Exact-Factorization Framework for Electron–Nuclear Dynamics in Electromagnetic Fields

The Exact Factorization (EF) theory aims at the separation of the nuclear and electronic degrees of freedom in the many-body (MB) quantum mechanical problem. Being formally equivalent to the solution of the MB Schrödinger equation, EF sets up a strategy for the construction of efficient approximations in the theory of the correlated electronic-nuclear motion. Here we extend the EF formalism to incorporate the case of a system under the action of electromagnetic fields. An important interplay between the physical magnetic and the Berry curvature fields is revealed and discussed within the fully nonadiabatic theory. In particular, it is a previously known property of the Born–Oppenheimer (BO) approximation that, for a neutral atom in a uniform magnetic field, the latter is compensated by the Berry curvature field in the nuclear equation of motion (Yin and Mead, Theoret. Chim. Acta1992,82, 397). From an intuitive argument that the atom must not be deflected by the Lorentz force from a straight line trajectory, it has been conjectured that the same compensation should occur within the EF theory as well. We prove that this property persists within the exact nonadiabatic theory, provided the atom is in its eigenstate. The latter finds itself in the conspicuous variance with the corresponding BO result, where the compensation holds for an arbitrary nuclear wave packet on a single BO potential surface.