Benchmark assessment of collinear, mixed-reference, and spin-adapted variants of spin-flip time-dependent density functional theory, for closed- and open-shell molecules

Spin-flip methods provide access to certain electronic states having multireference character while retaining single-reference cost. However, conventional spin-flip time-dependent density functional theory (SF-TDDFT) often suffers from severe spin contamination that may cause inaccurate state ordering or engender ambiguous state character. For singlet excited states, this is largely rectified by a "mixed-reference" formulation (MRSF-TDDFT), while a spin-adapted formalism (SA-SF-TDDFT) addresses spin contamination in a general way for arbitrary multiplicities. Here, we revisit SA-SF-TDDFT and demonstrate that it significantly improves the agreement with reference data compared to other variants and also relative to conventional (spin-conserving) linear response TDDFT. Overall, SA-SF-TDDFT proves to be the most accurate among these methods, for excitation energies of both closed-shell molecules and doublet radicals as well as for singlet-triplet gaps. However, SF methods exhibit a notable limitation in the case of linear and quasi-linear doublet radicals, due to degeneracies in the high-spin quartet reference state.