Excited State Properties from the Bethe--Salpeter Equation: State-to-State Transitions and Spin-Orbit Coupling

The formalism to calculate excited state properties from the $GW$-Bethe-Salpeter equation (BSE) method is introduced, providing convenient access to excited state absorption, excited state circular dichroism, and excited state optical rotation in the framework of the $GW$-BSE method. This is achieved using the second-order transition density, which can be obtained by solving a set of auxiliary equations similar to time-dependent density functional theory (TD-DFT). The proposed formulation therefore leads to no increase in the formal computational complexity when compared to the corresponding ground state properties. We further outline the calculation of fully relaxed spin-orbit coupling matrix elements within the $GW$-BSE method, allowing us to include perturbative corrections for spin-orbit coupling in aforementioned properties. These corrections are also extended to TD-DFT. Excited state absorption and perturbative spin-orbit coupling corrections within $GW$-BSE are evaluated for a selected set of molecular systems, yielding promising results.