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Physical Review A

Hartree-Fock states in the thermodynamic limit

MANUEL DE LLANO, Angel Ricardo Plastino

Published Apr 1, 1976• 23 citations
Publisher

Abstract

A two-parameter class of single-particle orbitals, giving rise to long-range order in the local (spatial and/or spin) density, is shown to satisfy the full Hartree-Fock (HF) equations for occupied states in the thermodynamic limit. For a $\ensuremath{\delta}$ interparticle potential, these states are stabler (have lower HF energy) than the usual plane-wave (or trivial) HF solutions, for sufficiently strong coupling and/or high density. Minimization of the energy with respect to the (new) free parameters leads to sometimes gradual (second-order transition) and sometimes abrupt (first-order transition) onset of order, accompanied by a "bifurcation" of the new energy state from the old. The existence of even lower-energy, nontrivial HF states is also mentioned. We discuss the relevance to neutron and nuclear matter, to the Pauling "close-packed spheron" model of nuclei, as well as to the electron-gas problem.

DOI
10.1103/physreva.13.1633

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Physical Review•Mar 15, 1955•467 citations
A two-parameter class of single-particle orbitals, giving rise to long-range order in the local (spatial and/or spin) density, is shown to satisfy the full Hartree-Fock (HF) equations for occupied states in the thermodynamic limit. For a \ensuremathδ\ensuremath{\delta}\ensuremathδ interparticle potential, these states are stabler (have lower HF energy) than the usual plane-wave (or trivial) HF solutions, for sufficiently strong coupling and/or high density. Minimization of the energy with respect to the (new) free parameters leads to sometimes gradual (second-order transition) and sometimes abrupt (first-order transition) onset of order, accompanied by a "bifurcation" of the new energy state from the old. The existence of even lower-energy, nontrivial HF states is also mentioned. We discuss the relevance to neutron and nuclear matter, to the Pauling "close-packed spheron" model of nuclei, as well as to the electron-gas problem.
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.
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
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)
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.
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.
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.