Efficient Algorithms for Analytical Gradients and Nonadiabatic Couplings of Second-Order Extended Multiconfiguration Quasi-Degenerate Perturbation Theory.

This work introduces efficient MPI-parallelized algorithms for the analytical calculation of energy gradients and nonadiabatic coupling matrix elements within extended multiconfiguration quasi-degenerate perturbation theory (XMCQDPT2). The algorithms are derived using a unified Lagrangian multiplier formalism combined with the resolvent-fitting approximation and are implemented in the Firefly quantum chemistry package. Agreement between analytical and numerical gradients validates the implementation, while benchmark calculations demonstrate its high computational efficiency for systems with large active spaces. Practical utility is illustrated through the optimization of conical intersection geometries and the mapping of potential energy surfaces in the branching spaces of the retinal protonated Schiff base, which is the biological chromophore of vision. This development provides a robust tool for exploring photochemical reaction pathways and nonadiabatic dynamics in complex molecular systems.