excited-state energies are indispensable for acquiring the normal modes for each electronic state, establishing the Duschinsky relation,¹²⁴ as well as incorporating the FC/HT and HT contributions.

1. Non-adiabatic coupling

The nonadiabatic coupling (NAC) is needed as long as the psBO approximation breaks down, which is quite common for radiationless processes, such as IC,¹²⁵ decay through CI,^126,127 and potentially ISC and RISC.¹ These nonadiabatic transitions are governed by ${\widehat{T}}_N$ mentioned in Eq. (18), and the NAC between two vibronic states could be derived as¹²⁸

$$\begin{array}{ccc}\hfill H_{\text{NAC}}& =\hfill & ⟨{\mathrm{\Phi }}_m(q,Q)\left|-,\frac{{\hslash }^2}{2},\sum _k,\frac{{\partial }^2}{\partial Q_k^2}\right|{\mathrm{\Phi }}_n(q,Q)⟩\hfill \\ \hfill & \sim \hfill & -{\hslash }^2\sum _k{d}_{mn}^k⟨{\mathrm{\Lambda }}_m(Q)\left|\frac{\partial {\mathrm{\Lambda }}_n(Q)}{\partial Q_k}\right.⟩,\hfill \end{array}$$ (23)

where ${\widehat{T}}_N$ is split into two momentum operators for electronic and vibrational states, respectively, and

$$d_{mn}^k=⟨{\mathrm{\Psi }}_m(q,Q)\left|\frac{\partial {\mathrm{\Psi }}_n(q,Q)}{\partial Q_k}\right.⟩$$ (24)

is the derivative coupling (DC). The DC associated with TDDFT excited states can be formulated using the equation-of-motion (EOM) and response theory^72,73,129 or pseudo-wavefunction approach.^74,130 The DC implementations with different DFT methods are summarized in Table I. In short, it is not necessary to explicitly solve for either orbital responses or amplitude derivatives in the computation of $d_{mn}^k$; only one z-vector needs to be solved for instead. One may refer to Ref. 131 for a review about the derivative couplings in TDDFT.

TABLE I.

Development of analytical gradient, Hessian, and nonadiabatic coupling (NAC) of various TDDFT and TDDFT/Classical embedding methods.

Method	Energy	Gradient	Hessian	NAC
TDDFT	Reference 61	References 67, 68, 108,	References 70 and 71	References 72–74
		110, and 150–154		and 155–159
SF-DFT	References 160–163	References 164 and 165	⋯	References 166 and 167
SA-TDDFT	References 130 and 168	Reference 168	⋯	References 130 and 169
TDDFT/PCM	References 22, 75, and 170–174	References 69, 114, and 115	Reference 116
TDDFT/MM	Many	Reference 175	Reference 175	References 176 and 177
TDDFT/MMpol	References 83, 92, and 178–181	References 111–113 and 182

2. Transition dipole moment and its nuclear derivatives

For studying radiative transitions, the transition dipole moment μ and frequently its nuclear derivative need to be calculated. Like SOCME in Eq. (22), μ can also be expanded as

$$\begin{array}{ll}\hfill ⟨{\mathrm{\Lambda }}_{Gg}|{\mu }^{GK}|{\mathrm{\Lambda }}_{Kk}⟩& ={\mu }^{GK}(Q=0)⟨{\mathrm{\Lambda }}_{Gg}|{\mathrm{\Lambda }}_{Kk}⟩\hfill \\ \hfill & +\sum _j\frac{\partial {\mu }^{GK}}{\partial Q_j}{\mid }_{Q=0}⟨{\mathrm{\Lambda }}_{Gg}|Q_j|{\mathrm{\Lambda }}_{Kk}⟩+\cdots .\hfill \end{array}$$ (25)

FC approximation just keeps the first term above and takes ${\mu }^{GK}=⟨{\mathrm{\Psi }}_G|\widehat{\mu }|{\mathrm{\Psi }}_K⟩$ as a constant. The contribution of the second term could also be significant especially when μ^GK is small. In these situations, one must invoke the FCHT approximation to take the first two terms into account.

