Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review C

Generalized Fermi sea for plane-wave Hartree-Fock theory. II. Semi-realistic interactions in three dimensions

Valdir C. Aguilera-Navarro, Robert Hartman Belehrad, MANUEL DE LLANO, M. S. Sandel, James P. Vary, O. Rojo

Published Sep 1, 1980• 11 citations
Publisher

Abstract

The possibility of abnormally occupying plane-wave Hartree-Fock determinants is studied analytically for several many-fermion systems interacting with short- and long-ranged pair potentials in three dimensions. Lower energy states are found in some cases, but they always depend sensitively on the shape of the potential as well as on the amount of space exchange present in the interaction.NUCLEAR STRUCTURE Plane-wave Hartree-Fock, nuclear matter.

DOI
10.1103/physrevc.22.1260

Related papers

Generalized Fermi sea for plane-wave Hartree-Fock theory: One dimensional model calculation

MANUEL DE LLANO, James P. Vary

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 $N$ 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.

Physical Review C•Mar 1, 1979•37 citations

Symmetry-breaking Hartree--Fock solutions in a many-body system

MANUEL DE LLANO

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

•Jun 1, 1973•0 citations

Stability of the Plane Wave Hartree-Fock Ground State

Fumiaki Iwamoto, K. Sawada

The condition that the plane wave state is the lowest Hartree-Fock state of a large fermion system with repulsive interactions, is generally investigated and applied to the systems with $\ensuremath{\delta}$-type interactions, finite-range interactions, and long-range Coulomb interactions. It is found that for finite-range interactions the plane wave state is stable in both the high-density and low-density regions, but may become unstable in the intermediate-density region. The high-density stable region disappears for $\ensuremath{\delta}$-type interactions, while the low-density stable region vanishes for long-range Coulomb interactions. Especially for an electron gas, the critical value ${r}_{s}$ is smaller than 4.5. A speculation concerning the transition of the true ground state is discussed.

Physical Review•May 1, 1962•27 citations

Non-trivial solutions of the Hartree--Fock equations for a very-many- particle system

MANUEL DE LLANO

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)

•Jan 1, 1973•0 citations

Hartree-Fock states in the thermodynamic limit. III. Low-density clustering effects

L. Döhnert, MANUEL DE LLANO, Angelo Plastino

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.

Physical Review A•Feb 1, 1978•14 citations

Structure of Nuclear Matter

A. W. Overhauser

Equations for explicit Hartree states of energy lower than that of conventional ground states for noninteracting Fermi gases are given for Fermi gases with attractive interactions. One-dimensional and three-dimensional cases are treated. Large static density waves are shown to exist in such Fermi gases, and the binding energy is derived therefrom. If the density waves are used with the variational theorem, their contribution to the binding energy is greater, and it is concluded that density waves are of great importance in nuclear matter. In the limit of weak attractive interactions, the Fermi surface will become a manyfaced polyhedron, each pair of opposite faces arising from a density wave. (D.L.C.)

Physical Review Letters•Apr 15, 1960•173 citations
The possibility of abnormally occupying plane-wave Hartree-Fock determinants is studied analytically for several many-fermion systems interacting with short- and long-ranged pair potentials in three dimensions. Lower energy states are found in some cases, but they always depend sensitively on the shape of the potential as well as on the amount of space exchange present in the interaction.NUCLEAR STRUCTURE Plane-wave Hartree-Fock, nuclear matter.
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.
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
The condition that the plane wave state is the lowest Hartree-Fock state of a large fermion system with repulsive interactions, is generally investigated and applied to the systems with $\ensuremath{\delta}$-type interactions, finite-range interactions, and long-range Coulomb interactions. It is found that for finite-range interactions the plane wave state is stable in both the high-density and low-density regions, but may become unstable in the intermediate-density region. The high-density stable region disappears for $\ensuremath{\delta}$-type interactions, while the low-density stable region vanishes for long-range Coulomb interactions. Especially for an electron gas, the critical value rs{r}_{s}rs​ is smaller than 4.5. A speculation concerning the transition of the true ground state is discussed.
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.
Equations for explicit Hartree states of energy lower than that of conventional ground states for noninteracting Fermi gases are given for Fermi gases with attractive interactions. One-dimensional and three-dimensional cases are treated. Large static density waves are shown to exist in such Fermi gases, and the binding energy is derived therefrom. If the density waves are used with the variational theorem, their contribution to the binding energy is greater, and it is concluded that density waves are of great importance in nuclear matter. In the limit of weak attractive interactions, the Fermi surface will become a manyfaced polyhedron, each pair of opposite faces arising from a density wave. (D.L.C.)