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Journal of Mathematical Physics

Ground state energy for the Hartree–Fock equations with Dirichlet boundary conditions

Joel F. Campbell

Published Apr 1, 1994• 3 citations
Publisher

Abstract

The Hartree–Fock equations for 2 N fermions in a one-dimensional box are solved in terms of a power series in the coupling constant for various values of N and the quantum numbers. A general formula is found for the total energy and wave function for N=1, which is accurate to sixth order. Another formula for the total energy is conjectured for any N, which is accurate to second order. Using these results, the ground state energy, energy of interaction, and boundary energy are found to second-order accuracy for arbitrary N.

DOI
10.1063/1.530602

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The Hartree–Fock equations for 2 N fermions in a one-dimensional box are solved in terms of a power series in the coupling constant for various values of N and the quantum numbers. A general formula is found for the total energy and wave function for N=1, which is accurate to sixth order. Another formula for the total energy is conjectured for any N, which is accurate to second order. Using these results, the ground state energy, energy of interaction, and boundary energy are found to second-order accuracy for arbitrary N.
Two infinite families of two-parameter generalized Overhauser orbitals are introduced and shown to satisfy explicitly, for occupied states, the self-consistent Hartree-Fock equations in the thermodynamic limit. For an attractive $\ensuremath{\delta}$ interaction, they give lower Hartree-Fock energy than the usual plane-wave solutions, even for relatively weak coupling and/or low density. The limiting members (possessing an infinite number of harmonics) of both families appear to tend to a "classical static lattice" state, via a second-order transition for one family and via first order for the other. The related density profiles and energy expressions are calculated as functions of the two new parameters. A direct variation with respect to these parameters was done numerically and results are presented graphically.
The quantum corrections to the two-dimensional Wigner crystal, for filling \ensuremath{\nu}\ensuremath{\le}1/3, are discussed by using a Hartree-Fock expansion based on wave functions which are (i) related to one another by magnetic translations, (ii) orthonormal, and (iii) strongly localized. Such wave functions are constructed in terms of Gaussians that are localized at the sites of a triangular (Wigner) lattice and have a small overlap c. The ground-state energy per particle is calculated by an expansion in \ensuremath{\surd}\ensuremath{\nu} and in \ensuremath{\delta}\ensuremath{\equiv}c1/4{\mathit{c}}^{1/4}c1/4, which is rapidly convergent and stable under the thermodynamical limit. In particular, in this limit the cancellation of the infrared divergences occur order by order in the above expansion. The accurate control on the approximations allows a clear-cut comparison with the energy obtained by the Laughlin ansatz on the ground state and the numerical results confirm that the Wigner-crystal picture is energetically favored with respect to the Laughlin state for \ensuremath{\nu}1/9.
A system of NNN antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\ensuremath{\rho}({x}_{1}, {x}_{2})$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R(\ensuremath{\nu}\ensuremath{\mu})$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\ensuremath{\omega}}_{j}}^{\ensuremath{'}}\ensuremath{-}{\ensuremath{\omega}}_{i}$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.
The Hartree-Fock theory for many fermions with plane-wave orbitals but with abnormal occupation is studied. A one-dimensional problem with an interparticle interaction of finite range which saturates the NNN body system at a fixed density and finite binding energy is solved explicitly. A direct variation of the resulting Hartree-Fock energy with respect to the additional variational parameters introduced is carried out numerically and results reminiscent of a gas-to-liquid phase transition are found.NUCLEAR STRUCTURE plane-wave Hartree-Fock; model calculation; nuclear matter.
The ground-state wave function and energy of a finite system of interacting fermions are expanded in terms of multiple-particle excitations on an uncorrelated zero-order state. The resulting set of coupled equations constitutes a systematic variational generalization of Hartree-Fock theory. Comparison is made with many-body perturbation theory and it is shown that to any order the theory incorporates an infinite number of perturbation theory terms. Solutions of the equations for ground-state atomic systems are discussed and related to previous work using many-body perturbation theory. It is shown that the sums of perturbation terms necessary for convergence are automatically included in the equations for two-particle excitations. Application of the equations to open-shell atoms is described.