Combining spin-adapted open-shell TD-DFT with spin–orbit coupling

The recently proposed spin-adapted time-dependent density functional theory (S-TD-DFT) is extended to the relativistic domain for fine-structure splittings of excited states of open-shell systems. Scalar-relativistic effects are treated to infinite order via the spin-free (sf) part of the exact two-component (X2C) Hamiltonian, whereas the spin–orbit couplings (SOC) between the scalar-excited states are treated perturbatively via an effective one-electron spin–orbit operator derived from the same X2C Hamiltonian. The calculated results for prototypical open-shell systems containing heavy elements reveal that the composite approach sf-X2C-S-TD-DFT-SOC is very promising. The fine-structure splitting of a spatially degenerate ground state can also be described properly by taking a non-degenerate excited state as the reference.