Skip to main content
NCBI home page
JACS Au logoLink to JACS Au
. 2023 Feb 1;3(2):358–367. doi: 10.1021/jacsau.2c00659

State Interaction Linear Response Time-Dependent Density Functional Theory with Perturbative Spin–Orbit Coupling: Benchmark and Perspectives

Can Liao †, Joseph M Kasper ‡, Andrew J Jenkins †, Ping Yang ‡, Enrique R Batista ‡, Michael J Frisch §, Xiaosong Li †,*
PMCID: PMC9975852  PMID: 36873704

Abstract

graphic file with name au2c00659_0005.jpg

Spin–orbit coupling (SOC) is an important driving force in photochemistry. In this work, we develop a perturbative spin–orbit coupling method within the linear response time-dependent density function theory framework (TDDFT-SO). A full state interaction scheme, including singlet–triplet and triplet–triplet coupling, is introduced to describe not only the coupling between the ground and excited states, but also between excited states with all couplings between spin microstates. In addition, expressions to compute spectral oscillator strengths are presented. Scalar relativity is included variationally using the second-order Douglas-Kroll-Hess Hamiltonian, and the TDDFT-SO method is validated against variational SOC relativistic methods for atomic, diatomic, and transition metal complexes to determine the range of applicability and potential limitations. To demonstrate the robustness of TDDFT-SO for large-scale chemical systems, the UV–Vis spectrum of Au25(SR)18– is computed and compared to experiment. Perspectives on the limitation, accuracy, and capability of perturbative TDDFT-SO are presented via analyses of benchmark calculations. Additionally, an open-source Python software package (PyTDDFT-SO) is developed and released to interface with the Gaussian 16 quantum chemistry software package to perform this calculation.

Keywords: spin−orbit coupling, TDDFT, fine-structure splitting, Breit-Pauli Hamiltonian, transition metal complex

1. Introduction

Spin–orbit coupling (SOC) is a cornerstone of modern chemistry, responsible for important photochemical phenomena such as intersystem crossing,1,2 relativistic spectroscopies (for example, L-edge X-ray absorption,3 magnetic circular dichroism4), and fine-structure splitting.5 Taking advantage of spin-forbidden processes allowed by SOC has contributed to recent advances throughout chemistry and materials science, especially in LED6 and solar cell technology.7,8 With growing popularity in exploring spin-forbidden processes and the ubiquity of nonmain group elements in modern chemistry, there is demand for an efficient SOC method suitable for atomically diverse and large systems.

The time-dependent density functional theory (TDDFT) has been the modern day workhorse for computational photochemistry,9,10 thanks to the balance between its predictive power and low computational scaling with respect to the system size. There are two types of approaches to include SOC in TDDFT: variational and perturbative. The state-of-the-art variational method that includes SOC in TDDFT employs the Dirac Hamiltonian and a four- or two-component spinor basis.5,11,12 Two-component approximations to the four-component Dirac Hamiltonian such as the exact two-component (X2C)5,13−25 Hamiltonian has been successful in reproducing experimental results while staying affordable for small systems.3,26,27 Although variational relativistic methods are the most complete theoretical treatment of SOC in TDDFT so far, there are challenges that make such an approach less practical for large chemical systems, including the requirements of using complex-valued arithmetic, noncollinear functional forms,28 and eigensolvers that are effective for a dense manifold of excited states.

In the perturbative approach, the zeroth-order wave function is first variationally determined in the absence of SOC and then SOC is introduced via state interaction to couple states of interest.29−43 The validity of the perturbative approach depends on the SOC term of the Hamiltonian being much smaller than the excitation energies of the scalar relativistic terms. As high-orders of perturbations are included or as the expansion space increases toward the full CI limit, the perturbative inclusion of SOC should converge to the variational limit. Compared to variational methods, perturbative approaches have some unique advantages, including using one-component real-valued wave functions and hence lower computational cost and less issues with self-consistent-field convergence, as well as identification of spin eigenfunctions in orbital analysis.

State interaction has been successfully adapted for linear response TDDFT through approximate mapping from TDDFT solutions to CIS wave functions,34,44−48 with a practical Python module developed by Gao et al.49 However, the limitation and applicability of perturbative spin–orbit approach is not well-known in the community. In this work, we introduce a complete state interaction TDDFT approach to include the SOC effect. The goal is to develop a perturbative TDDFT-SO method that is able to describe couplings between singlet and triplet, as well as between triplet and triplet states for both the ground and excited states. In addition, we explicitly couple spin microstates (for example, S = 1, MS = −1, 0, +1) in order to account for anisotropic spin–orbit effects in the full state interaction picture. State-to-state oscillator strengths are also evaluated in the TDDFT-SO framework. The method was implemented in a development version of the Gaussian software package,50 but is accompanied by an open-source Python module (PyTDDFT-SO)51 that takes Gaussian 16 TDDFT results as input and performs the complete state interaction illustrated in this work.

An extensive benchmark analysis is carried out and compared to variational relativistic X2C-TDDFT results5 to test the range of applicability of the perturbative TDDFT-SO method. Through this work, we provide the scientific community a practical tool for simulating spin–orbit driven photochemistry and a set of general theoretical guidance on the limitation and applicability of the perturbative spin–orbit approach.

2. Methodology

2.1. Linear Response Time-Dependent Density Functional Theory

The matrix equation of the linear response time-dependent density functional theory can be written as a non-Hermitian eigenvalue problem:9,10,52

2.1. 1
2.1. 2

where ωI is the excitation energy for the I-th excited state. The I-th column of X and Y represent particle-hole and hole-particle excitation amplitudes for the I-th excited state, respectively.45 The left-most matrix in eq 1, which is related to the orbital Hessian, is given by

2.1. 3
2.1. 4

where i, j index over occupied orbitals and a, b index over virtual orbitals. The scaling factor cHF modulates the amount of Hartree–Fock exact exchange in the DFT functional (cHF = 0 for pure functionals and cHF ∈ (0, 1] for hybrid functionals).

In the state interaction TDDFT with perturbative spin–orbit, the linear response equation is performed using a restricted Kohn–Sham reference. Because the restricted formalism allows the orbital Hessian to be entirely real-valued, the eigenvalue problem can be reduced to the halved-dimension form:52

2.1. 5
2.1. 6

The excited states are now represented by the columns of Z. The elements of ZI represent the contribution of a single orbital excitation toward the I-th excited state. A natural association between these orbital excitations and singly excited Slater determinants can be made. Hence, approximate CIS wave functions can be constructed using Z,44,45

2.1. 7

2.2. State Interaction with Perturbative Spin–Orbit

In state interaction, an effective Hamiltonian (H′) is constructed by perturbing the zeroth-order Hamiltonian (H0) with a spin–orbit coupling Hamiltonian (HSO)

2.2. 8
2.2. 9

In the case of LR-TDDFT, the zeroth-order Hamiltonian H0 is the diagonal matrix of excitation energies with the first diagonal element (ground state) being zero,34,44

2.2. 10

When the spin–orbit coupling is relatively weak compared to the energy difference between spin states, perturbation theory can be used to introduce spin–orbit coupling to a nonrelativistic or spin-free relativistic zeroth-order Hamiltonian. Among the most common spin–orbit operators is the Breit-Pauli spin–orbit operator,53,54

2.2. 11

where ZA is the charge of nucleus A, riA is the distance between electron i and nucleus A, l̂i is the orbital angular momentum operator of electron i, and ŝi is the spin angular momentum operator. Note that the angular momentum operators can be written in terms of the position and momentum operators so that l̂ij = r̂ij × p̂i, for example. These terms capture the interactions that give rise to the one-electron (the first term in eq 11) and two-electron spin–orbit couplings (the second term in eq 11). Since the two-electron spin–orbit contribution is computationally expensive to evaluate,55,56 previous attempts at computing spin–orbit matrix elements approximated two-electron SOC with an effective nuclear charge.31,46,49,57 However, these effective nuclear charges were only defined for some elements. Here, we use the Boettger factor,58 which is available for every element, to approximate two-electron SOC by scaling the one-electron SOC integrals.

