Comparison of the Hartree-Fock and Maximum Overlap Orbitals for a Simple Model

A simple model consisting of two particles in a common oscillator potential interacting with each other through a harmonic oscillator coupling is solved exactly for the singlet ground state in one dimension. The Hartree-Fock orbitals and the maximum overlap orbitals are then obtained. The expectation values of several operators are calculated from these independent particle models and compared with each other and with the true values to ascertain the validity of the two approximations.