  In the context of the exact factorization of the electron-nuclear wave
function, the coupling between electrons and nuclei beyond the adiabatic regime
is encoded (i) in the time-dependent vector and scalar potentials and (ii) in
the electron-nuclear coupling operator. The former appear in the
Schroedinger-like equation that drives the evolution of the nuclear degrees of
freedom, whereas the latter is responsible for inducing non-adiabatic effects
in the electronic evolution equation. As we have devoted previous studies to
the analysis of the vector and scalar potentials, in this paper we focus on the
properties of the electron-nuclear coupling operator, with the aim of
describing a numerical procedure to approximate it within a semiclassical
treatment of the nuclear dynamics.
