Analytical Gradient Theory for Quasidegenerate N-Electron Valence State Perturbation Theory (QD-NEVPT2)

Analytical nuclear gradient and derivative coupling theories for the quasidegenerate N-electron valence state perturbation theory using state-averaged orbitals and density matrices are formulated and implemented. In our implementation, the Lagrangian formalism is employed to derive the working expressions. By implementing a direct algorithm for the four-particle reduced density matrix derivatives, large active spaces up to (12e,12o) are routinely tractable. The applicability of the current algorithm is tested for optimizing the minimal energy conical intersections of the representative photochemical systems: ethylene, a retinal protonated Schiff base (penta-2,4-dieniminium cation, PSB3) model, and a green fluorescent protein chromophore (para-hydroxybenzilideneimidazolin-5-one, pHBI) model. For ethylene, we show that the optimized geometries reasonably agree with the previous geometries using the (extended) multistate second-order complete active space perturbation theory and the multireference configuration interaction with the singles and doubles method. For PSB3, we investigate the effect of the basis set selections, ranging from cc-pVDZ to cc-pV5Z, and the effect of noninvariance in describing conical intersections. For pHBI, we test two active spaces, (4e,3o) and (12e,11o), and survey the active-space dependence. We also discuss the computational cost, the parallel efficiency, and the future applicability of the current algorithm.