The diatomic Hartree model at dissociation

TL;DR

This paper analyzes the Hartree model for two electrons at large nuclear separation, revealing asymptotic eigenfunction behavior and quantifying the exponentially small energy gap caused by quantum tunneling.

Contribution

It provides a detailed asymptotic analysis of the two-electron Hartree model at dissociation, including bounds on the energy gap due to tunneling effects.

Findings

Eigenfunctions asymptotically superimpose one-nucleus minimizers

Exponential bounds on the energy gap between ground and first excited states

Quantitative description of quantum tunneling effects at large nuclear separation

Abstract

We study the Hartree model for two electrons with spin, living in the two-dimensional or three-dimensional space with Coulomb interactions and submitted to the potential induced by two nuclei of charge +1. In the limit where the nuclei move away from each other, we show that the two lowest eigenfunctions of the mean-field hamiltonian are asymptotically given by an even, respectively odd, superposition of the minimizer associated with the corresponding Hartree one nucleus model. We then give upper and lower bounds on the exponentially small gap between the first two eigenvalues, due to (nonlinear) quantum tunneling.