Phase Space Electronic Structure Theory: From Diatomic Lambda-Doubling to Macroscopic Einstein-de Haas

Λ-doubling of diatomic molecules is a subtle microscopic phenomenon that has long attracted the attention of experimental groups, insofar as rotation of molecular textit{nuclei} induces small energetic changes in the (degenerate) textit{electronic} state. A direct description of such a phenomenon clearly requires going beyond the Born-Oppenheimer approximation. Here we show that a phase space theory previously developed to capture electronic momentum and model vibrational circular dichroism - and which we have postulated should also describe the Einstein-de Haas effect, a macroscopic manifestation of angular momentum conservation - is also able to recover the Λ-doubling energy splitting (or Λ-splitting) of the NO molecule nearly quantitatively. The key observation is that, by parameterizing the electronic Hamiltonian in terms of both nuclear position $mathbf{X}$ and nuclear momentum $mathbf{P}$, a phase space method yields potential energy surfaces that explicitly include the electron-rotation coupling and correctly conserve angular momentum (which we show is essential to capture Λ-doubling). The data presented in this manuscript offers another small glimpse into the rich physics that one can learn from investigating phase space potential energy surfaces E_PS\(mathbf{X},mathbf{P}\) as a function of both nuclear position and momentum, all at a computational cost comparable to standard Born-Oppenheimer electronic structure calculations.