The one-electron SOC Hamiltonian can be written in second quantized form shown in eq 12

2.2. 12

where ϕpσ, ϕqσ′ are Kohn–Sham orbitals, and apσ†, aqσ′ are creation and annihilation operators, respectively. In the following equations, we dropped the “1e” notation for brevity. The working expressions can be obtained by partitioning the Hamiltonian into Cartesian components and applying the spin angular momentum operators on |ϕqσ′⟩ (see the Supporting Information, SI, for derivations using Wick’s theorem),

2.2. 13
2.2. 14
2.2. 15
2.2. 16
2.2. 17

At this point, the Boettger factor is applied to the hpqk. The angular-momentum-dependent factor is applied to the integrals in the atomic orbital (AO) basis,

2.2. 18

where lμ is the orbital angular momentum quantum number of orbital μ, Zμ is the charge of the nucleus at which orbital μ is centered, and Q(l) is the number of electrons in all filled shell with n ≤ l. That is, Q(0) = 0, Q(1) = 2, Q(2) = 10, and so forth.

In the restricted Kohn–Sham formalism, electronic states can either be singlets (S = 0; MS = 0) or triplets (S = 1; MS = −1, 0, +1). When obtaining the analytical expressions for spin–orbit Hamiltonian matrix elements, the states are treated as creation and annihilation operators acting on the Kohn–Sham ground state determinant

2.2. 19
2.2. 20
2.2. 21
2.2. 22

With the electronic states and SOC Hamiltonian clearly defined, expressions for the SOC matrix elements can be derived using Wick’s theorem. Here, we only present the final working expression where excited states and their spin quantum numbers are written as |IS,MS⟩.

The state interaction matrix elements between the closed-shell ground state |0⟩ and excited states are

2.2. 23
2.2. 24
2.2. 25

The state interaction matrix elements between excited states are

2.2. 26
2.2. 27
2.2. 28
2.2. 29
2.2. 30
2.2. 31

2.3. Oscillator Strength

After diagonalizing the effective Hamiltonian, we obtain states that include spin–orbit coupling, or spin–orbit adiabatic states,59 expressed as a linear combination of the unperturbed states,

2.3. 32

where C′ is the eigenvector of the effective Hamiltonian. This representation of spin–orbit adiabatic states allows us to obtain transition dipole moments between spin–orbit-coupled states via transformation of the unperturbed transition dipole moments,

2.3. 33
2.3. 34

Oscillator strengths can be computed using the spin–orbit transition dipole moment (in atomic units) between the ground state and spin–orbit adiabatic excited states

2.3. 35

and between spin–orbit adiabatic excited states,

2.3. 36

where ωI′ is the eigenvalue of the effective Hamiltonian.

3. Benchmark and Discussion

The state interaction TDDFT-SO method was implemented in the development version of the Gaussian quantum chemistry software package.50 An open-source code (PyTDDFT-SO) that can perform the same algorithm using Gaussian 1660 as the input is released under the authors GitHub Web site. The zeroth-order Hamiltonian includes the scalar relativity variationally in the ground state DFT calculations via the DKH2 transformation. In this benchmark study, atomic fine structure was compared along with excitation energies of diatomic molecules and transition metal complexes. The UV–Vis spectrum of the Au25 nanocluster was also generated by state interaction TDDFT-SO and compared. Benchmark calculations are compared to experiments and the variational X2C-TDDFT approach using the same SOC integrals.5,26,61

3.1. Atomic Fine Structure Splitting

In this section, the performance of TDDFT-SO was assessed for atomic cases. Fine structure splitting in the 3P and 3D excited state manifolds were calculated using TDDFT-SO and plotted against variational X2C-TDDFT results. The interaction space chosen for the atomic calculations included the entire manifold of interest, including both singlet and triplet states. Additional states beyond the manifold of interest had negligible effect on the fine structure splitting in the atomic calculations. For the 1S → 3P excitation, two types of electronic transitions are considered here: s2 → s1p1 and p6 → p5s1, plotted in Figure 1 and Figure 2. For the 1S → 3D excitation, we plot s2 → s1d1 and d10 → d9s1 transitions in Figure 1 and Figure 2, respectively.

Figure 1.

Figure 1

Open in a new tab

Comparison of excited state fine structure splitting calculated using TDDFT-SO and X2C-TDDFT for various atomic cases. 1S → 3P and 1S → 3D excitations arising from s2 → s1p1 and s2 → s1d1 transitions, respectively, are considered. Calculations were done using the ANO-RCC-VTZP basis set62−66 and the PBE0 functional.67

Figure 2.

Figure 2

Open in a new tab

Comparison of excited state fine structure splitting calculated using TDDFT-SO and X2C-TDDFT for various atomic cases. 1S → 3P and 1S → 3D excitations arising from p6 → p5s1 and d10 → d9s1 transitions, respectively, are considered. Calculations were done using the ANO-RCC-VDZ basis set62−64,66 and the PBE0 functional.67

TDDFT-SO is in excellent agreement with X2C-TDDFT for light elements, but tends to overestimate fine structure splitting as the atomic number increases. At Z > 56, the number of cases with an unsigned error greater than 0.10 eV drastically increased whereas for Z ≤ 56, such errors are only seen for highly charged species close to Z = 56 (Sb3+, Te4+). This is understandable as the perturbative spin–orbit treatment becomes inadequate when the coupling strength is comparable to the excited state energy gap. Perturbative methods are not expected to hold up at large perturbations. Unlike the 3P manifold, errors for in the 3D manifolds mostly stayed below 0.10 eV up to Z > 80. In addition, for the same atom, the error in fine structure splitting increases as the principal quantum number increases.34 For example, the TDDFT-SO computed Tl3+3P2/3P1 and 3P2/3P0 splittings arising from excitation of the 5p shell (Figure 2) only have an error of 2% and 11% compared to X2C-TDDFT results. In contrast, when the fine structure splitting is due to the excitation into the 6p shell in Tl+ (Figure 1), the percent error increases to 60% and 84%, respectively. This is likely due to the insufficient number of excited states that is needed to span the spin–orbit operator for outer orbitals.

Although the spin–orbit operator does not depend on the choice of DFT functional, the quality of the state interaction TDDFT-SO relies on the quality of the zeroth-order closed-shell TDDFT reference. For example, results for the Yb 3D1/3D3 splitting in Figure 1 significantly improved when the BHandH functional68 was used. Shown in Table 1 are the statistics of TDDFT-SO errors for various functionals. Error is defined as the difference in fine-structure splitting between TDDFT-SO and variational X2C-TDDFT calculations. The computed excitation energies are presented in the SI. For Z > 56, starting from the Ln series, all functionals exhibit a significant increase in error, although B3LYP performs slightly better than the other functionals tested, for these heavier elements. This study suggests that perturbative TDDFT-SO is generally reliable for predicting atomic fine structures for elements lighter than the Ln series, but one should be cautious for species with large atomic numbers.

Table 1. Error of TDDFT-SO Atomic Fine Structure Splitting (in eV) Using Different DFT Functionalsa.

  mean AE max AE std dev
B3LYP
Z ≤ 56 0.0270 0.3627 0.0651
Z > 56 0.1798 1.6568 0.3382
BHandH
Z ≤ 56 0.0300 0.3306 0.0606
Z > 56 0.2724 2.5637 0.5595
PBE
Z ≤ 56 0.0284 0.3811 0.0652
Z > 56 0.3727 2.4210 0.6526
PBE0
Z ≤ 56 0.0289 0.3261 0.0605
Z > 56 0.2581 2.4842 0.5171
SVWN
Z ≤ 56 0.0266 0.4043 0.0639
Z > 56 0.2813 1.6460 0.4281
Open in a new tab
a

Absolute error (AE) is defined to be the unsigned error of the TDDFT-SO result compared to X2C-TDDFT.

