An Exact Formulation of Phase-Space Electronic Structure Theory in One Dimension by Exploiting the Zero Curvature Condition

We show that, for the unique case of a system of electrons and nuclei in one dimension, the $Γ$ operator in phase space electronic structure theory has a zero non-Abelian curl. Using this fact, we are able to show that a phase space electronic structure framework in Wigner space has a one-to-one mapping to the exact fully quantum vibronic Hamiltonian. In other words, one loses no information by working within the framework of phase space electronic structure theory rather than Born-Huang theory; the only difference is that, whereas Born-Huang theory represents the total vibronic Hamiltonian as a quadratic order expansion in $\hbar$, the Hamiltonian becomes an infinite order expansion in $\hbar$ within a phase space electronic structure framework (a so-called "Generalized Born-Huang" approach). That being said, numerical results at first or second order demonstrate that, for a limited number of electronic states, phase space electronic structure theory strongly outperforms Born-Huang theory - a result that justifies a great deal more research into this novel electronic structure approach.