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 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. 
N. H. March, A. Cabo, F. Claro, G. G. N. Angilella