  Trajectory-based mixed quantum-classical approaches to coupled
electron-nuclear dynamics suffer from well-studied problems such as the lack of
(or incorrect account for) decoherence in the trajectory surface hopping method
and the inability of reproducing the spatial splitting of a nuclear wave packet
in Ehrenfest-like dynamics. In the context of electronic non-adiabatic
processes, these problems can result in wrong predictions for quantum
populations and in unphysical outcomes for the nuclear dynamics. In this paper
we propose a solution to these issues by approximating the coupled electronic
and nuclear equations within the framework of the exact factorization of the
electron-nuclear wave function. We present a simple quantum-classical scheme
based on coupled classical trajectories, and test it against the full quantum
mechanical solution from wave packet dynamics for some model situations which
represent particularly challenging problems for the above-mentioned traditional
methods.
