State-Interaction Approach for -Matrix Calculations in TDDFT: Ground-Excited State Couplings and beyond First-Order Spin-Orbit Effects.

We introduce a state-interaction approach for computing -matrices within time-dependent density functional theory (TDDFT) and the Tamm-Dancoff approximation (TDA), applied here for the first time. This method provides a detailed understanding of -shifts by explicitly accounting for spin-orbit couplings (SOC) and excitation energies, enabling the analysis of different SOC orders and their contributions. To evaluate its accuracy and reliability, we compare state-interaction TDDFT and TDA with the widely used one-component coupled-perturbed Kohn-Sham approach. Applications to a diverse set of systems, including light and heavy atom molecules as well as transition-metal complexes, demonstrate that both methods yield comparable results in the absence of heavy elements, while the state-interaction approach offers improved insights into SOC effects and their impact on -shifts.