7.3 Time-Dependent Density Functional Theory (TDDFT)7.3.5 Analytic Excited-State Hessian in TDDFT 7.3.7 Various TDDFT-Based Examples

7.3.6 Calculations of Spin-Orbit Couplings Between TDDFT States

(April 13, 2024)

Several options for computing spin-orbit couplings (SOCs) between TDDFT states are available: (i) one-electron part of the Breit-Pauli Hamiltonian, (ii) one-electron SOC using scaled nuclear charges; (iii) full SOC using the mean-field treatment of the two-electron part. Options (i) and (ii) are are available for both TDA and RPA variants (including TDHF and CIS states), for restricted Kohn-Sham references only. Option (iii) is available for both restricted and unrestricted variants, but only within TDA. Calculations of SOC for SF-TDDFT are also possible within TDA using option (iii).

The implementation of one-electron SOC, options (i) and (ii), is based on the following. The SOCs are computed by evaluating matrix elements of the one-electron part of the Breit-Pauli Hamiltonian:

\hat{H}_{\text{SO}}=-\frac{\alpha_{0}^{2}}{2}\sum_{i,A}\frac{Z_{A}}{r^{3}_{iA}% }\left(\mathbf{r}_{iA}\times\mathbf{p}_{i}\right)\bm{\cdot}\mathbf{s}_{i}

(7.16)

where $i$ denotes electrons, $A$ denotes nuclei, $\alpha_{0}=1/137.037$ is the fine structure constant, and $Z_{A}$ is the bare positive charge on nucleus $A$ . In the second quantization representation, the spin-orbit Hamiltonian in different directions can be expressed as


$\displaystyle\hat{H}_{\text{SO},x}$	$\displaystyle=-\frac{\alpha_{0}^{2}}{2}\sum_{pq}\tilde{L}_{x,pq}\frac{\hbar}{2% }\left(\hat{a}^{\dagger}_{p}\hat{a}_{\bar{q}}+\hat{a}^{\dagger}_{\bar{p}}\hat{% a}_{q}\right)$	(7.17a)
$\displaystyle\hat{H}_{\text{SO},y}$	$\displaystyle=-\frac{\alpha_{0}^{2}}{2}\sum_{pq}\tilde{L}_{y,pq}\frac{\hbar}{2% \text{i}}\left(\hat{a}^{\dagger}_{p}\hat{a}_{\bar{q}}-\hat{a}^{\dagger}_{\bar{% p}}\hat{a}_{q}\right)$	(7.17b)
$\displaystyle\hat{H}_{\text{SO},z}$	$\displaystyle=-\frac{\alpha_{0}^{2}}{2}\sum_{pq}\tilde{L}_{z,pq}\frac{\hbar}{2% }\left(\hat{a}^{\dagger}_{p}\hat{a}_{q}-\hat{a}^{\dagger}_{\bar{p}}\hat{a}_{% \bar{q}}\right)$	(7.17c)

where $\hat{\tilde{L}}_{\alpha}=\hat{L}_{\alpha}r^{-3}$ for $\alpha\in\{x,y,z\}$ and $\tilde{L}_{\alpha,pq}$ are the matrix elements of this operator. The single-reference ab initio excited states (within the TDA) are given by


$\displaystyle\big{\|}\Phi^{I}_{\text{singlet}}\big{\rangle}$	$\displaystyle=\sum_{i,a}s^{Ia}_{i}\left(\hat{a}^{\dagger}_{a}\hat{a}_{i}+\hat{% a}^{\dagger}_{\bar{a}}\hat{a}_{\bar{i}}\right)\|\Phi_{\text{HF}}\rangle$	(7.18a)
$\displaystyle\big{\|}\Phi^{I,M_{s}=0}_{\text{triplet}}\big{\rangle}$	$\displaystyle=\sum_{i,a}t^{Ia}_{i}\left(\hat{a}^{\dagger}_{a}\hat{a}_{i}-\hat{% a}^{\dagger}_{\bar{a}}\hat{a}_{\bar{i}}\right)\|\Phi_{\text{HF}}\rangle$	(7.18b)
$\displaystyle\big{\|}\Phi^{I,M_{s}=1}_{\text{triplet}}\big{\rangle}$	$\displaystyle=\sqrt{2}\sum_{i,a}t^{Ia}_{i}\;\hat{a}^{\dagger}_{a}\hat{a}_{\bar% {i}}\|\Phi_{\text{HF}}\rangle$	(7.18c)
$\displaystyle\big{\|}\Phi^{I,M_{s}=-1}_{\text{triplet}}\big{\rangle}$	$\displaystyle=\sqrt{2}\sum_{i,a}t^{Ia}_{i}\;\hat{a}^{\dagger}_{\bar{a}}\hat{a}% _{i}\|\Phi_{\text{HF}}\rangle$	(7.18d)

where $s^{Ia}_{i}$ and $t^{Ia}_{i}$ are singlet and triplet excitation coefficients of the $I$ th singlet or triplet state respectively, with the normalization

\sum_{ia}(s^{Ia}_{i})^{2}=\sum_{ia}(t^{Ia}_{i})^{2}=\frac{1}{2}\;.

