Multiple Solutions to the Hartree-Fock Problem. I. General Treatment of Two-Electron Closed-Shell Systems

Abstract

The number of fully self‐consistent, closed‐shell solutions to the two‐electron Hartree‐Fock problem is investigated. With N orbital basis functions the Hartree‐Fock wavefunction can be expanded as a superposition of $\text{N(N\hspace{0.17em}+\hspace{0.17em}1)\hspace{0.17em}/\hspace{0.17em}2}$ configurations. The constraints in the coefficients are analyzed exactly for the case $\text{N\hspace{0.17em}=\hspace{0.17em}2}$ . The free CI problem corresopnds to finding energy extrema on the surface of a sphere. The Hartree‐Fock state point is constrained to lie on a circle on this sphere. There will be two, three or four SCF solutions depending on the manner in which the circle of constraint crosses valleys and ridges on the sphere. This is shown explicitly by detailed calculations for the He atom. These calculations also reveal instances in which aufbau method of solution necessarily diverges. The general problem can be analyzed under the assumption of zero differential overlap. The Hartree‐Fock matrix is diagonal in this approximation, but the diagonal elements still involve the orbital coefficients. One can set an upper limit of ${\text{(3}}^N\text{\hspace{0.17em}−\hspace{0.17em}1)\hspace{0.17em}/\hspace{0.17em}2}$ to the number of acceptable solutions for this problem.