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This study describes variational energy expressions and analytical excited state energy gradients for time-dependent self-consistent field methods with polarizable solvent effects. Linear response, vertical excitation, and state-specific solvent models are examined. Enforcing a variational ground state energy expression in the state-specific model is found to reduce it to the vertical excitation model. Variational excited state energy expressions are then provided for the linear response and vertical excitation models and analytical gradients are formulated. Using semiempirical model chemistry, the variational expressions are verified by numerical and analytical differentiation with respect to a static external electric field. Analytical gradients are further tested by performing microcanonical excited state molecular dynamics with p-nitroaniline.

Solvent effects in time-dependent self-consistent field methods. II. Variational formulations and analytical gradients