The nuclear derivatives of the transition dipole moments between the ground and excited states can be evaluated as ${\mu }_{GE}^x=\mathrm{Tr}(\delta {\rho }^x\mu )+\mathrm{Tr}(\delta \rho {\mu }^x)$. Here, δρ and μ are the transition density matrix and the dipole moment matrix, respectively, in the basis of molecular or atomic orbitals. Observably, ${\mu }_{GE}^x$ is the geometry-dependent reference. δρ^x has already been evaluated during the analytical calculation of the excited-state Hessian, as showed in Ref. 70 (with TDA) and Ref. 71 (full TDDFT). Therefore, the computational time will be saved if one obtains ${\mu }_{GE}^x$ as a by-product of the calculation of the excited-state Hessian.

3. SOC matrix element between different electronic states and its nuclear derivatives

The SOC operator acts on both the angular and spin components of electronic states, leading to its defining characteristics of mixing the orbital and spin degrees of freedom, thus allowing electronic states of different multiplicities to couple. It plays an essential role in photoinduced phenomena that involves change in the spin multiplicities of electronic states as it governs the rates of phosphorescence emission¹³² and ISC/RISC processes.¹³³ Since the TDDFT has been routinely used in studying photoexcited molecular systems, it is desirable to calculate SOC at the TDDFT level of theory.

Given that SOC stems from relativistic Dirac theory, the SOC effects can be taken into account naturally in relativistic TDDFT,¹³⁴ such as the four- and two-component relativistic theories. This type of treatment is rigorous and of high quality but computational demanding. For molecules in which the SOC is weak and therefore serves as a perturbation, the perturbative approaches are readily applicable.^134,135 In this case, the pure-spin states are first obtained from scalar relativistic or non-relativistic LR-TDDFT;^136–144 then, one could calculate SOC matrix elements among these excited states.

The perturbational treatment is not so accurate as the variational approaches are. However, the perturbative SOC approach is less computational demanding and allows one to identify the weights of singlet and triplet characters in a spin-adiabatic state.¹⁴⁵ Beyond the linear-response regime, the calculation of SOCMEs has also been performed within quadratic-response TDDFT, which can be viewed as implicit sum-over-states compared to perturbation theory.^146,147 It is worth noting that Valentine and Li proposed a novel procedure to calculate SOCMEs from variational relativistic electronic structure theory.¹⁴⁸

As shown in Eq. (19), the calculation of ISC or RISC rate usually requires one to account for both the HT-type vibronic and the spin-vibronic effects.^1,149 The former vibronic effect requires one to know the nuclear derivatives of SOCME, whose analytical implementation is currently under development.

III. ELECTRONIC EXCITED STATES WITHIN TDDFT/PCM AND TDDFT/MM

When a system of interest is embedded into a condensed-phase medium, its potential energy surfaces are perturbed, which can potentially cause a substantial change in the molecular properties, spectral profiles, and various transition rates and the corresponding quantum yield.

Due to the formidable computational cost of the realistic complexity, a full-QM modeling of the electronic structure of a system and its environment is usually infeasible. For instance, QM modeling, especially plane-wave density functional theory, has been employed to describe plasmonic metal surfaces and adsorbed reactants in studies of photon-driven catalysis.^183–185 However, given its steeply increasing cost, routine ab initio QM modeling is typically limited to systems (or subsystems) smaller than 2 or 3 nm. This has inspired the development of methods for capturing the environment effects via lower-level theories, such as classical analytical models (specifically for metal nanoparticles),^186–191 implicit solvation models, polarizable or nonpolarizable MM force fields, and coarse-grained models.¹⁹²

When a chemical system is immersed in liquid solutions, its properties may change significantly, especially in polar solvents. The solvent effects are usually accounted for by a continuum solvent description, such as the family of PCM^6,7 or a mixed treatment that integrates different descriptions of the solvent, for example, representing the first solvation shell with explicit solvent molecules while using a continuum model for the bulk. In the PCM method, the electrostatic polarization interaction between solutes and solvents is determined via the induced apparent surface charge method for which one needs to specify a surface that defines the continuum boundary. TDDFT coupled with PCM for excitation energy calculations has become a general routine in most quantum chemistry software packages. The implementation of the analytical gradient^69,114,115 and Hessian¹¹⁶ of TDDFT/PCM excited states has also been realized by several groups. TDDFT/PCM has been shown to be successful in supporting the analysis of experimental data, providing useful insights for a better understanding of photophysical and photochemical pathways in a solution.

