Saddle-point character of the hartree-fock energy

Abstract

A discussion is given of the best way to improve a wave function, for the ground state of a two-electron atom, when starting with either the full Hartree-Fock wave function or else some analytic approximation to it (as in the Hylleraas-Eckart function for helium). In both cases the Hartree-Fock energy is a saddle point, and it is possible to give formulae from which we determine the direction of steepest descent towards the best open-shell function. Numerical calculations are made for the case of helium.