A Basis-Free Phase Space Electronic Hamiltonian That Recovers Beyond Born-Oppenheimer Electronic Momentum and Current Density

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).