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# 7.3.6 Calculations of Spin-Orbit Couplings Between TDDFT States

(July 14, 2022)

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)

or

 \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)

or

 $\big{\langle}\Phi^{I,M_{s}=\pm 1}_{\text{triplet}}\big{|}\hat{H}_{\text{SO}}% \big{|}\Phi^{J,M_{s}=\pm 1}_{\text{triplet}}\big{\rangle}=\pm\frac{\alpha_{0}^% {2}\hbar}{2}\left(\sum_{i,a,b}\tilde{L}_{z,ab}\;t^{Ia}_{i}\;t^{Jb}_{i}+\sum_{i% ,j,a}\tilde{L}_{z,ij}\;t^{Ia}_{i}\;t^{Ja}_{j}\right)\;.$ (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 7.10.20.4. The calculations of SOCs using option (iii)—with a mean-field treatment of the two-electron part—is implemented following the algorithm described in Ref.  962 Pokhilko P., Epifanovsky E., Krylov A. I.
J. Chem. Phys.
(2019), 151, pp. 034106.
and outlined in Section 7.10.20.4.

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, ADC, 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. Example 7.10 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
SYMMETRY             false
SYM_IGNORE           true
CIS_SINGLETS         true
CIS_TRIPLETS         true
CALC_SOC             true
SET_ITER             300
$end  Example 7.11 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 7.12 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