Testing Noncollinear Spin-Flip, Collinear Spin-Flip, and Conventional Time-Dependent Density Functional Theory for Predicting Electronic Excitation Energies of Closed-Shell Atoms

Abstract

Conventional time-dependent density functional theory (TDDFT) is based on a closed-shell Kohn–Sham (KS) singlet ground state with the adiabatic approximation, using either linear response (KS-LR) or the Tamm–Dancoff approximation (KS-TDA); these methods can only directly predict singly excited states. This deficiency can be overcome by using a triplet state as the reference in the KS-TDA approximation and “exciting” the singlet by a spin flip (SF) from the triplet; this is the method suggested by Krylov and co-workers, and we abbreviate this procedure as SF-KS-TDA. SF-KS-TDA can be applied either with the original collinear kernel of Krylov and co-workers or with a noncollinear kernel, as suggested by Wang and Ziegler. The SF-KS-TDA method does bring some new practical difficulties into play, but it can at least formally model doubly excited states and states with double-excitation character, so it might be more useful than conventional TDDFT (both KS-LR and KS-TDA) for photochemistry if these additional difficulties can be surmounted and if it is accurate with existing approximate exchange–correlation functionals. In the present work, we carried out calculations specifically designed to understand better the accuracy and limitations of the conventional TDDFT and SF-KS-TDA methods; we did this by studying closed-shell atoms and closed-shell monatomic cations because they provide a simple but challenging testing ground for what we might expect in studying the photochemistry of molecules with closed-shell ground states. To test their accuracy, we applied conventional KS-LR and KS-TDA and 18 versions of SF-KS-TDA (nine collinear and nine noncollinear) to the same set of vertical excitation energies (including both Rydberg and valence excitations) of Be, B⁺, Ne, Na⁺, Mg, and Al⁺. We did this for 10 exchange–correlation functionals of various types, both local and nonlocal. We found that the GVWN5 and M06 functionals with nonlocal kernels in spin-flip calculations can both have accuracy competitive to CASPT2 calculations. When the results were averaged over all 36 test energy differences, seven (GVWN5, M06, B3PW91, LRC-ωPBE, LRC-ωPBEh, PBE, and M06-2X) of the 10 studied density functionals had smaller mean unsigned errors for noncollinear calculations than the mean unsigned error of the best functional (M06-2X) for either conventional KS-TDA or KS-LR