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American Journal of Physics

Comparison of the Hartree-Fock and Maximum Overlap Orbitals for a Simple Model

W. P. Latham, Donald H. Kobe

Published Nov 1, 1973• 6 citations
Publisher

Abstract

A simple model consisting of two particles in a common oscillator potential interacting with each other through a harmonic oscillator coupling is solved exactly for the singlet ground state in one dimension. The Hartree-Fock orbitals and the maximum overlap orbitals are then obtained. The expectation values of several operators are calculated from these independent particle models and compared with each other and with the true values to ascertain the validity of the two approximations.

DOI
10.1119/1.1987540

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A simple model consisting of two particles in a common oscillator potential interacting with each other through a harmonic oscillator coupling is solved exactly for the singlet ground state in one dimension. The Hartree-Fock orbitals and the maximum overlap orbitals are then obtained. The expectation values of several operators are calculated from these independent particle models and compared with each other and with the true values to ascertain the validity of the two approximations.
Using a Hartree-Fock-like partitioning of the two-matrix in terms of the true correlated one-matrix, the corresponding approximate ground-state energy E0{E}_{0}E0​ is calculated for an exactly solvable closed-shell system proposed earlier by Moshinsky [Am. J. Phys. 36, 52 (1968)]. Comparisons of E0{E}_{0}E0​ with the Hartree-Fock ground-state energy EHF{E}_{\mathrm{HF}}EHF​ and the exact ground-state energy Eex{E}_{\mathrm{ex}}Eex​ are made. These comparisons provide valuable insight into the usefulness of a Hartree-Fock-like partitioning of the two-matrix with repulsive interparticle potentials. In particular, an explicit quantification for the applicability limit of Lieb's [Phys. Rev. Lett. 46, 457 (1981)] bound ${E}_{0}\ensuremath{\geqslant}{E}_{\mathrm{HF}}$ with unbounded oscillator potentials is given.
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.
We introduce an energy functional for ground-state electronic structure calculations. Its variables are the natural spin-orbitals of singlet many-body wave functions and their joint occupation probabilities deriving from controlled approximations to the two-particle density matrix that yield algebraic scaling in general, and Hartree-Fock scaling in its seniority-zero version. Results from the latter version for small molecular systems are compared with those of highly accurate quantum-chemical computations. The energies lie above full configuration interaction calculations, close to doubly occupied configuration interaction calculations. Their accuracy is considerably greater than that obtained from current density-functional theory approximations and from current functionals of the one-particle density matrix.
We present an orthonormal set of single-particle orbital functions explicitly satisfying the Hartree-Fock (HF) equations for occupied states, and which give low-density few-particle clustering effects in a many-fermion system. The initial study given here is in one and three dimensions for attractive $\ensuremath{\delta}$ pair interactions. In one dimension the numerical results compare favorably with the exact (lowest) HF state at zero density. The results turn out to be superior to the classic Overhauser HF orbitals, as well as to some more recent ones.