Spin–orbit couplings within spin-conserving and spin-flipping time-dependent density functional theory: Implementation and benchmark calculations

We present a new implementation for computing spin-orbit couplings (SOCs) within time-dependent density-functional theory (TD-DFT) framework in the stan- dard spin-conserving formulation as well in the spin-flip variant (SF-TD-DFT). This approach employs the Breit-Pauli Hamiltonian and Wigner-Eckart’s theorem ap- plied to the reduced one-particle transition density matrices, together with the spin–orbit mean-field (SOMF) treatment of the two-electron contributions. We use state-interaction procedure and compute the SOC matrix elements using zero-order non-relativistic states. Benchmark calculations using several closed-shell organic molecules, diradicals, and a single-molecule magnet (SMM) illustrate the efficiency of the SOC protocol. The results for organic molecules (described by standard TD- DFT) show that SOCs are insensitive to the choice of the functional or basis sets, as long as the states of the same characters are compared. In contrast, the SF-TD- DFT results for small diradicals (CH2, NH+2 , SiH2, and PH+2 ) show strong functional dependence. The spin-reversal energy barrier in a Fe(III) SMM computed using non- collinear SF-TD-DFT (PBE0, ωPBEh/cc-pVDZ) agrees well with the experimental estimate.