  The exact nuclear time-dependent potential energy surface arises from the
exact decomposition of electronic and nuclear motion, recently presented in [A.
Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)].
Such time-dependent potential drives nuclear motion and fully accounts for the
coupling to the electronic subsystem. We investigate the features of the
potential in the context of electronic non-adiabatic processes and employ it to
study the performance of the classical approximation on nuclear dynamics. We
observe that the potential, after the nuclear wave-packet splits at an avoided
crossing, develops dynamical steps connecting different regions, along the
nuclear coordinate, in which it has the same slope as one or the other
adiabatic surface. A detailed analysis of these steps is presented for systems
with different non-adiabatic coupling strength. The exact factorization of the
electron-nuclear wave-function is at the basis of the decomposition. In
particular, the nuclear part is the true nuclear wave-function, solution of a
time-dependent Schroedinger euqation and leading to the exact many-body density
and current density. As a consequence, the Ehrenfest theorem can be extended to
the nuclear subsystem and Hamiltonian, as discussed here with an analytical
derivation and numerical results.
