Molecular geometric phase from the exact electron-nuclear factorization

TL;DR

This paper demonstrates that the geometric phase in molecular systems can be accurately described using the exact electron-nuclear factorization, revealing new vector potentials and geometric effects beyond the Born-Oppenheimer approximation.

Contribution

It introduces a method to evaluate the molecular geometric phase using the exact electron-nuclear wavefunction, extending the concept beyond the traditional Born-Oppenheimer framework.

Findings

Exact electronic wavefunction yields a geometric phase as a complex exponential

Induced vector potentials contribute to nuclear currents and are gauge-invariant

Exact potential energy surface includes a term related to the Fubini-Study metric

Abstract

The Born-Oppenheimer electronic wavefunction $\Phi_R^{BO}(r)$ picks up a topological phase factor $\pm 1$, a special case of Berry phase, when it is transported around a conical intersection of two adiabatic potential energy surfaces in $R$-space. We show that this topological quantity reverts to a geometric quantity $e^{i\gamma}$ if the geometric phase $\gamma = \oint \mathrm{Im} \langle \Phi_R |\nabla_{\mu} \Phi_R\rangle \cdot d\mathbf{R}_{\mu}$ is evaluated with the conditional electronic wavefunction $\Phi_R(r)$ from the exact electron-nuclear factorization $\Phi_R(r)\chi(R)$ instead of the adiabatic function $\Phi_R^{BO}(r)$. A model of a pseudorotating molecule, also applicable to dynamical Jahn-Teller ions in bulk crystals, provides the first examples of induced vector potentials and molecular geometric phase from the exact factorization. The induced vector potential gives a contribution to the circulating nuclear current which cannot be removed by a gauge transformation. The exact potential energy surface is calculated and found to contain a term depending on the Fubini-Study metric for the conditional electronic wavefunction.