Electronic structure in phase space in a non-Born-Oppenheimer framework: Geometric forces and Moody-Shapere-Wilzcek theory revisited

We revisit the three-body problem in quantum mechanics in two and three dimensions, generating both exact eigenvalues and eigenvectors of the Hamiltonian and a series of approximate solutions as calculated with a variety of different schemes to separate heavy ("nuclear") and light ("electronic") particles. We show that, with minimal extra cost, one can go beyond the Born-Oppenheimer approximation by performing electronic structure calculations parameterized by both the nuclear position (${\mathbf X})$ and the nuclear momentum ($\mathbf{P}$), a so-called phase space theory of electronic structure. In particular, we demonstrate that such phase space electronic structure calculations correctly incorporate the non-inertial Coriolis and centrifugal forces felt by electrons in a moving nuclear frame, thus leading to far more accurate eigenenergies and electronic angular momenta than has been possible before. We also demonstrate that our approach naturally incorporates and generalizes the Moody-Shapere-Wilczek magnetic monopole for the non-abelian Berry curvature (now allowing for vibrational motion rather than a diatomic of fixed length). We argue that the resulting approach should be extremely useful for propagating dynamics where angular momentum flows between nuclei and electrons; in particular, if extended to include spin degrees of freedom, the present approach will offer a practical means to study chiral induced spin selectivity through the lens of chiral phonons and coupled nuclear-electronic motion.