A comparison of Hartree-Fock and exact diagonalization solutions for a model two-electron system

We compare the restricted (RHF) and unrestricted (UHF) Hartree-Fock solutions with the exact configuration interaction (CI) solutions for the model problem of two interacting electrons confined within an infinite spherical potential of radius R. An investigation of the ground state energy, electron density, and a population analysis of the wave functions reveals that the UHF solution compares favorably with the CI one in both the small and large R limits. The Coulson-Fischer transition (the point at which the RHF and UHF solutions are no longer degenerate) occurs at an r(s) identical with R/2(1/3) approximately 6 a.u., close to the crossover r(s) where the exact CI density acquires a minimum at the sphere center; this behavior captures the onset of Wigner molecule formation. In this regime, however, the HF densities and energies are not quantitatively accurate when compared to the CI ones. This failure is further quantified by computing Ec/E to determine the ability of each method at capturing the exact correlation energy; we conclude that the UHF solutions can be reasonably accurate at both high and low densities, but not at intermediate r(s).