  We present a novel quantum-classical approach to non-adiabatic dynamics,
deduced from the coupled electronic and nuclear equations in the framework of
the exact factorization of the electron-nuclear wave function. The method is
based on the quasi-classical interpretation of the nuclear wave function, whose
phase is related to the classical momentum and whose density is represented in
terms of classical trajectories. In this approximation, electronic decoherence
is naturally induced as effect of the coupling to the nuclei and correctly
reproduces the expected quantum behaviour. Moreover, the splitting of the
nuclear wave packet is captured as consequence of the correct approximation of
the time-dependent potential of the theory. This new approach offers a clear
improvement over Ehrenfest-like dynamics. The theoretical derivation presented
in the Letter is supported by numerical results that are compared to quantum
mechanical calculations.
