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Physical review. B, Solid state

Nonexistence of localized Hartree-Fock eigenfunctions in crystals

Curt A. Moyer

Published Apr 15, 1974• 4 citations
Publisher

Abstract

The possibility of localized canonical Hartree-Fock wave functions in periodic systems (crystals) is examined anew. It is argued that such solutions generally do not exist. The impact of nonexistence on concepts of dissociation and the metal-insulator (Mott) transition is assessed.
DOI
10.1103/physrevb.9.3555

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The question of the possible existence of localized canonical Hartree-Fock (HF) eigenfunctions in periodic systems is more subtle than has generally been recognized. The first analysis of this issue, by Kaplan and Argyres, is shown to be incomplete and thus inconclusive. We present a new argument which shows conclusively that localized canonical HF solutions cannot exist in periodic systems of physical interest. Other arguments, with varying emphasis on symmetry, intuition, and formal rigor, are also discussed. Consequences of this result for the understanding of Mott insulators will be discussed elsewhere.
The existence of localized canonical solutions of the Hartree--Fock equations is investigated. Application of a prescription proposed by Davidson provides the modification needed to settle the difficulties investigated by Brandow in the context of the theory of solid sup3sup 3sup3He.
Work related to proofs for the existence of energetically preferred, non- trivial (mostly periodic-density) solutions to the Hartree -- Fock equations is reviewed. Both long- and short-ranged interacting systems of N-bosons or - fermions (N >> 1) are considered, and stability criteria discussed. Also, some exact results associated with trivial (plane-wave) solutions are surveyed. The main conclusion is that, for any fixed physical density, a sufficiently strong interparticle coupling can induce the appearance of non-trivial HF states. These solutions provide a first step in studying the properties of many-particle systems, which range from astrophysical plasmas and quantum crystals to nuclear matter and helium at very low temperature. (auth)
Abstract. A simple and commonly employed approximate technique with which one can examine spatially disordered systems when strong electronic correlations are present is based on the use of real-space unrestricted self-consistent Hartree-Fock wave functions. In such an approach the disorder is treated exactly while the correlations are treated approximately. In this report we critique the success of this approximation by making comparisons between such solutions and the exact wave functions for the Anderson-Hubbard model. Due to the sizes of the complete Hilbert spaces for these problems, the comparisons are restricted to small one-dimensional chains, up to ten sites, and a 4x4 two-dimensional cluster, and at 1/2 filling these Hilbert spaces contain about 63,500 and 166 million states, respectively. We have completed these calculations both at and away from 1/2 filling. This approximation is based on a variational approach which minimizes the Hartree-Fock energy, and we have completed comparisons of the exact and Hartree-Fock energies. However, in order to assess the success of this approximation in reproducing ground-state correlations we have completed comparisons of the local charge and spin correlations, including the calculation of the overlap of the Hartree-Fock wave functions with those of the exact solutions. We find that this approximation reproduces the local charge densities to quite a high accuracy, but that the local spin correlations, as represented by 〈Si · Sj〉, are not as well represented. In addition to these comparisons, we discuss the properties of the spin degrees of freedom in the HF approximation, and where in the disorderinteraction phase diagram such physics may be important. PACS numbers: 71.27.+a,71.23.-k,71.10.-wCritiquing HF for the Anderson-Hubbard Model 2 1.