The PCM has also been applied to model nanoparticles (NPs), leading to the PCM-NP approach. In the quasi-static limit, the NP experiences an electrostatic potential at a given frequency, which induces polarization charges on the nanoparticle surface. The QM/PCM-NP approach, which explicitly includes mutual polarization effects between the molecule and NP, has been applied to calculate the absorption factor as well as the radiative and nonradiative decay rates.³

The all-atom polarizable molecular mechanical models have emerged in the last few decades. Through introducing induced charges and/or dipoles at each atom site of the MM region, these MMpol models can efficiently capture the electronic motion in solvents, proteins, and large MNPs (>10 nm).^193,194 Among the MMpol models for MNPs, of particular interest to us are the discrete interaction model (DIM)^83,195,196 and its coordination-dependent variant (cd-DIM)¹⁹³ with the induced charges and dipoles representing polarization effects. Several similar all-atom MMpol models have also been developed, such as the point–dipole interaction model^197,198 (including its combination with either electronegativity equalization¹⁹⁹ or charge-transfer²⁰⁰), charge–dipole interaction model,^201,202 and atomic dipole approximation model.¹⁹⁴ Using a polarizable force field in QM/MM calculations is advantageous in light of the fact that the electronic polarization of the MM region can be described, especially on the occasion when electronic excitation is involved in the QM region.

Comparing with the QM/PCM methods, QM/MM or QM/MMpol approaches, which allow us to incorporate the explicit atomic or molecular details of surrounding molecules, can be applied to more complex environments. There are three embedding schemes for QM/MM, namely, mechanical,²⁰³ electrostatic, and polarizable.²⁰⁴ It is essential to combine TDDFT with MM to explore the molecular excited-state structures and dynamical properties in complex condensed-phase systems. Table I summarizes the current status of TDDFT-related approaches. Currently, the analytic nuclear gradients of TDDFT/MM and TDDFT/MMpol excited states are made available by a few groups. The implementation of analytic second-order energy derivatives¹⁷⁵ for TDDFT/MM with a non-polarizable force field has also been realized. The newly proposed hybrid methods, TDDFT/DIM (for metal nanoparticles)^83,178 and TDDFT/MM with fluctuatingcharges and fluctuating dipoles (for solvents),¹⁸¹ have only been made available for excitation energy calculations. Note that within the hybrid schemes, NAC calculations have only been carried out using non-polarizable AMEBA,^176,177 and the incorporation of the environment effect in this model needs to be reviewed carefully.

IV. CHOICE OF TDDFT MODELS

In the TDDFT modeling of electronic transitions, one can choose among several variations of the TDDFT methodology for the problem in hand.

