Proposed definitions of the correlation energy density from a Hartree-Fock starting point: The two-electron Moshinsky model atom as an exactly solvable model

TL;DR

This paper explores new definitions of correlation energy density in quantum systems, using the exactly solvable two-electron Moshinsky model to compare analytical and numerical approaches from a Hartree-Fock perspective.

Contribution

It introduces two novel methods to define correlation energy density starting from Hartree-Fock theory, demonstrated through an exactly solvable model atom.

Findings

Analytical expression for correlation energy density in the Moshinsky model

Numerical analysis of correlation energy density relative to ground-state density

Comparison of two different correlation energy definitions

Abstract

In both molecular physics and condensed matter theory, deeper understanding of the correlation energy density epsilon_c (r) remains a high priority. By adopting Loewdin's definition of correlation energy as the difference between the exact and the Hartree-Fock values, here we propose two alternative routes to define this. One of these involves both exact and Hartree-Fock (HF) wavefunctions, while the second requires a coupling constant integration. As an exact analytical example of the first route, we treat the two-electron model atom of Moshinsky, for which both confinement potential and interactions are harmonic. Though the correlation energy density epsilon_c (r) is known analytically, we also investigate numerically its relation to the exact ground-state density in this example.