  Mixed quantum-classical mechanics descriptions are critical to modeling
coupled electron-nuclear dynamics, i.e. non-adiabatic molecular dynamics,
relevant to photochemical and photophysical processes. We argue that, for
polyatomic molecules, such mixed dynamics can not be efficiently described in
terms of a matrix gauge potential and develop the concept of a `length gauge'
effective Hamiltonian, which helps clarifying certain aspects for the popular
non-adiabatic computational approaches. In particular, within such an effective
Hamiltonain formalism one readily derives the momentum rescaling boundary
condition, used in the surface hopping algorithms. Furthermore, using the new
formalism, we introduce a coupled Gaussian wavepacket parameterization of the
nuclear wavefunction, which generalizes the Ehrenfest approach to account for
electron-nuclei correlations. We test this new approach, Ehrenfest-Plus, on the
standard set of model problems that probe electron-nuclear correlation in
non-adiabatic transitions. The high accuracy of our approach, combined with
mixed-quantum classical efficiency, opens a path for improved simulation of
non-adiabatic molecular dynamics in realistic molecular systems.