A. TDA vs full-TDDFT

Tamm–Dancoff approximation (TDA) amounts to the negligence of the Casida B matrix in the TDDFT working equations. As a result, all Y amplitudes become zero. Using perturbation theory, it is easy to show that TDA excitation energies would always be larger than corresponding TDDFT values with the leading difference being second-order: ${\omega }_m^{\text{TDA}}-{\omega }_m^{\text{TDDFT}}\approx {\sum }_n{\left|{\mathbf{X}}_m^{\text{TDA}}\cdot \mathbf{B}\cdot {\mathbf{X}}_n^{\text{TDA}}\right|}^2/({\omega }_m^{\text{TDA}}+{\omega }_n^{\text{TDA}})$. In general, B matrix elements are relatively small, so TDA produces results that closely resemble TDDFT values. When using pure functionals [such as Becke88-Lee-Yang-Parr (BLYP) and Perdew–Burke–Ernzerhof (PBE)] or conventional hybrid functionals (such as B3LYP and PBE0), however, full TDDFT tends to systematically underestimate the energy of excited states with a small charge-transfer or Rydberg character. In those cases, as shown in a recent TDDFT benchmarking study,²⁰⁵ TDA can produce slightly more accurate excitation energies than full TDDFT as shown in Fig. 1. On the other hand, for range-separated hybrid functionals, full TDDFT can produce more accurate results (compared to CC2 values). Full TDDFT usually encounters problems with triplet states because the DFT ground state is used as the reference, which in many cases even leads to triplet instabilities. The TDA can be used to circumvent the triplet instabilities. Furthermore, TDDFT obeys the Thomas–Reiche–Kuhn sum rule of the oscillator strengths²⁰⁶ and therefore yields more precise results for the oscillator strength and other related physical quantities. The TDA violates the sum rule, which leads to poor results in the calculated oscillator strengths. Interestingly, it is found that the TDA performs better in the calculation of NACs than the full TDDFT, contrary to the conjecture that the TDA might cause the NAC results to deteriorate and violate the sum rule.²⁰⁷ For the excited-state harmonic vibrational frequencies, however, full TDDFT does not seem to be advantageous since the numerical tests demonstrate that the accuracy of TDDFT with and without TDA is comparable to each other.⁷¹

FIG. 1.

Root-mean-square (RMS) and mean-signed-average (MSA) differences (in eV) between vertical excitation energies calculated at the TDDFT, TDA, and CC2²⁰⁸ levels using the def2-TZVP²⁰⁹ basis set. Excitation energies of the five lowest excited states of the 11 chromophores are included. Reproduced with permission from Shao et al., J. Chem. Theory Comput. 16, 587–600 (2019). Copyright 2019 American Chemical Society.

B. Spin–flip DFT

Spin–flip (SF) DFT, which can be viewed as a special version of TDA,^160–162 handles up to four lowest electronic states of a system with an even number of electrons (two closed-shell singlets, an open-shell singlet, and a m_z = 0 triplet) on the same footing, thus offering a balanced description for these states. These states are obtained by using a high-spin (m_z = 1 triplet) reference state and solving the eigenproblem within spin–flip excitations, where an electron is excited from an α occupied molecular orbital to a β virtual orbital. In general, for a system whose ground state has a total spin quantum number S (multiplicity 2S + 1), the reference needs to be in an S + 1 state in order to target the lowest-energy states with total spin quantum number S and S + 1 via spin–flip excitations. Compared to standard TDA and TDDFT, the spin–flip TDA model usually provides more accurate results for diradicals and other systems with multi-reference characters. After the proposition of the original SF-DFT,¹⁶⁰ there have been various improved variations, such as the spin-adapted SF-DFT method aiming to eliminate the spin-contamination issue.^163,210,211

When CI appears between the electronic ground state and an excited state, it is an advantage to use the SF method to provide a balanced description of the two electronic states. Gordon and Minezawa¹⁸⁰ used the SF-DFT combined with the effective fragment potential (EFP) and illustrated that polar solvents could change the CI geometry dramatically and strongly stabilize its energy. Herbert and Zhang¹⁶⁶ studied the CI point of the D₀ and D₁ surfaces of the H₃ molecule. They have demonstrated that the SF-configuration interaction singles (SF-CIS) and SF-BH&HLYP (the BH&HLYP functional, in which LYP correlation is combined with Becke’s “half and half” (BH&H) exchange functional, consisting of 50%Hartree-Fock exchange and 50% Becke88 exchange) methods could locate CI points that involve the reference (ground) state correctly, which is problematic for the spin-conserved counterparts of these methods.²¹² What’s more, equipped with analytic derivative couplings, their method could substantially reduce the cost of locating minimum-energy crossing points.¹⁶⁶

Analytic gradient^164,165 and nonadiabatic coupling^166,167 have been developed for gas-phase SF-DFT calculations, while the analytical Hessian is still under development. As far as we know, there has been no much discussion on SF-DFT/PCM or SF-DFT/MMpol models. Within a linear-response framework, however, a PCM or MMpol modeling of the complex environment is expected to yield no additional contribution to the A matrix (due to a transition density in the αβ block). Therefore, theoretically, it reduces to a “zeroth-order” model, where PCM surface charges or MMpol permanent/induced moments affect the results only by polarizing the KS orbitals.

