Analytical derivatives of the linear-response CASSCF excited-state energy

In this contribution, we present the development and implementation of analytical nuclear gradients for linear-response complete active space-self-consistent field (LR-CASSCF) excitation energies. This development provides an alternative framework for optimizing excited-state geometries within a multiconfigurational context and is shown to be both effective and competitive with existing methodologies. We derive the complete set of working equations and demonstrate that the computational cost of evaluating excited-state nuclear gradients scales no worse than that of ground-state CASSCF, though with a larger prefactor. The implementation is validated by comparison with numerical derivatives of the excitation energies. Furthermore, we compute excited-state dipole moments for a set of small molecules and perform geometry optimizations of tetrazine and short-chain polyenes, demonstrating the feasibility of characterizing and optimizing the lowest few singlet excited states of different electronic character.