The preceding discussion describes the conventional LR-TDDFT formalism, in which x iaσ has a single spin index because the ψ iσ → ψ aσ excitation is spin-conserving. For a spin-flipping transition, and under the usual assumption that f xc comes from a semilocal exchange-correlation functional, the coupling matrix ∂F iaσ /∂P jbσ in Eq. (11a) vanishes except for the HFX contribution, because only HFX can couple σ = α to σ = β. As noted later, the fact that the SF-TDDFT coupling matrix involves only the HFX term (and not the Coulomb term or the semilocal kernel f xc ) explains the observation, made already in the very first SF-TDDFT study of singlet-triplet gaps, that a hybrid functional with 50% HFX significantly outperforms B3LYP with its 20% HFX. This observation was later confirmed in application of SF-TDDFT to excitation energies. As a result, the "Becke half-and-half" functional BH&HLYP, consisting of 50% HFX and 50% semilocal B88 45 exchange ("BH&H"), in conjunction with LYP correlation, has become the standard choice for SF-TDDFT calculations.