(7.19)

The quantity $|\Phi_{\text{HF}}\rangle$ refers to the Hartree-Fock ground state. Thus the SOC constant from the singlet state to different triplet manifolds are

\big{\langle}\Phi^{I}_{\text{singlet}}\big{|}\hat{H}_{\text{SO}}\big{|}\Phi^{J% ,M_{s}=0}_{\text{triplet}}\big{\rangle}=\frac{\alpha_{0}^{2}\hbar}{2}\left(% \sum_{i,a,b}\tilde{L}_{z,ab}\;s^{Ia}_{i}\;t^{Jb}_{i}-\sum_{i,j,a}\tilde{L}_{z,% ij}\;s^{Ia}_{i}\;t^{Ja}_{j}\right)

(7.20)

\displaystyle\begin{aligned} \displaystyle\big{\langle}\Phi^{I}_{\text{singlet% }}\big{|}\hat{H}_{\text{SO}}\big{|}\Phi^{J,M_{s}=\pm 1}_{\text{triplet}}\big{% \rangle}&\displaystyle=\mp\frac{\alpha_{0}^{2}\hbar}{2\sqrt{2}}\left(\sum_{i,a% ,b}\tilde{L}_{x,ab}\;s^{Ia}_{i}\;t^{Jb}_{i}-\sum_{i,j,a}\tilde{L}_{x,ij}\;s^{% Ia}_{i}\;t^{Ja}_{j}\right)\\ &\displaystyle\qquad\qquad+\frac{\alpha_{0}^{2}\hbar}{2\sqrt{2}\text{i}}\left(% \sum_{i,a,b}\tilde{L}_{y,ab}\;s^{Ia}_{i}\;t^{Jb}_{i}-\sum_{i,j,a}\tilde{L}_{y,% ij}\;s^{Ia}_{i}\;t^{Ja}_{j}\right)\;.\end{aligned}

(7.21)

The SOC constant between different triplet manifolds can be obtained as

\displaystyle\begin{aligned} \displaystyle\big{\langle}\Phi^{I,M_{s}=0}_{\text% {triplet}}\big{|}\hat{H}_{\text{SO}}\big{|}\Phi^{J,M_{s}=\pm 1}_{\text{triplet% }}\big{\rangle}&\displaystyle=\mp\frac{\alpha_{0}^{2}\hbar}{2\sqrt{2}}\left(% \sum_{i,a,b}\tilde{L}_{x,ab}\;t^{Ia}_{i}\;t^{Jb}_{i}+\sum_{i,j,a}\tilde{L}_{x,% ij}\;t^{Ia}_{i}\;t^{Ja}_{j}\right)\\ &\displaystyle\qquad\qquad\frac{\alpha_{0}^{2}\hbar}{2\sqrt{2}\text{i}}\left(% \sum_{i,a,b}\tilde{L}_{y,ab}\;t^{Ia}_{i}\;t^{Jb}_{i}+\sum_{i,j,a}\tilde{L}_{y,% ij}\;t^{Ia}_{i}\;t^{Ja}_{j}\right)\end{aligned}

(7.22)

\big{\langle}\Phi^{I,M_{s}=\pm 1}_{\text{triplet}}\big{|}\hat{H}_{\text{SO}}% \big{|}\Phi^{J,M_{s}=\pm 1}_{\text{triplet}}\big{\rangle}=\pm\frac{\alpha_{0}^% {2}\hbar}{2}\left(\sum_{i,a,b}\tilde{L}_{z,ab}\;t^{Ia}_{i}\;t^{Jb}_{i}+\sum_{i% ,j,a}\tilde{L}_{z,ij}\;t^{Ia}_{i}\;t^{Ja}_{j}\right)\;.

(7.23)

Note that

\big{\langle}\Phi^{I,M_{s}=0}_{\text{triplet}}\big{|}\hat{H}_{\text{SO}}\big{|% }\Phi^{J,M_{s}=0}_{\text{triplet}}\big{\rangle}=0=\big{\langle}\Phi^{I,M_{s}=% \pm 1}_{\text{triplet}}\big{|}\hat{H}_{\text{SO}}\big{|}\Phi^{J,M_{s}=\mp 1}_{% \text{triplet}}\big{\rangle}\;.

