Spin-adapted spin-flip-down time-dependent density functional theory

Molecular systems with orbital (near-)degeneracy at the Fermi level tend to adopt a high-spin ground state. In these systems, one often finds low-lying electronic excitations with a lower total spin that can be reached from the ground state by a spin-flip-down excitation. In this work, we present three spin-adapted spin-flip-down time-dependent density functional theory (SFD-TD-DFT) approaches to calculate the excitation energies for these types of electronic transitions. These SFD-TD-DFT methods are based on a restricted open-shell Kohn-Sham (ROKS) formulation within the Tamm-Dancoff approximation (TDA), giving rise to the ROKS-SFD-TDA family of methods. The three methods differ in the kernel, having different two-electron coupling elements in the resulting working equations. In agreement with earlier work, we find that a noncollinear description of the kernel is vital for producing a decent description of these excitations. In terms of obtaining excitations with a definite spin, we present two fully spin-adapted ROKS-SFD-TDA methods that either stem from configuration interaction with single excitations (SF-CIS) or from the already existing equation-of-motion ansatz (SF-TDA). It is shown that the spin-adaptation in SF-CIS and SF-TDA gives rise to artificial double counting of correlation effects by incorporating double excitations. When discarding this double counting, one ends up with an excited state that is partly spin-adapted (only in the open-to-open configurations). This method is called quasi-spin-adapted SF-TDA (Q-SF-TDA) and is shown to be a stable and efficient method that performs similarly to spin-unrestricted SFD-TD-DFT.