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

## TL;DR

This paper shows how a phase space electronic structure theory can explain both microscopic Λ-doubling and macroscopic angular momentum effects like the Einstein–de Haas phenomenon.

## Contribution

A phase space method is shown to nonperturbatively capture Λ-splitting in NO without summing over states.

## Key findings

- The phase space method recovers the Λ-doubling energy splitting of the NO molecule nearly quantitatively.
- The method correctly conserves angular momentum and includes electron-rotation coupling in potential energy surfaces.
- The computational cost is comparable to standard Born–Oppenheimer calculations.

## Abstract

Λ-doubling of diatomic molecules is a subtle microscopic
phenomenon that has long attracted the attention of experimental groups,
insofar as rotation of molecular nuclei induces small
energetic changes in the (degenerate) 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 and nonperturbatively (without
a sum over states). The key observation is that, by parametrizing
the electronic Hamiltonian in terms of both nuclear position (X) and nuclear momentum (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 offer another small glimpse into the rich physics that
one can learn from investigating phase space potential energy surfaces E

PS
(X,P) as a function
of both nuclear position and momentum, all at a computational cost
comparable to standard Born–Oppenheimer electronic structure
calculations.