A Phase Space Electronic Structure View of The Solid State

We develop a phase space electronic structure view of the solid state, where electronic bands are parameterized by both nuclear position $\mathbf{Q}$ and nuclear momentum $\mathbf{\Pi}$ (rather than the standard practice of only $\mathbf{Q}$). Within this phase space, non-Born-Oppenheimer electronic structure context, we prove a nuclear wavevector ($\mathbf{q}$)-dependent version of Nafie's equality relating the change in electronic momentum as a function of nuclear momentum, $\partial \langle\mathbf{p}\rangle/\partial \mathbf{\Pi}_q$, to the change in electronic position as a function of nuclear position, $\partial \langle\mathbf{r}\rangle/\partial \mathbf{Q}q$. Furthermore, we show how information about electron-phonon interactions, specifically nuclear-induced electronic momentum, can be extracted from simple band structure calculations -- without Berry curvature calculations.