Stability of Hartree-Fock States

Abstract

The condition that must be satisfied by a Hartree-Fock wavefunction if it is to give an absolute minimum of the energy, is derived by variation of the one-electron density matrix. If the energy is not an absolute minimum, the state is unstable. Introducing spin explicitly into the equations, we find that there are two classes of variational functions which are particularly suitable in investigations of stability. The one variation is related to the alternate orbital transformation, while the other is connected with Hund's rule and the conditions for ferromagnetism. The first of these variations is used in two numerical examples. In the first example we investigate the stability of a restricted Hartree-Fock wavefunction for LiH relative to an unrestricted one. We find that for the chosen basis set, the restricted Hartree-Fock wavefunction is stable at the equilibrium internuclear distance (3.0 a.u.), but that at 4.0 a.u. it becomes unstable. The second example investigates the relative stability of the restricted and unrestricted Hartree-Fock approximations for the electron gas. It is shown that at a sufficiently low density, the unrestricted Hartree-Fock method gives a lower energy. The resulting state has a nonzero spin density. The importance of the stability condition in atomic, molecular, and solid-state problems is emphasized.