C. Spin-adiabatic DFT/TDDFT states

Spin-adiabatic (SA) TDDFT has gained attention only in recent years.^130,168 Compared with the spin-diabatic states, the spin-adiabatic counterparts have shown advantages¹³⁰ in Tully’s fewest switches surface hopping (FSSH) dynamics²¹³ to evaluate the rates and branching ratios of ISC processes where spin multiplicity changes. For a system with an even number of electrons, there are 4n + 1 spin-diabatic states: the ground state, the lowest n open-shell singlet excited-states, and the lowest 3n triplet excited-states (with m_s = −1, 0, 1). They could be coupled together via the SOC operator,

$${\mathbf{H}}^{(0)}+{\mathbf{V}}^{\text{SO}}=\left(\begin{array}{cccccc}\hfill E_{S_0}\hfill & \hfill 0\hfill & \hfill V_{S_0{T}_1}^{\text{SO}}\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill V_{S_0{T}_n}^{\text{SO}}\hfill \\ \hfill 0\hfill & \hfill E_{S_1}\hfill & \hfill V_{S_1{T}_1}^{\text{SO}}\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill V_{S_1{T}_n}^{\text{SO}}\hfill \\ \hfill V_{T_1{S}_0}^{\text{SO}}\hfill & \hfill V_{T_1{S}_1}^{\text{SO}}\hfill & \hfill E_{T_1}\hfill & \hfill \cdots \hfill & \hfill V_{T_1{S}_n}^{\text{SO}}\hfill & \hfill V_{T_1{T}_n}^{\text{SO}}\hfill \\ \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill & \hfill \cdots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill V_{S_n{T}_1}^{\text{SO}}\hfill & \hfill \cdots \hfill & \hfill E_{S_n}\hfill & \hfill V_{S_n{T}_n}^{\text{SO}}\hfill \\ \hfill V_{T_n{S}_0}^{\text{SO}}\hfill & \hfill V_{T_n{S}_1}^{\text{SO}}\hfill & \hfill V_{T_n{T}_1}^{\text{SO}}\hfill & \hfill \cdots \hfill & \hfill V_{T_n{S}_n}^{\text{SO}}\hfill & \hfill E_{T_n}\hfill \end{array}\right).$$ (26)

Note that the couplings between different triplet states could be nonzeros. Formally, all the spin-adiabatic states can be obtained by diagonalizing this (4n + 1) × (4n + 1) matrix.

Bellonzi et al.^130,168 implemented an efficient iterative procedure for diagonalizing the above coupling matrix between TDA singlets and triplets but without including the ground state. Furthermore, they formulated the analytical gradient of the obtained spin-adiabatic TDA states¹⁶⁸ and developed the nonadiabatic couplings between the spin-adiabatic TDA states.¹³⁰ This new and substantial advance is expected to facilitate the study of ISC and allow for a balanced treatment between ISC and IC events.

On the other hand, a future extension of this methodology to include the ground state as well in the coupling [Eq. (26)] would be desirable, which will provide a general framework for, for instance, the study of phosphorescence decay and the construction of spin-adiabatic potential energy surfaces for modeling spin-crossing reactions.²¹⁴ Furthermore, in the study of spin-crossing reactions, it was found to be a poor approximation to describe the lowest singlet and triplet states using the same set of molecular orbitals.²¹⁴ Therefore, it should be more suitable to have one set of optimized orbitals for the singlet state and another for the triplet state.

