State-Interaction Approach for <i>g</i>-Matrix Calculations in TDDFT: Ground-Excited State Couplings and beyond First-Order Spin–Orbit Effects

We introduce a state-interaction approach for computing <i>g</i>-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 <i>g</i>-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 <i>g</i>-shifts.