  We present an exact decomposition of the complete wavefunction for a system
of nuclei and electrons evolving in a time-dependent external potential. We
derive formally exact equations for the nuclear and electronic wavefunctions
that lead to rigorous definitions of a time-dependent potential energy surface
(TDPES) and a time-dependent geometric phase. For the $H_2^+$ molecular ion
exposed to a laser field, the TDPES proves to be a useful interpretive tool to
identify different mechanisms of dissociation.