V. CHALLENGES WITH TDDFT MODELING OF EXCITED STATES IN THE CONDENSED PHASE

A. Challenges with the modeling of system–environment interactions

To illustrate the challenges in describing several QM–MM interactions of an actual system, a pyridine–Ag₂₀ complex in Fig. 2 is used as a showcase. As shown in the following, this is a highly challenging system for classical embedding due to the strong interaction between the pyridine molecule and the metal cluster. Our primary focus is on the effect of the metal cluster on electronic excitations within the pyridine molecule. This complex is handled using a QM/DIM model adopted from Jensen et al.,^{83,84,178,215} which combines a DIM description for the Ag₂₀ cluster and a DFT or TDDFT description for pyridine. All calculations are carried out using a preliminary implementation²¹⁶ of the QM/DIM model within the Q-Chem software package.²¹⁷

FIG. 2.

Scheme of pyridine–silver₂₀ (Py–Ag₂₀) configuration.

1. Charge transfer between QM and MM regions

As indicated by charge populations in Table II, for the electronic ground state of Py–Ag₂₀, there is a charge migration (0.16–0.30 e⁻) from the pyridine molecule to the Ag₂₀ cluster. Energy decomposition analysis (EDA) based on absolutely localized molecular orbitals (ALMO)²¹⁸ for this complex also shows a substantial charge-transfer stabilization (−49.09 kJ/mol) together with a similar amount of polarization energy (−38.83 kJ/mol).

TABLE II.

Net charges on the pyridine molecule within Merz–Kollman ESP,²¹⁹ ChElPG,²²⁰ and fragment-based Hirshfeld (FBH)²²¹ charge partitioning schemes and energy decomposition analysis of the ground-state energy of the pyridine–silver₂₀ complex calculated with the ωB97X-D²²² functional and 6-31G(d) basis set and SRSC ECP.

QPy (a.u.)			ΔE (kJ/mol)
QESP	QChElPG	QFBH	ΔEFrozen	ΔEPol	ΔECT	ΔETotal
0.30	0.22	0.16	7.68	−38.83	−49.09	−80.23

Starting from this ground state with partial charge transfer, the lowest 200 excited states from a full-TDDFT calculation are mostly local excitations on the metal cluster with a few states featuring charge transfer between the molecule and the metal cluster. None of these low-lying states correspond to local excitations on the pyridine molecule. This poses a serious challenge to the modeling of electronic excitations in these complexes since the full or partial charge-transfer excitations are not incorporated into the existing hybrid QM/MM approaches, such as TDDFT/DIM. Here, state-selective optimization schemes^223–225 might help capture the local excited states in large systems.

2. Proper damping of QM/MM electrostatics

To show the effect of the damped QM/MM electrostatics in the QM/DIM modeling of the Py–Ag₂₀ complex, several frontier molecular orbital energies are plotted in Fig. 3. The orbital energies for the gas-phase pyridine and the Py–Ag₂₀ complex are obtained from DFT/DIM and ALMO-DFT²²⁶ calculations, respectively.

FIG. 3.

Frontier orbital energies (three occupied orbitals and two unoccupied orbitals) from DFT calculations of the gas-phase pyridine, DFT/DIM calculations (with Gaussian exponents of 0.4, 0.6, 0.8, and 1.0 a.u.), and ALMO-DFT calculations for the Py–Ag₂₀ complex. All calculations used the ωB97X-D functional, 6-31G(d) basis set, and SRSC ECP.

As shown in Fig. 3, all frontier MO energies of the pyridine molecule are shifted down upon its adsorption onto the Ag₂₀ cluster. The highest occupied molecule orbital (HOMO), which is the only σ-orbital out of the five orbitals, is subjected to the largest decrease in its energy, thus becoming HOMO − 1 or HOMO − 2 in DFT/DIM and ALMO-DFT calculations. Overall, we find that using a damping factor (α_m) of 0.6 or 0.8 a.u. for the induced dipoles in the DIM model best reproduces the relative energies between these frontier orbitals obtained from the ALMO-DFT reference. The five lowest excitation energies show a similar trend as the frontier orbitals with the increasing damping factor (see Fig. 4). In fact, TDDFT-TDA/DIM calculations with a damping factor α_m of 1.0 a.u. would best reproduce the results of the ALMO-TDDFT-TDA calculations,^97,227 which can be used as the benchmark of the local excited states.

