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.
where denotes electrons, denotes nuclei, is the fine structure constant. Z is the bare positive charge on nucleus A. In the second quantization representation, the spin-orbit Hamiltonian in different directions can be expressed as
where . The single-reference excited states (within the Tamm-Dancoff approximation) are given by
where and are singlet and triplet excitation coefficients of the singlet or triplet state respectively, with the normalization ; refers to the Hartree-Fock ground state. Thus the SOC constant from the singlet state to different triplet manifolds can be obtained as follows,
The SOC constant between different triplet manifolds can be obtained
Note that . The total (root-mean-square) spin-orbit coupling is given by
For RPA states, the SOC constant can simply be obtained by replacing () with () 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