Analytic Nuclear Gradients for Complex Potential Energy Surfaces: A Projected CAP Approach

The complex absorbing potential (CAP) technique is one of the commonly used non-Hermitian quantum mechanics approaches for characterizing electronic resonances. CAP, combined with various electronic structure methods, has shown promising results in quantifying the energies and widths of electronic resonances in molecular systems. While CAP-based methods can be used to map complex potential energy surfaces for resonance states, efficient exploration of these surfaces, e.g., geometry optimization or dynamical simulations, requires information on the nuclear gradient. Currently, the analytic nuclear gradients are only available for CAP-based Hartree-Fock and equation-of-motion coupled-cluster methods with single and double substitutions [Benda, Z.; Jagau, T.-C. J. Chem. Phys. 2017, 146, 031101]. Here, we provide a general approach that relies on the projected CAP technique and extends bound-state gradients and nonadiabatic couplings to resonances. This general approach can be used together with any electronic structure method, provided that analytic gradients and nonadiabatic couplings are available for bound states. We present the results for two quantum chemistry methods: state-averaged complete active space self-consistent field and multireference configuration interaction with single excitations. We test the accuracy of the introduced approximations and report equilibrium geometries for several representative temporary anion species (N₂^-, H₂CO^-, HCOOH^-, and C₂H₄^-).