3.2. Diatomic Molecules

This section assesses the performance of TDDFT-SO on diatomic molecules and the dependence on the size of state interaction space. The equilibrium bond lengths of Cu2, GaBr, and GaI were obtained from experimental data recorded in the NIST Webbook.69 Ag2 and Au2 bond lengths were optimized using the relativistic CRENBL effective core potential and its corresponding basis set.66,70,71 All TDDFT calculations were performed using the ANO-RCC-VDZP basis set63,64,66 and the PBE0 hybrid functional.67 Five different sizes of state interaction space (N = 5, 10, 15, 20, 50) were used with TDDFT-SO. For a closed-shell ground state, the TDDFT-SO calculation solves for N singlet and N triplet states, resulting in an interaction space of 4N + 1 states, including the ground state and each triplet microstate.

Table 2 tabulates the statistics of TDDFT-SO errors for excited state energy calculations of diatomic molecules. Error is defined as the difference in calculated excitation energy between TDDFT-SO and variational X2C-TDDFT. The computed excitation energies are presented in the SI. Table 2 shows that TDDFT-SO results are in good agreement with X2C-TDDFT even with the smallest interaction space N = 5. The agreement deteriorates with increasing Z. For Au2 (Z = 79) with N = 5, we observed a mean-average-error of ∼0.1 eV. Errors in TDDFT-SO calculated excitation energies can be improved with a larger interaction space. As the interaction space increases, the spin–orbit operator is more accurately represented in the expansion of zeroth-order eigenstates. In the limit of infinite order response theory (equivalent to the full CI limit), it should converge to the exact solution. Table 2 shows that as the interaction space increases, the standard deviation of TDDFT-SO errors decreases. With N = 50 (201 states), TDDFT-SO errors are only at the meV level for most diatomic molecules studied here.

Table 2. Error of TDDFT-SO Excitation Energies (in eV) of Diatomic Molecules Using Various Interaction Spacesa.

  interaction space
  N = 5 N = 10 N = 15 N = 20 N = 50
Cu2
mean AE 0.0282 0.0048 0.0048 0.0047 0.0047
max AE 0.1411 0.0106 0.0106 0.0107 0.0108
std dev 0.0441 0.0035 0.0037 0.0037 0.0038
Ag2
mean AE 0.0195 0.0137 0.0105 0.0113 0.0120
max AE 0.0536 0.0536 0.0195 0.0202 0.0218
std dev 0.0184 0.0160 0.0070 0.0072 0.0075
Au2
mean AE 0.1280 0.1098 0.1339 0.1464 0.1586
max AE 0.5142 0.1510 0.1644 0.1814 0.1957
std dev 0.1510 0.0473 0.0266 0.0269 0.0330
GaBr
mean AE 0.0114 0.0058 0.0062 0.0064 0.0069
max AE 0.0904 0.0200 0.0225 0.0225 0.0233
std dev 0.0297 0.0072 0.0077 0.0077 0.0079
GaI
mean AE 0.0464 0.0384 0.0392 0.0396 0.0419
max AE 0.2248 0.0702 0.0705 0.0707 0.0746
std dev 0.0708 0.0221 0.0223 0.0224 0.0229
Open in a new tab
a

Absolute error (AE) is defined to be the unsigned error of the TDDFT-SO result compared to X2C-TDDFT.

Readers should note that all microstates (S = 1, MS = −1, 0, +1) that belong to a same triplet manifold must be included in the interaction space. Failing to do so will cause unphysical degeneracy and Kramers’ symmetry breaking.

3.3. Transition Metal Complexes

In this section, we further assess the performance of TDDFT-SO by applying the method to transition metal complexes. High symmetry complexes, consisting of late-row transition metals (Mo, W, Pd, Ru, and Os), were chosen to showcase the interplay between spin–orbit coupling and the ligand field. All geometries were optimized using the PBE0 hybrid functional67 with the CRENBL effective core potential66,70,71 and its corresponding basis set. The TDDFT calculations were performed using the PBE0 hybrid functional with the relativistically optimized double-ζ Sapporo-2012 basis set including diffuse functions66,72,73 for metal centers and the nonrelativistic Pople 6-311G(d,p) basis sets66,74−76 for ligands.

The computed excitation energies of the lowest several excited states for each transition metal complex are presented in the SI. Table 3 shows the statistics of errors defined as the difference in excitation energy between TDDFT-SO and X2C-TDDFT. The results are consistent with those obtained for atomic and diatomic species with the sixth-row elements, W (Z = 74) and Os (Z = 76), exhibiting relatively large yet acceptable errors (less than 0.1 eV) compared to the variational X2C-TDDFT calculations.

Table 3. Error of TDDFT-SO Excitation Energies (in eV) of Transition Metal Complexes Using Various Interaction Spacesa.

  interaction space
  N = 6 N = 10 N = 15 N = 20 N = 60
Mo(CO)6
mean AE 0.0099 0.0027 0.0027 0.0027 0.0027
max AE 0.0368 0.0068 0.0070 0.0070 0.0070
std dev 0.0120 0.0021 0.0022 0.0022 0.0022
W(CO)6
mean AE 0.0101 0.0171 0.0182 0.0187 0.0188
max AE 0.0399 0.0293 0.0299 0.0299 0.0300
std dev 0.0130 0.0070 0.0074 0.0071 0.0071
PdCl62–
mean AE 0.0042 0.0031 0.0014 0.0013 0.0023
max AE 0.0074 0.0052 0.0031 0.0032 0.0055
std dev 0.0023 0.0017 0.0011 0.0011 0.0016
RuO4
mean AE 0.0087 0.0086 0.0084 0.0077 0.0078
max AE 0.0163 0.0159 0.0150 0.0130 0.0130
std dev 0.0064 0.0063 0.0056 0.0054 0.0045
OsO4
mean AE 0.0692 0.0665 0.0540 0.0425 0.0286
max AE 0.0968 0.0948 0.0885 0.0649 0.0600
std dev 0.0234 0.0239 0.0262 0.0218 0.0232
Open in a new tab
a

Absolute error (AE) is defined to be the unsigned error of the TDDFT-SO result compared to X2C-TDDFT.

The TDDFT-SO errors shown in transition metal complexes are smaller than those for diatomic molecules. In transition metal complexes with light-element ligands, the metal center is the main contributor of the spin–orbit coupling. In contrast, the diatomic molecules presented in Table 2 include two elements having significant spin–orbit characters, resulting in many strongly coupled states. As a result, for a similar size of interaction space, TDDFT-SO has a smaller error for transition metal complexes with a single main spin–orbit center than diatomic molecules with two spin–orbit centers.

3.4. UV–Vis Spectrum of Au25(SH)18–

As a low-scaling method, the state interaction TDDFT-SO method is uniquely suited for studying large scale systems, such as metal nanoclusters. Figure 3 compares perturbative TDDFT-SO spectrum of a Au25 cluster with variational two-component TDDFT (2c-TDDFT) where scalar relativity and spin–orbit coupling were captured through the CRENBL effective core potential.70,71,77−79 The nonrelativistic TDDFT (no SOC term) spectrum is also included for comparison. All electrons were treated explicitly in the TDDFT-SO calculation using the DKH-optimized triple-ζ Jorge basis set with polarization functions for the Au atoms66,80 and the nonrelativistic Pople 6-31G(d,p)66,75,81 and 6-31G basis sets66,82 for S and H atoms, respectively. The PBE0 hybrid functional67 was used in both the TDDFT-SO and two-component TDDFT calculations. Although the experimental spectrum83 was obtained using Au25(SPET)18–, whereas the computed spectra used the model system Au25(SH)18, it is known in the literature that the choice of ligands minimally affects the fingerprint band.84

Figure 3.

Figure 3

Open in a new tab

UV–Vis spectrum of Au25(SR)18–. Computed spectra are shifted, broadened, and normalized to align with the large peak at 1.90 eV. Excited states from TDDFT-SO are indicated by blue sticks, where the height is scaled by the oscillator strength. The full-width half-max was set to 0.07 eV. Experimental spectrum reproduced from ref (83). Copyright 2011 American Chemical Society.