FIG. 4.

First five excitation energies from TDDFT-TDA calculations of the gas-phase pyridine, TDDFT-TDA/DIM calculations (with Gaussian exponents of 0.4, 0.6, 0.8, and 1.0 a.u.), and ALMO-TDDFT-TDA calculations for the Py–Ag₂₀ complex. All calculations used the ωB97X-D functional, 6-31G(d) basis, and SRSC ECP.

3. Description of QM/MM van der Waals interactions

The van der Waals (vdW) interactions between the QM and MM regions are crucial for obtaining accurate potential energy profiles. In our implementation of the QM/DIM method, a distance-dependent Lennard-Jones (i.e., 12–6) potential proposed by Payton et al.²¹⁵ is adopted with a small modification. Without the repulsive interaction, the optimized Py–Ag bond would become very short due to a strongly attractive QM/MM electrostatic interaction.

To avoid a system-dependent tuning of QM/MM vdW parameters as we have performed with the ɛ₀ and ɛ₁ parameters for the nitrogen atom, one should make the vdW parameters of the QM atoms dependent on the electronic structure (densities, molecular orbitals, or atomic charges) of the QM region.^228–230 Within the ab initio QM/MM framework, one can tap into the Tkatchenko–Scheffler dispersion model,²³¹ where the C₆ parameter of an atom is computed from its Hirshfeld-weight-based atomic volume in the molecule. This has led to two density-dependent QM/MM vdW models by Giovannini et al.²²⁹ and by Curutchet et al.,²³⁰ respectively. We expect that these models be further tested and developed before becoming widely adopted in condensed-phase QM/MM simulations. Alternatively, one can predefine localized molecular orbitals on the MM atoms and interact them with orbitals from the QM region to acquire dispersion and repulsion interactions. This has been implemented in DFT/EFP models.^232–234

B. Challenges with the computational costs

TDDFT modeling of condensed-phase excited-state processes can become very expensive, especially with a large QM region or a long simulation (1 ps or longer). The first bottleneck of these modelings come from the QM calculation itself, namely, the computation of DFT and TDDFT energy gradient/Hessian and state-to-state derivative couplings. It is time-consuming to evaluate many Fock-like or Fock-derivative-like matrices with some of them involving the third and fourth derivatives of the exchange–correlation functional. Two ways can be pursued to accelerate these calculations. First, following Grimme and co-workers, one can approximate the two-electron integrals and transition density, which has led to the simplified TDA scheme for a large system.^235–237 Second, one should be able to extrapolate SCF orbitals, TDDFT amplitudes, and z-vectors from previous geometries during a geometry optimization or molecular dynamics simulation. To extrapolate SCF orbitals, the Fock matrix extrapolation scheme from Pulay, Herbert, and others^238,239 and extended Lagrangian from Niklasson^240,241 can already reduce the SCF cycles. For excited state simulations with TDDFT energies/forces, one should also be able to extrapolate amplitudes and z-vectors in a way similar to the handling of MP2 z-vectors by Steele and Tully.²⁴²

For TDDFT/MMpol with an extended (solvent, protein, or metal-cluster) environment, a second bottleneck comes from the solution of a large number of induced charges/dipoles during each SCF iteration.^3,76,243,244 Several efforts have been devoted to the reduction of this computational cost.^93,94 First, in order to avoid the $\mathcal{O}(N^3)$ complexity associated with directly solving linear equations for induced charges or dipoles, iterative solvers can be employed, such as Jacobi iterations (JI)^245–247 and preconditioned conjugate gradient (PCG)²⁴⁸ and the truncated version of JI²⁴⁹ and PCG.^250,251 For large systems, Nocito and Beran proposed a divide-and-conquer Jacobi iteration (DA-JI) method^252,253 and also modified an always stable predictor-corrector (ASPC) algorithm.²⁵⁴ Second, one can truncate the mutual polarization among MM atoms such as in the 3-AMOEBA scheme.²⁵⁵ Finally, the extended Lagrangian method can be employed to remove the need to fully converge MM induced dipoles at each geometry.^256–258

