Analytic Gradient for Spin-Flip TDDFT Using Noncollinear Functionals in the Multicollinear Approach.

Spin-flip time-dependent density functional theory (TDDFT) is an efficient tool for describing ground and excited states, especially when they exhibit significant multiconfigurational effects. Currently, most implementations and applications rely on collinear functionals. Using noncollinear functionals in spin-flip TDDFT is a more natural and appropriate choice, which preserves energy degeneracy and spin symmetry better. However, its development has been hindered by numerical instabilities in the second-order functional derivatives. We recently proposed a new approach for constructing noncollinear functionals, the multicollinear approach, which provides numerically stable higher-order functional derivatives and has been applied to spin-flip TDDFT calculations. In this study, we apply the multicollinear approach to the spin-flip TDDFT analytic gradient to address its main challenge: calculating the third-order derivatives of noncollinear functionals. Since these derivatives for generalized gradient approximation (GGA) and meta-GGA functionals may have been implemented for the first time, we validated the implementation through benchmark tests, comparing the results with numerical gradients and the analytic gradients of spin-conserving excited states with the same energy. Finally, we demonstrate the application of spin-flip TDDFT analytic gradients in optimizing the geometries of the lower singlet states, including double-excitation states, and calculating the adiabatic excitation energies and dissociation curves. The analytic gradients of spin-flip TDDFT also serve as a reference for studying analytic derivative couplings and provide a new potential option for on-the-fly molecular dynamics simulations. Furthermore, this work provides useful references for the study of other first-order properties and the development of relativistic TDDFT gradients.