Because they possess a large number of degrees of freedom, Jahn–Teller distortion causes most nanoclusters with an even number of electrons to have a closed-shell ground state.85 A study by Jiang et al. showed that spin–orbit coupling plays a large role in Au25(SR)18– nanocluster valence excitations.86Figure 3 shows that the splitting of the “fingerprint” absorption band at 1.60–2.00 eV was due to spin–orbit coupling. This splitting is clearly absent in the nonrelativistic TDDFT calculation. The TDDFT-SO spectrum compared remarkably, in both relative peak position and intensity, to results from experiment and 2c-TDDFT.

3.5. Partial Failure of State Interaction TDDFT-SO

As in all perturbative spin–orbit coupling treatment, when the spin–orbit strength is comparable in magnitude to ligand field or other electron–electron repulsion effects, the state interaction picture becomes inadequate. This failure is particularly acute when the interaction space is small and when the excitation is only limited to the first order. Figure 4 shows the molecular orbital and state energy diagrams of PtCl62–. The energy difference between the spin-free t2g (Pt-d) and t1g (Cl-p) orbitals in this case is computed to be ∼0.09 eV. The spin–orbit coupling (∼0.39 eV) for the t2g manifold splits the t2g manifold into two groups with two levels higher in energy than t1g. Electronic excitations from the occupied orbitals to the unoccupied space in TDDFT gives rise to the state energy diagram (right side of Figure 4) which is labeled with double group notations. These are energy ordering expected from a variational X2C-TDDFT calculation.

Figure 4.

Figure 4

Open in a new tab

Left: Molecular orbital (MO) diagram of PtCl62–. Each level is a single spin–orbital, with electrons represented by vertical bars. The MO diagram without SOC was obtained from the restricted Kohn–Sham reference used in TDDFT-SO. The MO diagram with SOC was obtained from the X2C Kohn–Sham reference used in X2C-TDDFT. Right: State diagram of PtCl6. Each state is described by its irreducible representation in G × SU(2), where G is the spatial symmetry group of the molecule.

Table 4 shows the TDDFT-SO computed ordering of excited states in PtCl62– compared to variational X2C-TDDFT results. It is evident that the TDDFT-SO incorrectly predicted the ordering of excited states. It is understandable that TDDFT-SO will have difficulty in resolving small (<20 meV) energy difference (for example, A2g vs 3T2g) given the statistics of errors presented in the previous section. However, we also observed an incorrect ordering for states with a large gap (A1g vs Eg in Table 4) This error, however, cannot be reconciled by simply increasing the interaction space. Accurately capturing spin–orbit coupling in TDDFT-SO may require constructing the effective Hamiltonian in an interaction space closer to the infinite-order response theory limit since linear response TDDFT can only provide a full CIS equivalent interaction space. Strong spin–orbit coupling will still require excitation operators beyond singles.

Table 4. Comparison between Excitation Energies of PtCl62– Calculated Using X2C-TDDFT and TDDFT-SO with an Interaction Space of N Singlet and N Triplet Spin-Free Statesa.

state TDDFT-SO
 X2C-TDDFT
  N = 6 N = 10 N = 15 N = 20 N = 60  
Eg 1.8010 1.7851 1.7835 1.7835 1.7624 1.8648
T2g 1.8697 1.8584 1.8573 1.8574 1.8334 1.9492
T1g 1.9167 1.9015 1.9006 1.9006 1.8780 1.9691
T1g 2.3104 2.3096 2.3082 2.3082 2.2640 2.3607
A2g 2.3867 2.3542 2.3590 2.3590 2.3358 2.3646
T2g 2.3727 2.3565 2.3572 2.3573 2.3261 2.4046
A1g 2.5220 2.5131 2.4966 2.4966 2.4630 2.4051
Eg 2.3552 2.3578 2.3513 2.3513 2.3180 2.4121
T1g 2.5832 2.5630 2.5627 2.5627 2.5232 2.5824
Open in a new tab
a

Each state is described by its irreducible representation in G × SU(2), where G is the spatial symmetry group of the molecule. States with large singlet-triplet mixing is written as a direct sum of the singlet and triplet irreducible representations. Notation was adopted from Altmann and Herzig.87 The ordering of highlighted states are incorrectly predicted by TDDFT-SO with a >0.1 eV error.

4. Conclusions and Perspective

TDDFT-SO, a perturbative SOC method for TDDFT, was developed within the state interaction framework. Scalar relativity was included variationally in the restricted Kohn–Sham reference. The TDDFT excited states are used as the zeroth-order wave function with one-electron spin–orbit operator in the TDDFT-SO formalism. Analytical expressions are presented to compute the spin–orbit coupled Hamiltonian matrix with zeroth-order wave function being spin microstates from a linear response TDDFT calculation. Two-electron SOC was approximated using the Boettger factor. Expressions to compute oscillator strengths between spin–orbit adiabatic states are also presented. In addition to the implementation in the development version of the Gaussian software package, an open-source Python code (PyTDDFT-SO) was developed to interface with Gaussian 16 to perform this method.

TDDFT-SO was tested against X2C-TDDFT for atoms, diatomic molecules, and transition metal complexes. TDDFT-SO results for atomic fine structure splitting agreed very well overall with X2C-TDDFT for light elements, but the performance deteriorates toward late-row elements as expected. Studies of diatomic molecules and transition metal complexes show that the accuracy of the TDDFT-SO approach can be improved by increasing the size of the interaction space.

The motivation for this work was to increase the accessibility of SOC methods in TDDFT for large systems. TDDFT-SO was able to produce the UV–Vis spectrum of the Au25(SR)18– nanocluster. The spectrum generated by TDDFT-SO is nearly identical to the X2C-TDDFT result and is in good agreement with experiment. While the method is widely applicable to most spin–orbit-driven chemical processes in light elements, the benchmark case of PtCl6 exemplifies the limitation of the state interaction TDDFT-SO when the spin–orbit strength is stronger than the energy splitting as TDDFT-SO incorrectly predicted the ordering of excited states. Caution must be taken when applying the TDDFT-SO approach to late-row elements:

  • Perturbative TDDFT-SO generally performs well for light element Z ≤ 56. Starting from the Ln series, TDDFT-SO error increases significantly, although increasing the interaction space size can aid reducing the error.

  • Readers should note that all microstates (S = 1, MS = −1, 0, +1) that belong to a same triplet manifold must be included in the interaction space. Failing to do so will cause degeneracy and Kramers’ symmetry breaking.

  • When the molecular system include multiple spin–orbit centers, a large interaction space is needed to produce satisfactory result using TDDFT-SO.

  • The TDDFT-SO approach requires a closed-shell ground state reference. This is because the analytical expressions for perturbative spin–orbit operators are defined in the spin eigenspace. When the ground state is of an open-shell character, variational relativistic TDDFT methods5,26,61,88 should be used.

The TDDFT-SO approach developed here also works with interior spin-free states using the energy-specific algorithm.89−91 In principle, the algorithm also works with effective-core-potentials (ECP), provided that the ECP is parametrized and calibrated for use with perturbative spin–orbit coupling, which requires an accurate description of the valence orbitals close to the nucleus. These important topics will be future investigations.

Acknowledgments

The development of state interaction spin–orbit methods for computing excited state couplings is supported by the Air Force Office of Scientific Research (grant no. FA9550-21-1-0344). Computations were facilitated through the use of advanced computational, storage, and networking infrastructure provided by the Hyak supercomputer system at the University of Washington, funded by the Student Technology Fee. Part of this work was conducted at Los Alamos National Laboratory, which is operated for the National Nuclear Security Administration of DOE by Triad National Security, LLC (Contract 89233218CNA000001). E.R.B. and P.Y. gratefully acknowledge the U.S. Department of Energy, Office of Science (DOE OS), Basic Energy Sciences (BES) Heavy Element Chemistry Program (HEC) at LANL (Contract DE-AC52-06NA25396). C.L. acknowledges support from the Seaborg Institute at LANL. J.M.K. gratefully acknowledges support from a Director’s Postdoctoral Fellowship at LANL.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacsau.2c00659.

