  This work identifies geometric effects on dynamics due to nonadiabatic
couplings in Born Oppenheimer systems and provides a systematic method for
deriving corrections to mixed quantum-classical methods. Specifically, an exact
path integral formulation of the quantum nonadiabatic dynamics of Born
Oppenheimer systems is described. Stationary phase approximations to the
propagator for full quantum dynamics are derived. It is shown that quantum
corrections to mixed quantum classical methods can be obtained through
stationary phase approximations to the full quantum dynamics. A rigorous
description of the quantum corrections due to electronic nonadiabatic coupling
on the nuclear dynamics within the Ehrenfest framework is obtained. The fewest
switches surface hopping method is shown to be obtained as a quasiclassical
approximation to the dynamics and natural semiclassical extensions to include
classically forbidden nonadiabatic transitions are suggested.
