  The quantum dynamics of electron-nuclear systems is analyzed from the
perspective of the exact factorization of the wavefunction, with the aim of
defining gauge invariant equations of motion for both the nuclei and the
electrons. For pure states this is accomplished with a quantum hydrodynamical
description of the nuclear dynamics and electronic density operators tied to
the fluid elements. For statistical mixtures of states the exact factorization
approach is extended to two limiting situations that we call "type-n" and
"type-e" mixtures, depending on whether the nuclei or the electrons are,
respectively, in an intrinsically mixed state. In both cases a fully gauge
invariant formulation of the dynamics is obtained again in hydrodynamic form
with the help of mechanical momentum moments (MMMs). Nuclear MMMs extend in a
gauge invariant way the ordinary momentum moments of the Wigner distribution
associated with a density matrix of positional variables, electron MMMs are
operator-valued and represent a generalization of the (conditional) density
operators used for pure states. The theory presented here bridges exact quantum
dynamics with several mixed quantum-classical approaches currently in use to
tackle non-adiabatic molecular problems, offering a foundation for systematic
improvements. It further connects to non-adiabatic theories in condensed-phase
systems. As an example, we re-derive the finite-temperature theory of
electronic friction of Dou, Miao \& Subotnik (Phys. Rev. Lett. 119, 046001
(2017)) from the dynamics of "type-e" mixtures and discuss possible
improvements.
