# 7.3.6 Calculations of Spin-Orbit Couplings Between TDDFT States

Calculations of spin-orbit couplings (SOCs) for TDDFT states within the Tamm-Dancoff approximation or RPA (including TDHF and CIS states) is available. We employ the one-electron Breit Pauli Hamiltonian to calculate the SOC constant between TDDFT states.

 $\displaystyle\hat{H}_{\textrm{SO}}$ $\displaystyle=$ $\displaystyle{-\frac{\alpha_{0}^{2}}{2}\sum\limits_{i,A}\frac{Z_{A}}{r^{3}_{iA% }}\left(\mathbb{r}_{iA}\times\mathbb{p}_{i}\right)\cdot\mathbb{s}_{i}}$ (7.16)

where ${i}$ denotes electrons, ${A}$ denotes nuclei, $\alpha_{0}=137.037^{-1}$ is the fine structure constant. 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

 $\displaystyle{\hat{H}_{SO_{x}}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{0}^{2}}{2}\sum\limits_{pq}{\tilde{L_{x}}}_{pq}% \cdot\frac{\hbar}{2}\left(a^{\dagger}_{p}a_{\bar{q}}+a^{\dagger}_{\bar{p}}a_{q% }\right)$ (7.17) $\displaystyle{\hat{H}_{SO_{y}}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{0}^{2}}{2}\sum\limits_{pq}{\tilde{L_{y}}}_{pq}% \cdot\frac{\hbar}{2i}\left(a^{\dagger}_{p}a_{\bar{q}}-a^{\dagger}_{\bar{p}}a_{% q}\right)$ (7.18) $\displaystyle{\hat{H}_{SO_{z}}}$ $\displaystyle=$ $\displaystyle-\frac{\alpha_{0}^{2}}{2}\sum\limits_{pq}{\tilde{L_{z}}}_{pq}% \cdot\frac{\hbar}{2}\left(a^{\dagger}_{p}a_{q}-a^{\dagger}_{\bar{p}}a_{\bar{q}% }\right)$ (7.19)

where $\tilde{L_{\alpha}}=L_{\alpha}/r^{3}\left(\alpha=x,y,z\right)$. The single-reference $ab\ initio$ excited states (within the Tamm-Dancoff approximation) are given by

 $\displaystyle|\Phi^{I}_{\textrm{singlet}}\rangle$ $\displaystyle=$ $\displaystyle\sum\limits_{i,a}s^{Ia}_{i}\left(a^{\dagger}_{a}a_{i}+a^{\dagger}% _{\bar{a}}a_{\bar{i}}\right)|\Phi_{\textrm{HF}}\rangle$ (7.20) $\displaystyle|\Phi^{I,m_{s}=0}_{\textrm{triplet}}\rangle$ $\displaystyle=$ $\displaystyle\sum\limits_{i,a}t^{Ia}_{i}\left(a^{\dagger}_{a}a_{i}-a^{\dagger}% _{\bar{a}}a_{\bar{i}}\right)|\Phi_{\textrm{HF}}\rangle$ (7.21) $\displaystyle|\Phi^{I,m_{s}=1}_{\textrm{triplet}}\rangle$ $\displaystyle=$ $\displaystyle\sum\limits_{i,a}\sqrt{2}t^{Ia}_{i}a^{\dagger}_{a}a_{\bar{i}}|% \Phi_{\textrm{HF}}\rangle$ (7.22) $\displaystyle|\Phi^{I,m_{s}=-1}_{\textrm{triplet}}\rangle$ $\displaystyle=$ $\displaystyle\sum\limits_{i,a}\sqrt{2}t^{Ia}_{i}a^{\dagger}_{\bar{a}}a_{i}|% \Phi_{\textrm{HF}}\rangle$ (7.23)

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 $\sum\limits_{ia}{s^{Ia}_{i}}^{2}=\sum\limits_{ia}{t^{Ia}_{i}}^{2}=\frac{1}{2}$; $|\Phi_{\textrm{HF}}\rangle$ refers to the Hartree-Fock ground state. Thus the SOC constant from the singlet state to different triplet manifolds can be obtained as follows,

 $\displaystyle\langle\Phi^{I}_{\textrm{singlet}}|{\hat{H}_{\textrm{SO}}|}\Phi^{% J,m_{s}=0}_{\textrm{triplet}}\rangle$ $\displaystyle=$ $\displaystyle\frac{\alpha_{0}^{2}\hbar}{2}\left(\sum\limits_{i,a,b}{\tilde{L_{% z}}}_{ab}s^{Ia}_{i}t^{Jb}_{i}-\sum\limits_{i,j,a}{\tilde{L_{z}}}_{ij}s^{Ia}_{i% }t^{Ja}_{j}\right)$ (7.24) $\displaystyle\langle\Phi^{I}_{\textrm{singlet}}|{\hat{H}_{\textrm{SO}}|}\Phi^{% J,m_{s}=\pm 1}_{\textrm{triplet}}\rangle$ $\displaystyle=$ $\displaystyle\mp\frac{\alpha_{0}^{2}\hbar}{2\sqrt{2}}\left(\sum\limits_{i,a,b}% {\tilde{L_{x}}}_{ab}s^{Ia}_{i}t^{Jb}_{i}-\sum\limits_{i,j,a}{\tilde{L_{x}}}_{% ij}s^{Ia}_{i}t^{Ja}_{j}\right)$ (7.25) $\displaystyle+\frac{\alpha_{0}^{2}\hbar}{2\sqrt{2}i}\left(\sum\limits_{i,a,b}{% \tilde{L_{y}}}_{ab}s^{Ia}_{i}t^{Jb}_{i}-\sum\limits_{i,j,a}{\tilde{L_{y}}}_{ij% }s^{Ia}_{i}t^{Ja}_{j}\right)$