  • Derivations of SOC matrix element using Wick’s theorem, atomic fine-structure splitting computed with different DFT functionals, excitation energies of diatomic molecules, and excitation energies of transition metal complexes (PDF)

  • Input geometries for all molecular species (TXT)

Author Contributions

CRediT: Can Liao data curation, investigation, methodology, software, validation, writing-original draft; Joseph M. Kasper conceptualization, resources, writing-original draft, writing-review & editing; Andrew J. Jenkins methodology, software, supervision, writing-review & editing; Ping Yang funding acquisition, investigation, project administration, writing-review & editing; Enrique R. Batista funding acquisition, investigation, project administration, resources, writing-review & editing; Michael J. Frisch software, writing-review & editing; Xiaosong Li conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, software, supervision, validation, writing-original draft, writing-review & editing.

The authors declare no competing financial interest.

Supplementary Material

au2c00659_si_001.pdf (1.2MB, pdf)
au2c00659_si_002.txt (4.4KB, txt)

References

  1. Marian C. M. Spin-Orbit Coupling and Intersystem Crossing in Molecules. WIREs Comput. Mol. Sci. 2012, 2, 187–203. 10.1002/wcms.83. [DOI] [Google Scholar]
  2. Valentine A. J. S.; Li X. Intersystem Crossings in Late-Row Elements: A Perspective. J. Phys. Chem. Lett. 2022, 13, 3039–3046. 10.1021/acs.jpclett.2c00207. [DOI] [PubMed] [Google Scholar]
  3. Kasper J. M.; Stetina T. F.; Jenkins A. J.; Li X. Ab Initio Methods for L-Edge X-Ray Absorption Spectroscopy. Chem. Phys. Rev. 2020, 1, 011304. 10.1063/5.0029725. [DOI] [Google Scholar]
  4. Sun S.; Li X. Relativistic Effects in Magnetic Circular Dichroism: Restricted Magnetic Balance and Temperature Dependence. J. Chem. Theory Comput. 2020, 16, 4533–4542. 10.1021/acs.jctc.0c00287. [DOI] [PubMed] [Google Scholar]
  5. Egidi F.; Sun S.; Goings J. J.; Scalmani G.; Frisch M. J.; Li X. Two-Component Non-Collinear Time-Dependent Spin Density Functional Theory for Excited State Calculations. J. Chem. Theory Comput. 2017, 13, 2591–2603. 10.1021/acs.jctc.7b00104. [DOI] [PubMed] [Google Scholar]
  6. Uoyama H.; Goushi K.; Shizu K.; Nomura H.; Adachi C. Highly Efficient Organic Light-Emitting Diodes from Delayed Fluorescence. Nature 2012, 492, 234–238. 10.1038/nature11687. [DOI] [PubMed] [Google Scholar]
  7. Wu W.; Guo H.; Wu W.; Ji S.; Zhao J. Organic Triplet Sensitizer Library Derived from a Single Chromophore (BODIPY) with Long-Lived Triplet Excited State for Triplet-Triplet Annihilation Based Upconversion. J. Org. Chem. 2011, 76, 7056–7064. 10.1021/jo200990y. [DOI] [PubMed] [Google Scholar]
  8. Pristash S. R.; Corp K. L.; Rabe E. J.; Schlenker C. W. Heavy-Atom-Free Red-to-Yellow Photon Upconversion in a Thiosquaraine Composite. ACS Appl. Energy Mater. 2020, 3, 19–28. 10.1021/acsaem.9b01808. [DOI] [Google Scholar]
  9. Casida M. E. In Recent Advances in Density Functional Methods (Part I); Chong D., Ed.; World Scientific: Singapore, 1995; p 155. [Google Scholar]
  10. Dreuw A.; Head-Gordon M. Single-Reference ab Initio Methods for the Calculation of Excited States of Large Molecules. Chem. Rev. 2006, 37, 832–839. 10.1002/chin.200605293. [DOI] [PubMed] [Google Scholar]
  11. Gao J.; Liu W.; Song B.; Liu C. Time-Dependent Four-Component Relativistic Density Functional Theory for Excitation Energies. J. Chem. Phys. 2004, 121, 6658–6666. 10.1063/1.1788655. [DOI] [PubMed] [Google Scholar]
  12. Gao J.; Zou W.; Liu W.; Xiao Y.; Peng D.; Song B.; Liu C. Time-Dependent Four-Component Relativistic Density-Functional Theory for Excitation Energies. II. The Exchange-Correlation Kernel. J. Chem. Phys. 2005, 123, 054102. 10.1063/1.1940609. [DOI] [PubMed] [Google Scholar]
  13. Kutzelnigg W.; Liu W. Quasirelativistic Theory Equivalent to Fully Relativistic Theory. J. Chem. Phys. 2005, 123, 241102. 10.1063/1.2137315. [DOI] [PubMed] [Google Scholar]
  14. Liu W.; Peng D. Infinite-Order Quasirelativistic Density Functional Method Based on the Exact Matrix Quasirelativistic Theory. J. Chem. Phys. 2006, 125, 044102. 10.1063/1.2222365. [DOI] [PubMed] [Google Scholar]
  15. Peng D.; Liu W.; Xiao Y.; Cheng L. Making Four- and Two-Component Relativistic Density Functional Methods Fully Equivalent Based on the Idea of From Atoms to Molecule. J. Chem. Phys. 2007, 127, 104106. 10.1063/1.2772856. [DOI] [PubMed] [Google Scholar]
  16. Ilias M.; Saue T. An Infinite-Order Relativistic Hamiltonian by a Simple One-Step Transformation. J. Chem. Phys. 2007, 126, 064102. 10.1063/1.2436882. [DOI] [PubMed] [Google Scholar]
  17. Liu W.; Peng D. Exact Two-component Hamiltonians Revisited. J. Chem. Phys. 2009, 131, 031104. 10.1063/1.3159445. [DOI] [PubMed] [Google Scholar]
  18. Liu W. Ideas of Relativistic Quantum Chemistry. Mol. Phys. 2010, 108, 1679–1706. 10.1080/00268971003781571. [DOI] [Google Scholar]
  19. Saue T. Relativistic Hamiltonians for Chemistry: A Primer. ChemPhysChem 2011, 12, 3077–3094. 10.1002/cphc.201100682. [DOI] [PubMed] [Google Scholar]
  20. Li Z.; Xiao Y.; Liu W. On the Spin Separation of Algebraic Two-Component Relativistic Hamiltonians. J. Chem. Phys. 2012, 137, 154114. 10.1063/1.4758987. [DOI] [PubMed] [Google Scholar]
  21. Peng D.; Middendorf N.; Weigend F.; Reiher M. An Efficient Implementation of Two-Component Relativistic Exact-Decoupling Methods for Large Molecules. J. Chem. Phys. 2013, 138, 184105. 10.1063/1.4803693. [DOI] [PubMed] [Google Scholar]
  22. Egidi F.; Goings J. J.; Frisch M. J.; Li X. Direct Atomic-Orbital-Based Relativistic Two-Component Linear Response Method for Calculating Excited-State Fine Structures. J. Chem. Theory Comput. 2016, 12, 3711–3718. 10.1021/acs.jctc.6b00474. [DOI] [PubMed] [Google Scholar]
