Analytic energy gradients for unrestricted mixed-reference spin-flip time-dependent density functional theory

Analytic nuclear gradients for individual low-lying singlet and triplet states within mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) based on an unrestricted Kohn-Sham (UKS) reference (UMRSF-TDDFT) were derived and implemented. The mixed-reference response restores the missing spin-complementary configurations and substantially attenuates the effects of the unequal UKS α/β orbitals. Relative to the restricted-open-shell (RO)-MRSF gradient, the major change is that the UKS reference supplies independently optimized α and β spatial-orbital sets. Corresponding orbitals in these generally unequal sets must first be paired by an orbital alignment. Differentiation of this additional alignment generates two contributions absent from RO-MRSF: spin-resolved alignment-induced orbital-gradient terms ΔGf,σalign in the orbital gradient and an explicit overlap contribution -Tr(W^alignS^ξ). The aligned response is also nonstationary to relevant occupied-occupied (α) and virtual-virtual (β) rotations, requiring an independent-spin, full-block Z-vector. When the SOMOs are exactly degenerate, the minimum-norm solution sets the undetermined null-mode response component to zero without affecting the gradient. Compared with the RO-MRSF gradient, the formal scaling is unchanged, but the independent α/β response approximately doubles the principal coupled Z-vector dimension. Despite substantial differences between UKS and restricted-open-shell Kohn-Sham (ROKS) frontier orbitals, the UMRSF and RO-MRSF structures and potential-energy surfaces show close agreement.