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

(February 4, 2022)

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)
		$\displaystyle+\frac{\alpha_{0}^{2}\hbar}{2\sqrt{2}i}\left(\sum\limits_{i,a,b}{% \tilde{L_{y}}}_{ab}s^{Ia}_{i}t^{Jb}_{i}-\sum\limits_{i,j,a}{\tilde{L_{y}}}_{ij% }s^{Ia}_{i}t^{Ja}_{j}\right)$	(7.25)

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

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/FALSE values will work, EOM-CC code has more variants of SOC evaluations. For details, consult with EOM section.

Note: SOC calculations are only implemented for restricted CIS/TDDFT at this time. SOCs can also be computed using effective nuclear charges, as described in Section 7.10.21.2. Such calculations are activated using CALC_SOC=4.

Example 7.10 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

View output

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7.3.6 Calculations of Spin-Orbit Couplings Between TDDFT States