  23. Goings J. J.; Kasper J. M.; Egidi F.; Sun S.; Li X. Real Time Propagation of the Exact Two Component Time-Dependent Density Functional Theory. J. Chem. Phys. 2016, 145, 104107. 10.1063/1.4962422. [DOI] [PubMed] [Google Scholar]
  24. Konecny L.; Kadek M.; Komorovsky S.; Malkina O. L.; Ruud K.; Repisky M. Acceleration of Relativistic Electron Dynamics by Means of X2C Transformation: Application to the Calculation of Nonlinear Optical Properties. J. Chem. Theory Comput. 2016, 12, 5823–5833. 10.1021/acs.jctc.6b00740. [DOI] [PubMed] [Google Scholar]
  25. Liu J.; Cheng L. Relativistic Coupled-Cluster and Equation-of-Motion Coupled-Cluster Methods. WIREs Comput. Mol. Sci. 2021, 11, 1536. 10.1002/wcms.1536. [DOI] [Google Scholar]
  26. Stetina T. F.; Kasper J. M.; Li X. Modeling L2,3-Edge X-ray Absorption Spectroscopy with Linear Response Exact Two-Component Relativistic Time-Dependent Density Functional Theory. J. Chem. Phys. 2019, 150, 234103. 10.1063/1.5091807. [DOI] [PubMed] [Google Scholar]
  27. Kasper J. M.; Li X. Natural Transition Orbitals for Complex Two-Component Excited State Calculations. J. Comput. Chem. 2020, 41, 1557–1563. 10.1002/jcc.26196. [DOI] [PubMed] [Google Scholar]
  28. Goings J. J.; Egidi F.; Li X. Current Development of Non-collinear Electronic Structure Theory. Int. J. Quantum Chem. 2018, 118, e25398. 10.1002/qua.25398. [DOI] [Google Scholar]
  29. Heß B. A.; Marian C. M.; Wahlgren U.; Gropen O. A Mean-field Spin-orbit Method Applicable to Correlated Wavefunctions. Chem. Phys. Lett. 1996, 251, 365–371. 10.1016/0009-2614(96)00119-4. [DOI] [Google Scholar]
  30. Berning A.; Schweizer M.; Werner H.-J.; Knowles P. J.; Palmieri P. Spin-Orbit Matrix Elements for Internally Contracted Multireference Configuration Interaction Wavefunctions. Mol. Phys. 2000, 98, 1823–1833. 10.1080/00268970009483386. [DOI] [Google Scholar]
  31. Koseki S.; Fedorov D. G.; Schmidt M. W.; Gordon M. S. Spin-Orbit Splittings in the Third-Row Transition Elements: Comparison of Effective Nuclear Charge and Full Breit-Pauli Calculations. J. Phys. Chem. A 2001, 105, 8262–8268. 10.1021/jp011677r. [DOI] [Google Scholar]
  32. Malmqvist P. A.; Roos B.; Schimmelpfennig B. The Restricted Active Space (RAS) State Interaction Approach with Spin–Orbit Coupling. Chem. Phys. Lett. 2002, 357, 230–240. 10.1016/S0009-2614(02)00498-0. [DOI] [Google Scholar]
  33. Roos B.; Malmqvist P. A. Relativistic Quantum Chemistry: The Multiconfigurational Approach. Phys. Chem. Chem. Phys. 2004, 6, 2919–2927. 10.1039/b401472n. [DOI] [Google Scholar]
  34. Wang F.; Ziegler T. A Simplified Relativistic Time-Dependent Density-Functional Theory Formalism for the Calculations of Excitation Energies Including Spin-Orbit Coupling Effect. J. Chem. Phys. 2005, 123, 154102. 10.1063/1.2061187. [DOI] [PubMed] [Google Scholar]
  35. Wang F.; Gauss J.; van Wüllen C. Closed-shell Coupled-cluster Theory with Spin-orbit Coupling. J. Chem. Phys. 2008, 129, 064113. 10.1063/1.2968136. [DOI] [PubMed] [Google Scholar]
  36. Wang Z.; Wang F. Spin-Orbit Coupling and Electron Correlation at Various Coupled-cluster Levels for Closed-shell Diatomic Molecules. Phys. Chem. Chem. Phys. 2013, 15, 17922–17928. 10.1039/c3cp51749g. [DOI] [PubMed] [Google Scholar]
  37. Epifanovsky E.; Klein K.; Stopkowicz S.; Gauss J.; Krylov A. Spin-Orbit Couplings within the Equation-of-Motion Coupled-Cluster Framework: Theory, Implementation, and Benchmark Calculations. J. Chem. Phys. 2015, 143, 064102. 10.1063/1.4927785. [DOI] [PubMed] [Google Scholar]
  38. Cheng L.; Wang F.; Stanton J. F.; Gauss J. Perturbative Treatment of Spin-orbit-coupling within Spin-free Exact Two-component Theory using Equation-of-motion Coupled-cluster Methods. J. Chem. Phys. 2018, 148, 044108. 10.1063/1.5012041. [DOI] [PubMed] [Google Scholar]
  39. Mussard B.; Sharma S. One-Step Treatment of Spin–Orbit Coupling and Electron Correlation in Large Active Spaces. J. Chem. Theory Comput. 2018, 14, 154–165. 10.1021/acs.jctc.7b01019. [DOI] [PubMed] [Google Scholar]
  40. Bokhan D.; Trubnikov D. N.; Perera A.; Bartlett R. J. Spin-orbit Split Ionized and Electron-attached States using Explicitly-Correlated Equation-of-motion Coupled-cluster Singles and Doubles Eigenvectors. Chem. Phys. Lett. 2019, 730, 372–377. 10.1016/j.cplett.2019.06.036. [DOI] [Google Scholar]
  41. Vidal M. L.; Pokhilko P.; Krylov A. I.; Coriani S. Equation-of-Motion Coupled-Cluster Theory to Model L-Edge X-ray Absorption and Photoelectron Spectra. J. Phys. Chem. Lett. 2020, 11, 8314–8321. 10.1021/acs.jpclett.0c02027. [DOI] [PubMed] [Google Scholar]
  42. Meitei O. R.; Houck S. E.; Mayhall N. J. Spin–Orbit Matrix Elements for a Combined Spin-Flip and IP/EA Approach. J. Chem. Theory Comput. 2020, 16, 3597–3606. 10.1021/acs.jctc.0c00103. [DOI] [PubMed] [Google Scholar]
  43. Carreras A.; Jiang H.; Pokhilko P.; Krylov A. I.; Zimmerman P. M.; Casanova D. Calculation of Spin–orbit Couplings using RASCI Spinless One-particle Density Matrices: Theory and Applications. J. Chem. Phys. 2020, 153, 214107. 10.1063/5.0029146. [DOI] [PubMed] [Google Scholar]
  44. Li Z.; Suo B.; Zhang Y.; Xiao Y.; Liu W. Combining Spin-Adapted Open-Shell TD-DFT with Spin–Orbit Coupling. Mol. Phys. 2013, 111, 3741–3755. 10.1080/00268976.2013.785611. [DOI] [Google Scholar]
  45. Franco de Carvalho F.; Curchod B. F. E.; Penfold T. J.; Tavernelli I. Derivation of Spin-Orbit Couplings in Collinear Linear-Response TDDFT: A Rigorous Formulation. J. Chem. Phys. 2014, 140, 144103. 10.1063/1.4870010. [DOI] [PubMed] [Google Scholar]
  46. de Souza B.; Farias G.; Neese F.; Izsák R. Predicting Phosphorescence Rates of Light Organic Molecules Using Time-Dependent Density Functional Theory and the Path Integral Approach to Dynamics. J. Chem. Theory Comput. 2019, 15, 1896–1904. 10.1021/acs.jctc.8b00841. [DOI] [PMC free article] [PubMed] [Google Scholar]
  47. Bellonzi N.; Alguire E.; Fatehi S.; Shao Y.; Subotnik J. E. TD-DFT Spin-adiabats with Analytic Nonadiabatic Derivative Couplings. J. Chem. Phys. 2020, 152, 044112. 10.1063/1.5126440. [DOI] [PMC free article] [PubMed] [Google Scholar]
