How Good Is the Hartree-Fock Approximation? II. The Case of Closed Shells

Abstract

In the present paper, we analyze the problem of an arbitrary number of particles A in a harmonic-oscillator common potential interacting through a harmonic-oscillator force. The particles have spin 12 and obey Fermi statistics. The exact lowest eigenstate and energy are derived for values of A corresponding to closed subshells. An approximate analysis is also made using determinantal wave functions formed from single-particle harmonic-oscillator states, giving the eigenstates and energies of a simple form of the Hartree-Fock approximation. The exact and Hartree-Fock energies and eigenstates are compared both when potential and force have the same sign, and when they have opposite sign. We discuss the results for models which we call pseudonuclei and pseudoatoms, and we indicate some of the inferences for the real thing.