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.

${\widehat{H}}_{\text{SO}}$

$=$

$-{\displaystyle \frac{{\alpha }_0^2}{2}}{\displaystyle \sum _{i,A}}{\displaystyle \frac{Z_A}{r_{iA}^3}}\left({𝕣}_{iA}\times {𝕡}_i\right)\cdot {𝕤}_i$

(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

	${\widehat{H}}_{S{O}_x}$	$=$	$-{\displaystyle \frac{{\alpha }_0^2}{2}}{\displaystyle \sum _{pq}}{\widetilde{L_x}}_{pq}\cdot {\displaystyle \frac{\mathrm{\hslash }}{2}}\left(a_p^{†}{a}_{\overline{q}}+a_{\overline{p}}^{†}{a}_q\right)$		(7.17)
	${\widehat{H}}_{S{O}_y}$	$=$	$-{\displaystyle \frac{{\alpha }_0^2}{2}}{\displaystyle \sum _{pq}}{\widetilde{L_y}}_{pq}\cdot {\displaystyle \frac{\mathrm{\hslash }}{2i}}\left(a_p^{†}{a}_{\overline{q}}-a_{\overline{p}}^{†}{a}_q\right)$		(7.18)
	${\widehat{H}}_{S{O}_z}$	$=$	$-{\displaystyle \frac{{\alpha }_0^2}{2}}{\displaystyle \sum _{pq}}{\widetilde{L_z}}_{pq}\cdot {\displaystyle \frac{\mathrm{\hslash }}{2}}\left(a_p^{†}{a}_q-a_{\overline{p}}^{†}{a}_{\overline{q}}\right)$		(7.19)

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

	$|{\mathrm{\Phi }}_{\text{singlet}}^I⟩$	$=$	${\displaystyle \sum _{i,a}}{s}_i^{Ia}\left(a_a^{†}{a}_i+a_{\overline{a}}^{†}{a}_{\overline{i}}\right)|{\mathrm{\Phi }}_{\text{HF}}⟩$		(7.20)
	$|{\mathrm{\Phi }}_{\text{triplet}}^{I,m_s=0}⟩$	$=$	${\displaystyle \sum _{i,a}}{t}_i^{Ia}\left(a_a^{†}{a}_i-a_{\overline{a}}^{†}{a}_{\overline{i}}\right)|{\mathrm{\Phi }}_{\text{HF}}⟩$		(7.21)
	$|{\mathrm{\Phi }}_{\text{triplet}}^{I,m_s=1}⟩$	$=$	${\displaystyle \sum _{i,a}}\sqrt{2}{t}_i^{Ia}{a}_a^{†}{a}_{\overline{i}}|{\mathrm{\Phi }}_{\text{HF}}⟩$		(7.22)
	$|{\mathrm{\Phi }}_{\text{triplet}}^{I,m_s=-1}⟩$	$=$	${\displaystyle \sum _{i,a}}\sqrt{2}{t}_i^{Ia}{a}_{\overline{a}}^{†}{a}_i|{\mathrm{\Phi }}_{\text{HF}}⟩$		(7.23)

where $s_i^{Ia}$ and $t_i^{Ia}$ are singlet and triplet excitation coefficients of the $I^{th}$ singlet or triplet state respectively, with the normalization $\sum _{ia}s_i^{Ia}{}^2=\sum _{ia}t_i^{Ia}{}^2=\frac{1}{2}$; $|{\mathrm{\Phi }}_{\text{HF}}⟩$ refers to the Hartree-Fock ground state. Thus the SOC constant from the singlet state to different triplet manifolds can be obtained as follows,

	$⟨{\mathrm{\Phi }}_{\text{singlet}}^I|{\widehat{H}}_{\text{SO}}|{\mathrm{\Phi }}_{\text{triplet}}^{J,m_s=0}⟩$	$=$	${\displaystyle \frac{{\alpha }_0^2\mathrm{\hslash }}{2}}\left({\displaystyle \sum _{i,a,b}}{\widetilde{L_z}}_{ab}{s}_i^{Ia}{t}_i^{Jb}-{\displaystyle \sum _{i,j,a}}{\widetilde{L_z}}_{ij}{s}_i^{Ia}{t}_j^{Ja}\right)$		(7.24)
	$⟨{\mathrm{\Phi }}_{\text{singlet}}^I|{\widehat{H}}_{\text{SO}}|{\mathrm{\Phi }}_{\text{triplet}}^{J,m_s=\pm 1}⟩$	$=$	$\mp {\displaystyle \frac{{\alpha }_0^2\mathrm{\hslash }}{2\sqrt{2}}}\left({\displaystyle \sum _{i,a,b}}{\widetilde{L_x}}_{ab}{s}_i^{Ia}{t}_i^{Jb}-{\displaystyle \sum _{i,j,a}}{\widetilde{L_x}}_{ij}{s}_i^{Ia}{t}_j^{Ja}\right)$		(7.25)
			$+{\displaystyle \frac{{\alpha }_0^2\mathrm{\hslash }}{2\sqrt{2}i}}\left({\displaystyle \sum _{i,a,b}}{\widetilde{L_y}}_{ab}{s}_i^{Ia}{t}_i^{Jb}-{\displaystyle \sum _{i,j,a}}{\widetilde{L_y}}_{ij}{s}_i^{Ia}{t}_j^{Ja}\right)$