  48. Kotaru S.; Pokhilko P.; Krylov A. I. Spin–Orbit Couplings within Spin-Conserving and Spin-Flipping Time-Dependent Density Functional Theory: Implementation and Benchmark Calculations. J. Chem. Phys. 2022, 157, 224110. 10.1063/5.0130868. [DOI] [PubMed] [Google Scholar]
  49. Gao X.; Bai S.; Fazzi D.; Niehaus T.; Barbatti M.; Thiel W. Evaluation of Spin-Orbit Couplings with Linear-Response Time- Dependent Density Functional Methods. J. Chem. Theory Comput. 2017, 13, 515–524. 10.1021/acs.jctc.6b00915. [DOI] [PubMed] [Google Scholar]
  50. Frisch M. J.; Trucks G. W.; Schlegel H. B.; Scuseria G. E.; Robb M. A.; Cheeseman J. R.; Scalmani G.; Barone V.; Petersson G. A.; Nakatsuji H.; Li X.; Caricato M.; Marenich A. V.; Bloino J.; Janesko B. G.; Gomperts R.; Mennucci B.; Hratchian H. P.; Ortiz J. V.; Izmaylov A. F.; Sonnenberg J. L.; Williams-Young D.; Ding F.; Lipparini F.; Egidi F.; Goings J.; Peng B.; Petrone A.; Henderson T.; Ranasinghe D.; Zakrzewski V. G.; Gao J.; Rega N.; Zheng G.; Liang W.; Hada M.; Ehara M.; Toyota K.; Fukuda R.; Hasegawa J.; Ishida M.; Nakajima T.; Honda Y.; Kitao O.; Nakai H.; Vreven T.; Throssell K.; Montgomery J. A. Jr.; Peralta J. E.; Ogliaro F.; Bearpark M. J.; Heyd J. J.; Brothers E. N.; Kudin K. N.; Staroverov V. N.; Keith T. A.; Kobayashi R.; Normand J.; Raghavachari K.; Rendell A. P.; Burant J. C.; Iyengar S. S.; Tomasi J.; Cossi M.; Millam J. M.; Klene M.; Adamo C.; Cammi R.; Ochterski J. W.; Martin R. L.; Morokuma K.; Farkas O.; Foresman J. B.; Fox D. J.. Gaussian Development Version Revision J.14+.
  51. Liao C.; Li X.. PyTDDFT-SO. https://github.com/xsligroup/PyTDDFT-SO (accessed 2022–11–17, git hash 282ba6c04ec18f104ed4983fe14137eee93c9cc8), 2022.
  52. Stratmann R. E.; Scuseria G. E.; Frisch M. J. An Efficient Implementation of Time-Dependent Density-Functional Theory for the Calculation of Excitation Energies of Large Molecules. J. Chem. Phys. 1998, 109, 8218–8224. 10.1063/1.477483. [DOI] [Google Scholar]
  53. Dyall K. G.; Faegri K.. Introduction to Relativistic Quantum Chemistry; Oxford University Press, Inc.: Oxford, U.K., 2007. [Google Scholar]
  54. Reiher M.; Wolf A.. Relativistic Quantum Chemistry, 2nd ed.; Wiley-VCH: Weinheim, 2015. [Google Scholar]
  55. Sun S.; Stetina T. F.; Zhang T.; Hu H.; Valeev E. F.; Sun Q.; Li X. Efficient Four-Component Dirac–Coulomb–Gaunt Hartree–Fock in the Pauli Spinor Representation. J. Chem. Theory Comput. 2021, 17, 3388–3402. 10.1021/acs.jctc.1c00137. [DOI] [PubMed] [Google Scholar]
  56. Sun S.; Ehrman J. N.; Sun Q.; Li X. Efficient Evaluation of the Breit Operator in the Pauli Spinor Basis. J. Chem. Phys. 2022, 157, 064112. 10.1063/5.0098828. [DOI] [PubMed] [Google Scholar]
  57. Chiodo S. G.; Russo N. One-Electron Spin-Orbit Contribution by Effective Nuclear Charges. J. Comput. Chem. 2009, 30, 832–839. 10.1002/jcc.21113. [DOI] [PubMed] [Google Scholar]
  58. Boettger J. C. Approximate Two-Electron Spin-Orbit Coupling Term for Density-Functional-Theory DFT Calculations Using the Douglas-Kroll-Hess Transformation. Phys. Rev. B 2000, 62, 7809–7815. 10.1103/PhysRevB.62.7809. [DOI] [Google Scholar]
  59. Valentine A. J. S.; Li X. Toward the Evaluation of Intersystem Crossing Rates with Variational Relativistic Methods. J. Chem. Phys. 2019, 151, 084107. 10.1063/1.5113815. [DOI] [PubMed] [Google Scholar]
  60. Frisch M. J.; Trucks G. W.; Schlegel H. B.; Scuseria G. E.; Robb M. A.; Cheeseman J. R.; Scalmani G.; Barone V.; Petersson G. A.; Nakatsuji H.; Li X.; Caricato M.; Marenich A. V.; Bloino J.; Janesko B. G.; Gomperts R.; Mennucci B.; Hratchian H. P.; Ortiz J. V.; Izmaylov A. F.; Sonnenberg J. L.; Williams-Young D.; Ding F.; Lipparini F.; Egidi F.; Goings J.; Peng B.; Petrone A.; Henderson T.; Ranasinghe D.; Zakrzewski V. G.; Gao J.; Rega N.; Zheng G.; Liang W.; Hada M.; Ehara M.; Toyota K.; Fukuda R.; Hasegawa J.; Ishida M.; Nakajima T.; Honda Y.; Kitao O.; Nakai H.; Vreven T.; Throssell K.; Montgomery J. A. Jr.; Peralta J. E.; Ogliaro F.; Bearpark M. J.; Heyd J. J.; Brothers E. N.; Kudin K. N.; Staroverov V. N.; Keith T. A.; Kobayashi R.; Normand J.; Raghavachari K.; Rendell A. P.; Burant J. C.; Iyengar S. S.; Tomasi J.; Cossi M.; Millam J. M.; Klene M.; Adamo C.; Cammi R.; Ochterski J. W.; Martin R. L.; Morokuma K.; Farkas O.; Foresman J. B.; Fox D. J.. Gaussian Inc 16, Revision A.03;Gaussian Inc.: Wallingford CT, 2016. [Google Scholar]
  61. Kasper J. M.; Lestrange P. J.; Stetina T. F.; Li X. Modeling L2,3-Edge X-ray Absorption Spectroscopy with Real-Time Exact Two-Component Relativistic Time-Dependent Density Functional Theory. J. Chem. Theory Comput. 2018, 14, 1998–2006. 10.1021/acs.jctc.7b01279. [DOI] [PubMed] [Google Scholar]
  62. Veryazov V.; Widmark P.-O.; Roos B. O. Relativistic Atomic Natural Orbital Type Basis Sets for the Alkaline and Alkaline-Earth Atoms Applied to the Ground-State Potentials for the Corresponding Dimers. Theor. Chem. Acc. 2004, 111, 345–351. 10.1007/s00214-003-0537-0. [DOI] [Google Scholar]
  63. Roos B. O.; Lindh R.; Malmqvist P.-A.; Veryazov V.; Widmark P.-O. Main Group Atoms and Dimers Studied with a New Relativistic ANO Basis Set. J. Phys. Chem. A 2004, 108, 2851–2858. 10.1021/jp031064+. [DOI] [Google Scholar]
  64. Roos B. O.; Lindh R.; Malmqvist P.-A.; Veryazov V.; Widmark P.-O. MNew Relativistic ANO Basis Sets for Transition Metal Atoms. J. Phys. Chem. A 2005, 109, 6575–6579. 10.1021/jp0581126. [DOI] [PubMed] [Google Scholar]
  65. Roos B. O.; Lindh R.; Malmqvist P.-A.; Veryazov V.; Widmark P.-O.; Borin A. C. New Relativistic Atomic Natural Orbital Basis Sets for Lanthanide Atoms with Applications to the Ce Diatom and LuF3. J. Phys. Chem. A 2008, 112, 11431–11435. 10.1021/jp803213j. [DOI] [PubMed] [Google Scholar]
