  We show that a phase space electronic Hamiltonian
$\hat{H}_{PS}(\mathbf{X},\mathbf{P})$, parameterized by both nuclear position
$\mathbf{X}$ and momentum $\mathbf{P}$, can recover not just experimental
vibrational circular dichroism (VCD) signals, but also a meaningful electronic
current density that explains the features of the VCD rotatory strengths.
Combined with earlier demonstrations that such Hamiltonians can also recover
qualitatively correct electronic momenta with electronic densities that
approximately satisfy a continuity equation, the data would suggest that we
have isolated a meaningful alternative approach to electronic structure theory,
one that entirely avoids Born-Oppenheimer theory and frozen nuclei. While the
dynamical implications of such a phase space electronic Hamiltonian are not yet
known, we hypothesize that, by offering classical trajectories the conserve the
total angular momentum (unlike Born-Oppenheimer theory), this new phase space
electronic structure Hamiltonian may well explain some fraction of the
chiral-induced spin selectivity effect.
