  In the paper we derive a semiclassical model for surface hopping allowing
quantum dynamical non-adiabatic transition between different potential energy
surfaces in which cases the classical Born-Oppenheimer approximation breaks
down. The model is derived using the Wigner transform and Weyl quantization,
and the central idea is to evolve the entire Wigner matrix rather than just the
diagonal entries as was done previously in the adiabatic case. The off-diagonal
entries of the Wigner matrix suitably describe the non-adiabatic transition,
such as the Berry connection, for avoided crossings. We study the numerical
approximation issues of the model, and then conduct numerical experiments to
validate the model.