  66. Pritchard B. P.; Altarawy D.; Didier B.; Gibson T. D.; Windus T. L. New Basis Set Exchange: An Open, Up-to-Date Resource for the Molecular Sciences Community. J. Chem. Inf. Model. 2019, 59, 4814–4820. 10.1021/acs.jcim.9b00725. [DOI] [PubMed] [Google Scholar]
  67. Adamo C.; Barone V. Toward Reliable Density Functional Methods Without Adjustable Parameters: The PBE0 Model. J. Chem. Phys. 1999, 110, 6158–6170. 10.1063/1.478522. [DOI] [Google Scholar]
  68. Becke A. D. A New Mixing of Hartree–Fock and Local Density-Functional Theories. J. Chem. Phys. 1993, 98, 1372–1377. 10.1063/1.464304. [DOI] [Google Scholar]
  69. Huber K. P.; Herzberg G. H. In NIST Chemistry WebBook, NIST Standard Reference Database Number 69; Linstrom P., Mallard W., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, p 20899, (retrieved January 15, 2022).
  70. LaJohn L. A.; Ross R. B.; Atashroo T.; Ermler W. C.; Christiansen P. A. Ab Initio Relativistic Effective Potentials with Spin-Orblt Operators. III. Rb Through Xe. J. Chem. Phys. 1987, 87, 2812–2824. 10.1063/1.453069. [DOI] [Google Scholar]
  71. LaJohn L. A.; Ross R. B.; Powers J. M.; Atashroo T.; Ermler W. C.; Christiansen P. A. Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. IV. Cs Through Rn. J. Chem. Phys. 1990, 93, 6654–6670. 10.1063/1.458934. [DOI] [Google Scholar]
  72. Noro T.; Sekiya M.; Koga T. Segmented Contracted Basis Sets for Atoms H Through Xe: Sapporo-(DK)-nZP Sets (n = D, T, Q). Theor. Chem. Acc. 2012, 131, 1124. 10.1007/s00214-012-1124-z. [DOI] [Google Scholar]
  73. Noro T.; Sekiya M.; Koga T. Sapporo-(DKH3)-nZP (n = D, T, Q) Sets for the Sixth Period s-, d-, and p-Block Atoms. Theor. Chem. Acc. 2013, 132, 1363. 10.1007/s00214-013-1363-7. [DOI] [Google Scholar]
  74. Krishnan R.; Binkley J. S.; Seeger R.; Pople J. A. Self-Consistent Molecular Orbital Methods. XX. A Basis Set for Correlated Wave Functions. J. Chem. Phys. 1980, 72, 650–654. 10.1063/1.438955. [DOI] [Google Scholar]
  75. Francl M. M.; Pietro W. J.; Hehre W. J.; Binkley J. S.; Gordon M. S.; DeFrees D. J.; Pople J. A. Self-Consistent Molecular Orbital Methods. XXIII. A Polarization-Type Basis Set for Second-Row Elements. J. Chem. Phys. 1982, 77, 3654–3665. 10.1063/1.444267. [DOI] [Google Scholar]
  76. McLean A. D.; Chandler G. S. Contracted Gaussian Basis Sets for Molecular Calculations. I. Second Row Atoms, Z = 11–18. J. Chem. Phys. 1980, 72, 5639–5648. 10.1063/1.438980. [DOI] [Google Scholar]
  77. Pacios L. F.; Christiansen P. A. Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. I. Li Through Ar. J. Chem. Phys. 1985, 82, 2664–2671. 10.1063/1.448263. [DOI] [Google Scholar]
  78. Hurley M.; Pacios L. F.; Ross R. B.; Ermler W. C.; Christiansen P. A. Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. II. K Through Kr. J. Chem. Phys. 1986, 84, 6840–6853. 10.1063/1.450689. [DOI] [Google Scholar]
  79. Ross R. B.; Gayen S.; Ermler W. C. Ab Initio Relativistic Effective Potentials with Spin-Orbit Operators. V. Ce Through Lu. J. Chem. Phys. 1994, 100, 8145–8155. 10.1063/1.466809. [DOI] [Google Scholar]
  80. Martins L. S. C.; Jorge F. E.; Machado S. F. All-Electron Segmented Contraction Basis Sets of Triple Zeta Valence Quality for the Fifth-Row Elements. Mol. Phys. 2015, 113, 3578–3586. 10.1080/00268976.2015.1040095. [DOI] [Google Scholar]
  81. Gordon M. S.; Binkley J. S.; Pople J. A.; Pietro W. J.; Hehre W. J. Self-Consistent Molecular-Orbital Methods. 22. Small Split-Valence Basis Sets for Second-Row Elements. J. Am. Chem. Soc. 1982, 104, 2797–2803. 10.1021/ja00374a017. [DOI] [Google Scholar]
  82. Ditchfield R.; Hehre W. J.; Pople J. A. Self-Consistent Molecular-Orbital Methods. IX. An Extended Gaussian- Type Basis for Molecular-Orbital Studies of Organic Molecules. J. Chem. Phys. 1971, 54, 724–728. 10.1063/1.1674902. [DOI] [Google Scholar]
  83. Devadas M. S.; Bairu S.; Qian H.; Sinn E.; Jin R.; Ramakrishna G. Temperature-Dependent Optical Absorption Properties of Monolayer-Protected Au25 and Au38 Clusters. J. Phys. Chem. Lett. 2011, 2, 2752–2758. 10.1021/jz2012897. [DOI] [Google Scholar]
  84. Kang X.; Chong H.; Zhu M. Hanbao Au25(SR)18: The Captain of the Great Nanocluster Ship. Nanoscale 2018, 10, 10758–10834. 10.1039/C8NR02973C. [DOI] [PubMed] [Google Scholar]
  85. Cheshnovsky O.; Brucat P. J.; Yang S.; Pettiette C. L.; Craycraft M. J.; Smalley R. E. In Physics and Chemistry of Small Clusters; Jena P., Ed.; Plenum Press: New York, 1987. [Google Scholar]
  86. Jiang D.-e.; Kuhn M.; Tang Q.; Weigend F. Superatomic Orbitals under Spin–Orbit Coupling. J. Phys. Chem. Lett. 2014, 5, 3286–3289. 10.1021/jz501745z. [DOI] [PubMed] [Google Scholar]
  87. Altmann S. L.; Herzig P.. Point-Group Theory Tables; Clarendon Press: Oxford, 2011. [Google Scholar]
  88. Petrone A.; Williams-Young D. B.; Sun S.; Stetina T. F.; Li X. An Efficient Implementation of Two-Component Relativistic Density Functional Theory with Torque-Free Auxiliary Variables. Euro. Phys. J. B 2018, 91, 169. 10.1140/epjb/e2018-90170-1. [DOI] [Google Scholar]
  89. Liang W.; Fischer S. A.; Frisch M. J.; Li X. Energy-Specific Linear Response TDHF/TDDFT for Calculating High-Energy Excited States. J. Chem. Theory Comput. 2011, 7, 3540–3547. 10.1021/ct200485x. [DOI] [PubMed] [Google Scholar]
  90. Lestrange P. J.; Nguyen P. D.; Li X. Calibration of Energy-Specific TDDFT for Modeling K-Edge XAS Spectra of Light Elements. J. Chem. Theory Comput. 2015, 11, 2994–2999. 10.1021/acs.jctc.5b00169. [DOI] [PubMed] [Google Scholar]
  91. Peng B.; Lestrange P. J.; Goings J. J.; Caricato M.; Li X. Energy-Specific Equation-of-Motion Coupled-Cluster Methods for High-Energy Excited States: Application to K-Edge X-Ray Absorption Spectroscopy. J. Chem. Theory Comput. 2015, 11, 4146–4153. 10.1021/acs.jctc.5b00459. [DOI] [PubMed] [Google Scholar]

Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

au2c00659_si_001.pdf (1.2MB, pdf)
au2c00659_si_002.txt (4.4KB, txt)

Articles from JACS Au are provided here courtesy of American Chemical Society

RESOURCES