Conical Intersections and Electronic Momentum as Viewed from Phase Space Electronic Structure Theory

We investigate the structure of a prototypical two-state conical intersection (BeH₂) using a phase space electronic Hamiltonian ${\hat{H}}_{PS}(R,P)$ that goes beyond the Born-Oppenheimer framework. By parametrizing the electronic Schrödinger equation by both nuclear position (R) and momentum (P), 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 ±P_min ≠ 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 BeH₂, this electronic momentum (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 still have a lot to learn about conical intersections when it comes to electronic momentum, and it 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.