  We present a phase-space electronic Hamiltonian $\hat{H}_{PS}$ (parameterized
by both nuclear position $\mathbf{X}$ and momentum $\mathbf{P}$) that boosts
each electron into the moving frame of the nuclei that are closest in real
space -- without presuming the existence of an atomic orbital basis. We show
that $(i)$ quantum-classical dynamics along such a Hamiltonian maintains
momentum conservation and $(ii)$ diagonalizing such a Hamiltonian can recover
the electronic momentum and electronic current density reasonably well. In
conjunction with other reports in the literature that such a phase-space
approach can also recover vibrational circular dichroism (VCD) spectra, we
submit that the present phase-space approach offers a testable and powerful
approach to post-Born-Oppenheimer electronic structure theory. Moreover, the
approach is inexpensive and can be immediately applied to simulations of chiral
induced spin selectivity experiments (where the transfer of angular momentum
between nuclei and electrons is considered critical).