(7.24)

The total (root-mean-square) spin-orbit coupling is


$\displaystyle\big{\langle}\Phi^{I}_{\text{singlet}}\big{\|}\hat{H}_{\text{SO}}% \big{\|}\Phi^{J}_{\text{triplet}}\big{\rangle}$	$\displaystyle=\left(\sum_{M_{s}=0,\pm 1}\big{\\|}\big{\langle}\Phi^{I}_{\text{% singlet}}\big{\|}\hat{H}_{\text{SO}}\big{\|}\Phi^{J,M_{s}}_{\text{triplet}}\big{% \rangle}\big{\\|}^{2}\right)^{1/2}$	(7.25a)
$\displaystyle\big{\langle}\Phi^{I}_{\text{triplet}}\big{\|}\hat{H}_{\text{SO}}% \big{\|}\Phi^{J}_{\text{triplet}}\big{\rangle}$	$\displaystyle=\left(\sum_{M_{s}=0,\pm 1}\big{\\|}\big{\langle}\Phi^{I,M_{s}}_{% \text{triplet}}\big{\|}\hat{H}_{\text{SO}}\big{\|}\Phi^{J,M_{s}}_{\text{triplet}% }\big{\rangle}\big{\\|}^{2}\right)^{1/2}\;.$	(7.25b)

For RPA states, the SOC constant can simply be obtained by replacing $s^{Ia}_{i}\;t^{Jb}_{j}$ with $X^{Ia}_{i,\text{triplet}}X^{Jb}_{j,\text{triplet}}+Y^{Ia}_{i,\text{singlet}}Y^% {Jb}_{j,\text{triplet}}$ and $t^{Ia}_{i}\;t^{Jb}_{j}$ with $X^{Ia}_{i,\text{triplet}}X^{Jb}_{j,\text{triplet}}+Y^{Ia}_{i,\text{triplet}}Y^% {Jb}_{j,\text{triplet}}$ .

The calculation of SOCs using effective nuclear charges, option (ii), is described in Section 7.11.21.4. The calculations of SOCs using option (iii)—with a mean-field treatment of the two-electron part—is implemented following the algorithm described in Refs. ⁹⁹⁰ Pokhilko P., Epifanovsky E., Krylov A. I.
J. Chem. Phys.
(2019), 151, pp. 034106. Link ^, ⁶⁴⁴ Kotaru S., Pokhilko P., Krylov A. I.
J. Chem. Phys.
(2022), 157, pp. 224110. Link and outlined in Section 7.11.21.4.

The SOC calculation is activated by $rem variable CALC_SOC: CALC_SOC = 1 activates option (i), CALC_SOC = 4 activates option (ii), and CALC_SOC=2 activates option (iii).

Note: Setting CALC_SOC = TRUE activates a one-electron calculation using old algorithm, i.e., option (i).

CALC_SOC

CALC_SOC
       Controls whether to calculate the SOC constants for EOM-CC, RAS-CI, CIS, TDDFT/TDA and TDDFT/RPA.
TYPE:
       INTEGER/LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       FALSE Do not perform the SOC calculation. TRUE Perform the SOC calculation.
RECOMMENDATION:
       Although TRUE and FALSE values will work, EOM-CC code has more variants of SOC evaluations. For details, consult with the EOM section. For TDDFT/CIS, one can use values 1, 2, and 4, as explained above.

Examples 7.3.6, 7.3.6, and 7.3.6 illustrate calculations of SOCs for (SF)-TDDFT states using the above features. These calculations can also be carried out for CIS states by modifying METHOD appropriately.

The libwfa analysis of the spinless one-particle transition density matrices (as described in Ref. ⁹⁹² Pokhilko P., Krylov A. I.
J. Phys. Chem. Lett.
(2019), 10, pp. 4857–4862. Link ) is implemented for the TDDFT and SF-TDDFT calculations. To activate this analysis, use the following: CALC_SOC = 2, STATE_ANALYSIS = TRUE, MOLDEN_FORMAT = TRUE, and NTO_PAIRS = N (see Section 10.2 for details of the libwfa package). Note: This analysis differs from the NTO analysis of the regular transition density matrix between the singlet and triplet states.

Example 7.13 Calculation of one-electron SOCs for water molecule using TDDFT/B3LYP within the TDA.

$comment
   This sample input calculates the spin-orbit coupling constants for water
   between its ground state and its TDDFT/TDA excited triplets as well as the
   coupling between its TDDFT/TDA singlets and triplets.  Results are given in
   cm-1.
$end

$molecule
   0 1
   H       0.000000    -0.115747     1.133769
   H       0.000000     1.109931    -0.113383
   O       0.000000     0.005817    -0.020386
$end

$rem
   EXCHANGE             b3lyp
   BASIS                6-31G
   CIS_N_ROOTS          4
   CIS_CONVERGENCE      8
   MAX_SCF_CYCLES       600
   MAX_CIS_CYCLES       50
   SCF_ALGORITHM        diis
   MEM_STATIC           300
   MEM_TOTAL            2000
integral_symmetry false
point_group_symmetry False
   CIS_SINGLETS         true
   CIS_TRIPLETS         true
   CALC_SOC             true
   SET_ITER             300
$end

Example 7.14 Calculation of full SOCs for water molecule including mean-field treatment of the two-electron part of the Breit-Pauli Hamilton and UHF/TDDFT/B3LYP within the TDA.

