Hartree-Fock Theory: Slater Determinants of Minimum Energy

Abstract

A theorem is proved (on the basis of physically reasonable assumptions) to the effect that in Hartree-Fock (HF) theory an electron can always be added to an $N$-electron system without raising its energy. The theorem is applied to show that the conventional HF wave function is unstable for certain negative atomic ions (${\mathrm{H}}^{\ensuremath{-}}$, ${\mathrm{Li}}^{\ensuremath{-}}$ ${\mathrm{B}}^{\ensuremath{-}}$, ${\mathrm{N}}^{\ensuremath{-}}$, ${\mathrm{N}}^{\mathrm{---}}$, ${\mathrm{O}}^{\ensuremath{-}}$, ${\mathrm{O}}^{\mathrm{---}}$, ${\mathrm{Na}}^{\ensuremath{-}}$, ${\mathrm{P}}^{\ensuremath{-}}$), and that the typical extent of the instability is chemically significant. A consequence of the instability is that the lowest level in HF theory does not have the symmetry of the exact ground level. This is the case, in particular, for the closed-shell ions ${\mathrm{H}}^{\ensuremath{-}}$, ${\mathrm{Li}}^{\ensuremath{-}}$, ${\mathrm{O}}^{\mathrm{---}}$, and ${\mathrm{Na}}^{\ensuremath{-}}$---a result contrary to some earlier expectations. The two-electron one-center system with fixed nuclear charge $\mathrm{Ze}$ ($Z$ not restricted to integer values) is investigated in detail on the basis of a simplified model defined by specifying restrictions which the allowable determinantal wave functions must satisfy. The orbitals of the determinantal wave functions of the model are of the form ${\ensuremath{\psi}}_{i}={v}_{i}{\ensuremath{\chi}}_{i}$, $i=1,2$, where ${\ensuremath{\chi}}_{i}$ is a spin function and ${v}_{i}$ is essentially of $1s$ hydrogenic form. As $Z$ decreases, the model shows explicitly the onset of the instability described above. It further shows that for $Zg1$, any true minimum-energy determinant ${D}_{g}$ is bound (although the interesting question of whether ${D}_{g}$ is bound for $Zl~1$, apparently raised here for the first time, remains unanswered). A precise analogy between the electron problem and certain spin problems is defined. This analogy sheds light on the HF approximation. It leads to examples which show strikingly that it is important not to always impose symmetry restrictions on an otherwise restricted wave function; the analogy also makes clear, in a simple way, certain points fundamental to an understanding of HF theory. Some possible implications of the results of this paper for solid-state calculations are discussed.