Conical Intersections and Electronic Momentum As Viewed From Phase Space Electronic Structure Theory

We investigate the structure of a prototypical two-state conical intersection (BeH$_2$) using a phase space electronic Hamiltonian $\hat{H}_{PS}(\bR,\bP)$ that goes beyond the Born-Oppenheimer framework. By parameterizing the electronic Schr{\"o}dinger equation by both nuclear position ($\bR$) and momentum ($\bP$), we solve for quantum electronic states in a moving frame that can break time reversal symmetry and, as a result, the branching plane of the conical intersection within a phase space framework now has dimension three (rather than dimension two as found within the standard Born-Oppenheimer framework). Moreover, we note that, if one fixes a geometry in real space that lies in the conical intersection seam and scans over the corresponding momentum space, one finds a double well (with minima at $\pm \bP_{min} \ne 0$), indicating that the stationary electronic states of the phase space electronic Hamiltonian carry electronic momentum -- a feature that cannot be captured by a Born-Oppenheimer electronic state. Interestingly, for $BeH_2$, this electronic momenta (as calculated with full configuration interaction) agrees with what is predicted by approximate complex restricted Hartree-Fock calculations, indicating a physical interpretation of complex instabilities in modern electronic structure calculations. Altogether, this study suggests that we have still have a lot to learn about conical intersections when it comes to electronic momentum, and highlights the urgent need for more experiments to probe what photochemical observables can and/or cannot be captured by standard electronic structure that isolates conical intersections within the Born-Oppenheimer framework.