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

In both molecular physics and condensed matter theory, deeper understanding of the correlation energy density epsilon(c)(r) remains a high priority. By adopting Lowdin'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) wave functions, 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.