The SOC constant between different triplet manifolds can be obtained

	$⟨{\mathrm{\Phi }}_{\text{triplet}}^{I,m_s=0}|{\widehat{H}}_{\text{SO}}|{\mathrm{\Phi }}_{\text{triplet}}^{J,m_s=\pm 1}⟩$	$=$	$\mp {\displaystyle \frac{{\alpha }_0^2\mathrm{\hslash }}{2\sqrt{2}}}\left({\displaystyle \sum _{i,a,b}}{\widetilde{L_x}}_{ab}{t}_i^{Ia}{t}_i^{Jb}+{\displaystyle \sum _{i,j,a}}{\widetilde{L_x}}_{ij}{t}_i^{Ia}{t}_j^{Ja}\right)$		(7.26)
			$+{\displaystyle \frac{{\alpha }_0^2\mathrm{\hslash }}{2\sqrt{2}i}}\left({\displaystyle \sum _{i,a,b}}{\widetilde{L_y}}_{ab}{t}_i^{Ia}{t}_i^{Jb}+{\displaystyle \sum _{i,j,a}}{\widetilde{L_y}}_{ij}{t}_i^{Ia}{t}_j^{Ja}\right)$
	$⟨{\mathrm{\Phi }}_{\text{triplet}}^{I,m_s=\pm 1}|{\widehat{H}}_{\text{SO}}|{\mathrm{\Phi }}_{\text{triplet}}^{J,m_s=\pm 1}⟩$	$=$	$\pm {\displaystyle \frac{{\alpha }_0^2\mathrm{\hslash }}{2}}\left({\displaystyle \sum _{i,a,b}}{\widetilde{L_z}}_{ab}{t}_i^{Ia}{t}_i^{Jb}+{\displaystyle \sum _{i,j,a}}{\widetilde{L_z}}_{ij}{t}_i^{Ia}{t}_j^{Ja}\right)$		(7.27)

Note that $⟨{\mathrm{\Phi }}_{\text{triplet}}^{I,m_s=0}|{\widehat{H}}_{\text{SO}}|{\mathrm{\Phi }}_{\text{triplet}}^{J,m_s=0}⟩=⟨{\mathrm{\Phi }}_{\text{triplet}}^{I,m_s=\pm 1}|{\widehat{H}}_{\text{SO}}|{\mathrm{\Phi }}_{\text{triplet}}^{J,m_s=\mp 1}⟩=0$. The total (root-mean-square) spin-orbit coupling is given by

	$⟨{\mathrm{\Phi }}_{\text{singlet}}^I|{\widehat{H}}_{\text{SO}}|{\mathrm{\Phi }}_{\text{triplet}}^J⟩$	$=$	$\sqrt{{\displaystyle \sum _{m_s=0,\pm 1}}{\parallel ⟨{\mathrm{\Phi }}_{\text{singlet}}^I|{\widehat{H}}_{\text{SO}}|{\mathrm{\Phi }}_{\text{triplet}}^{J,m_s}⟩\parallel }^2}$		(7.28)
	$⟨{\mathrm{\Phi }}_{\text{triplet}}^I|{\widehat{H}}_{\text{SO}}|{\mathrm{\Phi }}_{\text{triplet}}^J⟩$	$=$	$\sqrt{{\displaystyle \sum _{m_s=0,\pm 1}}{\parallel ⟨{\mathrm{\Phi }}_{\text{triplet}}^{I,m_s}|{\widehat{H}}_{\text{SO}}|{\mathrm{\Phi }}_{\text{triplet}}^{J,m_s}⟩\parallel }^2}$		(7.29)

For RPA states, the SOC constant can simply be obtained by replacing $s_i^{Ia}{t}_j^{Jb}$ ($t_i^{Ia}{t}_j^{Jb}$) with $X_{i,\text{singlet}}^{Ia}{X}_{j,\text{triplet}}^{Jb}+Y_{i,\text{singlet}}^{Ia}{Y}_{j,\text{triplet}}^{Jb}$ ($X_{i,\text{triplet}}^{Ia}{X}_{j,\text{triplet}}^{Jb}+Y_{i,\text{triplet}}^{Ia}{Y}_{j,\text{triplet}}^{Jb}$) Setting the $rem variable CALC_SOC = TRUE will enable the SOC calculation for all calculated TDDFT states.

$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