  It was recently shown [Phys. Rev. Lett. 105, 123002 (2010)] that the complete
wavefunction for a system of electrons and nuclei evolving in a time-dependent
external potential can be exactly factorized into an electronic wavefunction
and a nuclear wavefunction. The concepts of an exact time-dependent potential
energy surface (TDPES) and exact time-dependent vector potential emerge
naturally from the formalism. Here we present a detailed description of the
formalism, including a full derivation of the equations that the electronic and
nuclear wavefunctions satisfy. We demonstrate the relationship of this exact
factorization to the traditional Born-Oppenheimer expansion. A one-dimensional
model of the H$_2^+$ molecule in a laser field shows the usefulness of the
exact TDPES in interpreting coupled electron-nuclear dynamics: we show how
features of its structure indicate the mechanism of dissociation. We compare
the exact TDPES with potential energy surfaces from the time-dependent
Hartree-approach, and also compare traditional Ehrenfest dynamics with
Ehrenfest dynamics on the exact TDPES.
