NAC-TDDFT: Time-Dependent Density Functional Theory for Nonadiabatic Couplings

First-order nonadiabatic coupling (NAC) matrix elements (fo-NACMEs) are the basic quantities in theoretical descriptions of electronically processes that ubiquitous molecular physics and chemistry. Given large size systems chemical interests, time-dependent density functional theory (TDDFT) is usually first choice methods. However, lack many-electron wave functions TDDFT renders formulation NAC-TDDFT for fo-NACMEs conceptually difficult. Because this, various variants have been proposed literature from different standing points, including Hellmann-Feynman-like expression auxiliary/pseudo-wave function (AWF)-, equation-of-motion (EOM)-, perturbation (TDPT)-based formulations. Based on critical analyses, following conclusions made here: (1) The expression, which rooted exact theory, hardly useful due to huge demand basis sets. (2) Although most popular, AWF not theoretically founded become ambiguous particularly between two excited states, although they do agree with EOM TDPT under Tamm-Dancoff approximation. (3) variant rigorous but suffers numerical instabilities one hand does differ a significant extent other hand. (4) As such, ground states solely right practice. These formal analyses fully supported by experimentations, taking azulene as showcase. proper implementation also highlighted, showing computationally very much same analytic energy gradients DFT TDDFT, respectively. Possible future developments highlighted. Its extensions spin-adapted open-shell treatment spin-orbit couplings (which another source force processes) warranted near future.