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:
where denotes electrons, denotes nuclei, is the fine structure constant, and is the bare positive charge on nucleus . In the second quantization representation, the spin-orbit Hamiltonian in different directions can be expressed as
where for and are the matrix elements of this operator. The single-reference ab initio excited states (within the TDA) are given by
where and are singlet and triplet excitation coefficients of the th singlet or triplet state respectively, with the normalization
The quantity refers to the Hartree-Fock ground state. Thus the SOC constant from the singlet state to different triplet manifolds are
The SOC constant between different triplet manifolds can be obtained as
The total (root-mean-square) spin-orbit coupling is
For RPA states, the SOC constant can simply be obtained by replacing with and with .
The calculation of SOCs using effective nuclear charges, option (ii), is described in Section 220.127.116.11.
The calculations of SOCs using option (iii)—with a mean-field treatment of the two-electron part—is implemented following the algorithm described in
J. Chem. Phys.
(2019), 151, pp. 034106. and outlined in Section 18.104.22.168.
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).
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.
$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
$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
$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