C. Challenges with the interpretability of computational results

Fundamentally, we are interested in the effects of various internal and external perturbations on excited-state processes, namely, how they shift absorption/emission wavelengths and how they modulate the radiative/nonradiative transition rates and thus change the overall quantum yield. Internally, a system can be “perturbed” by chemical modifications, such as substitutions and combination of different donors/bridges/acceptors. Externally, a system can be placed in different solvents, protein mutants, or metal clusters of various shapes, sizes, and compositions.

TDDFT modeling has proven to be extremely valuable in these studies. In addition to the TDDFT calculations themselves, many efforts have been devoted to the challenging task of analyzing the results of these calculations. Given a computed TDDFT excited state, one can now readily (a) visualize the density change from the ground state and compute the corresponding charge populations; (b) visualize the canonical orbitals and natural transition orbitals associated with the excitation; and (c) characterize it as being a local excitation, a charge-transfer excitation, or a hybrid.

The characterization of the lowest excited states of a system is especially important in many design problems, such as new thermally activated delayed fluorescence (TADF) molecules for which a hybridized local and intramolecular charge-transfer (HLCT) excitation might optimally enhance the reverse internal system crossing while retaining a decent fluorescence intensity. The Λ values from Peach et al.²⁵⁹ and charge transfer numbers (CTN) from Plasser et al.^260,261 have been widely used for such characterizations.

While the electronic density change upon an excitation is commonly visualized, it was only until recently when one can start to analyze the excitation energy distribution, which can also be of interest. For several example systems, we showed that it is feasible to compute and visualize the LR-TDDFT excitation energy density.²⁶² It has also provided a way for one to formulate the effective energies of the hole and particle for a charge-transfer excitation.

In the design of new fluorophores/dyes, guiding principles or inexpensive tools are needed to help quickly predict how a chemical modification or environmental change (as mentioned above) might affect the photophysical properties of a fluorophore (dye) molecule. As for many of these fluorophores, the HOMO → LUMO transition or other transitions between frontier orbitals play the dominant role; we expect the ALMO-based orbital interaction analysis^96,100 to be increasingly used to analyze or predict substituent effects or donor–acceptor interactions. On the other hand, the substitution/mutation effects on the rate of radiative and non-radiative transitions are less well-studied and poorly understood, thus providing opportunities for the development of new analysis tools.

VI. SUMMARY

In this Perspective, we have provided an overview on the successes and challenges of TDDFT-related approaches for modeling the excited-state properties and processes. A particular emphasis is placed on which electronic structure properties are accessible with which methods rather than on one single topic, such as the accuracy of excitation energies.

First, we introduced the main electronic structure quantities required by the description of molecular excited-state properties and processes, such as the analytic energy derivatives of the TDDFT excited state, and the NAC and SOC matrix elements. Second, we gave a summary of the recent development of hybrid embedding methods for the description of molecular excited states in complex environments, noting that the evaluation of NAC and SOCME accounting for the environment effects, the analytic nuclear derivative of SOCME, and the analytic Hessian of TDDFT/MMpol have not been successfully implemented into the electronic structure packages. Third, we discussed the advantages of three variations of the TDDFT methods. Specifically, TDA might slightly outperform full TDDFT with pure or conventional hybrid functionals; spin–flip DFT is useful for systems with some multi-reference characters (such as regions around conical interactions); and spin-adiabatic TDDFT could be helpful for studying spin-crossing reactions or the intersystem crossing events. In the end, we elaborated on the challenges in accurately describing interactions between the QM and MM regions, reducing the computational costs associated with modeling excited states in complex environments, and making the results of excited-state calculations physically more transparent and easier to interpret. These challenges have afforded many new opportunities for the development of advanced computational and analysis techniques to treat excited-state-involved processes in complex environments in the future.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

DATA AVAILABILITY

The data that support the findings of this study are available within the article.