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Physical review. B, Solid state

Nonexistence of localized Hartree-Fock eigenfunctions in periodic systems

B. H. Brandow

Published Oct 15, 1975• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.12.3464

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Physical Review C•Nov 1, 1986•1 citation
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.
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)
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.
The self-consistent field method utilized for solving the Hartree-Fock (HF) problem and the closely related Kohn-Sham problem is typically thought of as one of the cheapest methods available to quantum chemists. This intuition has been developed from the numerous applications of the self-consistent field method to a large variety of molecular systems. However, as characterized by its worst-case behavior, the HF problem is NP-complete. In this work, we map out boundaries of the NP-completeness by investigating restricted instances of HF. We have constructed two new NP-complete variants of the problem. The first is a set of Hamiltonians whose translationally invariant Hartree-Fock solutions are trivial, but whose broken symmetry solutions are NP-complete. Second, we demonstrate how to embed instances of spin glasses into translationally invariant Hartree-Fock instances and provide a numerical example. These findings are the first steps towards understanding in which cases the self-consistent field method is computationally feasible and when it is not.
Systems of N(>>1) fermions (or bosons) interacting via either repulsive long-ranged forces (e.g., Coulomb) or short-ranged forces (e.g., repulsive barrier plus attractive Yukawa tail) are clearly of interest in problems from nuclear physics to astrophysics. By comparing expectation values for determinants (or permanents) of plane-wave orbitals in the lowest kinetic-energy state, and Wannier-like orbitals localized in the points of a perfect lattice, the conditions on the coupling and particle density for the existence of energetically-preferred, spatially-periodic, nontrivial (i.e.. nonplane-wave) solutions to the Hartree - Fock equations were deduced. That an electron gas in a crystalline confirmation, the Wigner solid,'' has lower energy than the fluid (plane-waves) configuration at low densities (or all densities, if the charge is increased indefinitely) can be shown to imply the existence of lower-energy, spatially-periodic HF solutions. Analogously, for the short-ranged force case cited before one can show very simply that a determinant (or permanent) of nonover-lapping Wannier orbitals gives lower energy for high enough repulsive barrier in an intermediate range of densities (or all densities if the barrier height is increased indefinitely to the hard core limit) in three dimensions. Lower-energy non-trivial solutions therefore emerge under these conditions, giving a finite energy per particle evenmore » in the hard core limit. These results differ qualitatively from the one-dimensional case first studied by Overhauser and agree with those conjectured for 3-dimensions by several subsequent authors. The inclusion of attractive forces can, moreover, provide a condensed (negative energy) spatiallyperiodic state lower than the trivial state. One can furthermore show that there will exist a whole family of aperiodic singledeterminantal states of higher (but finite) energy than the perfect hep lattice state but lower still than the trivial state. (auth)« less