  We discuss nuclear dynamics at molecule-metal interfaces including
non-equilibrium molecular junctions. Starting from the many-body states
(pseudoparticle) formulation of the molecule-metal system in the molecular
vibronic basis, we introduce gradient expansion in order to reduce the
adiabatic nuclear dynamics (that is, nuclear dynamics on a single molecular
potential surface) into its semi-classical form while maintaining the effect of
the non-adiabatic electronic transitions between different molecular charge
states. This yields a set of equations for the nuclear dynamics in the presence
of these non-adiabatic transitions, which reproduce surface hopping formulation
in the limit of small metal-molecule coupling (where broadening of the
molecular energy levels can be disregarded) and Ehrenfest dynamics (motion on
the potential of mean force) when information on the different charging states
is traced out, which is relevant when this coupling is strong.