The SOC constant between different triplet manifolds can be obtained

 $\displaystyle\langle\Phi^{I,m_{s}=0}_{\textrm{triplet}}|{\hat{H}_{\textrm{SO}}% |}\Phi^{J,m_{s}=\pm 1}_{\textrm{triplet}}\rangle$ $\displaystyle=$ $\displaystyle\mp\frac{\alpha_{0}^{2}\hbar}{2\sqrt{2}}\left(\sum\limits_{i,a,b}% {\tilde{L_{x}}}_{ab}t^{Ia}_{i}t^{Jb}_{i}+\sum\limits_{i,j,a}{\tilde{L_{x}}}_{% ij}t^{Ia}_{i}t^{Ja}_{j}\right)$ (7.26) $\displaystyle+\frac{\alpha_{0}^{2}\hbar}{2\sqrt{2}i}\left(\sum\limits_{i,a,b}{% \tilde{L_{y}}}_{ab}t^{Ia}_{i}t^{Jb}_{i}+\sum\limits_{i,j,a}{\tilde{L_{y}}}_{ij% }t^{Ia}_{i}t^{Ja}_{j}\right)$ $\displaystyle\langle\Phi^{I,m_{s}=\pm 1}_{\textrm{triplet}}|{\hat{H}_{\textrm{% SO}}|}\Phi^{J,m_{s}=\pm 1}_{\textrm{triplet}}\rangle$ $\displaystyle=$ $\displaystyle\pm\frac{\alpha_{0}^{2}\hbar}{2}\left(\sum\limits_{i,a,b}{\tilde{% L_{z}}}_{ab}t^{Ia}_{i}t^{Jb}_{i}+\sum\limits_{i,j,a}{\tilde{L_{z}}}_{ij}t^{Ia}% _{i}t^{Ja}_{j}\right)$ (7.27)

Note that $\langle\Phi^{I,m_{s}=0}_{\textrm{triplet}}|{\hat{H}_{\textrm{SO}}|}\Phi^{J,m_{% s}=0}_{\textrm{triplet}}\rangle=\langle\Phi^{I,m_{s}=\pm 1}_{\textrm{triplet}}% |{\hat{H}_{\textrm{SO}}|}\Phi^{J,m_{s}=\mp 1}_{\textrm{triplet}}\rangle=0$. The total (root-mean-square) spin-orbit coupling is given by

 $\displaystyle\langle\Phi^{I}_{\textrm{singlet}}|{\hat{H}_{\textrm{SO}}|}\Phi^{% J}_{\textrm{triplet}}\rangle$ $\displaystyle=$ $\displaystyle\sqrt{\sum\limits_{m_{s}=0,\pm 1}\|\langle\Phi^{I}_{\textrm{% singlet}}|{\hat{H}_{\textrm{SO}}|}\Phi^{J,m_{s}}_{\textrm{triplet}}\rangle\|^{% 2}}$ (7.28) $\displaystyle\langle\Phi^{I}_{\textrm{triplet}}|{\hat{H}_{\textrm{SO}}|}\Phi^{% J}_{\textrm{triplet}}\rangle$ $\displaystyle=$ $\displaystyle\sqrt{\sum\limits_{m_{s}=0,\pm 1}\|\langle\Phi^{I,m_{s}}_{\textrm% {triplet}}|{\hat{H}_{\textrm{SO}}|}\Phi^{J,m_{s}}_{\textrm{triplet}}\rangle\|^% {2}}$ (7.29)

For RPA states, the SOC constant can simply be obtained by replacing $s^{Ia}_{i}t^{Jb}_{j}$ ($t^{Ia}_{i}t^{Jb}_{j}$) with $X^{Ia}_{i,{\textrm{singlet}}}X^{Jb}_{j,{\textrm{triplet}}}+Y^{Ia}_{i,{\textrm{% singlet}}}Y^{Jb}_{j,{\textrm{triplet}}}$ ($X^{Ia}_{i,{\textrm{triplet}}}X^{Jb}_{j,{\textrm{triplet}}}+Y^{Ia}_{i,{\textrm{% triplet}}}Y^{Jb}_{j,{\textrm{triplet}}}$) Setting the $rem variable CALC_SOC = TRUE will enable the SOC calculation for all calculated TDDFT states. CALC_SOC Controls whether to calculate the SOC constants for EOM-CC, ADC, TDDFT/TDA and TDDFT. TYPE: INTEGER/LOGICAL DEFAULT: FALSE OPTIONS: FALSE Do not perform the SOC calculation. TRUE Perform the SOC calculation. RECOMMENDATION: Although TRUE/FALSE values will work, EOM-CC code has more variants of SOC evaluations. For details, consult with EOM section. Example 7.9 Calculation of SOCs for water molecule using TDDFT/B3LYP functional 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