  The decomposition of electronic and nuclear motion presented in~[A. Abedi, N.
T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)] yields a
time-dependent potential that drives the nuclear motion and fully accounts for
the coupling to the electronic subsystem. Here we show that propagation of an
ensemble of independent classical nuclear trajectories on this exact potential
yields dynamics that are essentially indistinguishable from the exact quantum
dynamics for a model non-adiabatic charge transfer problem. We point out the
importance of step and bump features in the exact potential that are critical
in obtaining the correct splitting of the quasiclassical nuclear wave packet in
space after it passes through an avoided crossing between two Born-Oppenheimer
surfaces, and analyze their structure. Lastly, an analysis of the exact
potentials in the context of trajectory surface hopping procedure is presented,
including preliminary investigations of velocity-adjustment, and the
force-induced decoherence effect.
