Non-adiabatic perturbation theory of the exact factorization

We present a novel nonadiabatic perturbation theory (NAPT) for correlated systems of electrons and nuclei beyond the Born-Oppenheimer (BO) approximation. The essence of the method is to exploit the smallness of the electronic-to-nuclear mass ratio by treating the electron-nuclear correlation terms in the electronic equation of motion of the exact factorization framework as perturbation. We prove that any finite-order truncation of the NAPT preserves the normalization of the conditional electronic factor as well as the gauge covariance of the resulting perturbative equations of motion. As a particularly sensitive test of the usefulness of NAPT, we obtain nonadiabatic corrections to the BO Berry phase in Jahn-Teller systems with a conical intersection. Lowest-order NAPT well captures the difference between the traditional BO Berry phase, which is a quantized path-independent topological phase and the exact Berry phase, which is a path-dependent geometric phase.