$comment
Calculation of full SOCs for water molecule inlcuding mean-field treatment
of the two-electron part of the Breit-Pauli Hamiltonian and Wigner-Eckart theorem.
UHF/TDDFT/B3LYP/6-31G within the TDA.
$end

$molecule
0 1
H       0.000000    -0.115747     1.133769
H       0.000000     1.109931    -0.113383
O       0.000000     0.005817    -0.020386
$end

$rem
jobtype                 sp
unrestricted            true
method                  b3lyp
basis                   6-31G
cis_n_roots             4
cis_convergence         8
cis_singlets            true
cis_triplets            true
calc_soc                2
$end

Example 7.15 Calculation of SOCs for methylene using non-collinear SF-TDDFT/PBE0.

$comment
Calculation of SOCs for methylene using non-collinear SF-TDDFT/PBE0,
with tight convergence.
$end

$molecule
0 3
H1
C  H1 1.0775
H2 C  1.0775 H1 133.29
$end

$rem
method = pbe0
basis = cc-pvtz
scf_convergence = 12
cis_convergence = 12
THRESH = 14
cis_n_roots = 2
calc_soc = 2                    Compute full SOC with mean-field treatment of 2el part
WANG_ZIEGLER_KERNEL =  TRUE     Important for 1,1 diradicals
spin_flip = TRUE
$end


$\displaystyle\big{\|}\Phi^{I}_{\text{singlet}}\big{\rangle}$	$\displaystyle=\sum_{i,a}s^{Ia}_{i}\left(\hat{a}^{\dagger}_{a}\hat{a}_{i}+\hat{% a}^{\dagger}_{\bar{a}}\hat{a}_{\bar{i}}\right)\|\Phi_{\text{HF}}\rangle$	(7.18a)
$\displaystyle\big{\|}\Phi^{I,M_{s}=0}_{\text{triplet}}\big{\rangle}$	$\displaystyle=\sum_{i,a}t^{Ia}_{i}\left(\hat{a}^{\dagger}_{a}\hat{a}_{i}-\hat{% a}^{\dagger}_{\bar{a}}\hat{a}_{\bar{i}}\right)\|\Phi_{\text{HF}}\rangle$	(7.18b)
$\displaystyle\big{\|}\Phi^{I,M_{s}=1}_{\text{triplet}}\big{\rangle}$	$\displaystyle=\sqrt{2}\sum_{i,a}t^{Ia}_{i}\;\hat{a}^{\dagger}_{a}\hat{a}_{\bar% {i}}\|\Phi_{\text{HF}}\rangle$	(7.18c)
$\displaystyle\big{\|}\Phi^{I,M_{s}=-1}_{\text{triplet}}\big{\rangle}$	$\displaystyle=\sqrt{2}\sum_{i,a}t^{Ia}_{i}\;\hat{a}^{\dagger}_{\bar{a}}\hat{a}% _{i}\|\Phi_{\text{HF}}\rangle$	(7.18d)


$\displaystyle\big{\langle}\Phi^{I}_{\text{singlet}}\big{\|}\hat{H}_{\text{SO}}% \big{\|}\Phi^{J}_{\text{triplet}}\big{\rangle}$	$\displaystyle=\left(\sum_{M_{s}=0,\pm 1}\big{\\|}\big{\langle}\Phi^{I}_{\text{% singlet}}\big{\|}\hat{H}_{\text{SO}}\big{\|}\Phi^{J,M_{s}}_{\text{triplet}}\big{% \rangle}\big{\\|}^{2}\right)^{1/2}$	(7.25a)
$\displaystyle\big{\langle}\Phi^{I}_{\text{triplet}}\big{\|}\hat{H}_{\text{SO}}% \big{\|}\Phi^{J}_{\text{triplet}}\big{\rangle}$	$\displaystyle=\left(\sum_{M_{s}=0,\pm 1}\big{\\|}\big{\langle}\Phi^{I,M_{s}}_{% \text{triplet}}\big{\|}\hat{H}_{\text{SO}}\big{\|}\Phi^{J,M_{s}}_{\text{triplet}% }\big{\rangle}\big{\\|}^{2}\right)^{1/2}\;.$	(7.25b)

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7.3.6 Calculations of Spin-Orbit Couplings Between TDDFT States