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

World Scientific series in 20th century physics

Recent Progress in Constructing Nonlocal Energy Density Functionals

Norman Henry March, I. A. Howard, V. E. Van Doren

Published Sep 1, 2009• 0 citations
Publisher

Abstract

Both model results and some exact relations concerning kinetic, exchange, and correlation energy density functionals will be presented. As to single-particle kinetic energy, a model of harmonic confinement, motivated by recent experimental activity on Fermion vapors, of an arbitrary number of Fermions will be used to construct exact nonlocal kinetic energy density functionals in both one and three dimensions. Then, some further exact results concerning nuclear cusp conditions for exchange only, and for exchange plus correlation, will be discussed. Finally, in Appendices, the one-shell harmonic confinement is generalized to include Coulomb repulsion in the two-electron Hookean atom and some results on the Pauli potentials for many-Fermion systems are obtained. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

DOI
10.1142/9789814271783_0085

Related papers

Complete functional theory for the fermion density of independent particles subject to harmonic confinement in d dimensions for an arbitrary number of closed shells

I. A. Howard, N. H. March, Luis Miguel Nieto

In earlier work, expressions have been constructed for the single-particle kinetic-energy functional ${T}_{s}[\ensuremath{\rho}]$ for independent fermions subject to harmonic confinement in low dimensions, with $\ensuremath{\rho}$ the particle density. Here, the differential equation for $\ensuremath{\rho}$ is first obtained in d dimensions for an arbitrary number of closed shells. Then, by using the known Euler-Lagrange equation, the functional derivative $\ensuremath{\delta}{T}_{s}/\ensuremath{\delta}\ensuremath{\rho}(\mathbf{r})$ is constructed. ${T}_{s}[\ensuremath{\rho}]$ itself is proved to take the form of a linear combination of three pieces: (i) a von Weizs\"acker inhomogeneity kinetic energy, but with the original coefficient reduced by a dimensionality factor $1/d,$ (ii) a Thomas-Fermi kinetic energy, and (iii) a truly nonlocal contribution that, however, is shown to involve only the density $\ensuremath{\rho}$ itself and its first derivative. Thus, for this model, which is currently highly relevant to the interpretation of experiments on the evaporative cooling of dilute, and hence almost noninteracting, fermions, a complete density-functional theory now exists.

Physical Review A•Nov 22, 2002•25 citations

Single-particle kinetic energy in density-functional theory: Harmonic confinement in two and three dimensions

N. H. March

Using the known perturbation series for the idempotent Dirac density matrix in powers of a given one-body potential $V(\mathbf{r})$, Stoddart and March (SM) generated a corresponding series for the kinetic energy density. While the general term of the SM series is known, a summation has not been achieved to date. A contribution to solve this problem is made here by exhibiting an explicit form of the above kinetic energy density for Fermions filling an arbitrary number of closed shells, when the confinement is harmonic. This example is of considerable current interest because of ongoing experiments on ultracold atomic gases of Fermions.

Physical Review A•Apr 20, 2006•2 citations

Many‐fermion systems: Differential equations and kinetic energy functionals for different confining potentials

I. A. Howard, N. H. March

Abstract Determination of the kinetic energy functional is one of the central problems in the density functional theory of a many‐fermion system. We consider several many‐fermion systems to be characterized by simple confining potentials: the harmonic oscillator potential in one and three dimensions, the Coulomb potential, and the potential V(x) = −Uosech2(x/L). We discuss for each of these cases the differential equation that determines the fermion density in the given potential, and the corresponding kinetic energy density. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

International Journal of Quantum Chemistry•Nov 18, 2002•2 citations

The Pauli potential in one-dimensional density functional theory: general result for two-level systems and specific example forNharmonically confined Fermions

I. A. Howard, N. H. March

There is ongoing interest in the kinetic energy functional T s [ρ] in density functional theory. The present study lies in this area and concerns the Pauli potential V P [ρ]. A differential equation is obtained here for V P ( x ) in one dimension for a general two-level system. Also, as a specific example, such a functional of ρ( x ), the ground-state Fermion density, is given for the case of N Fermions which are harmonically confined.

Journal of Physics A Mathematical and General•Jun 5, 2003•3 citations

Some tests of an approximate density functional for the ground-state kinetic energy of a fermion system

Conyers Herring, Manoj Chopra

Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density \ensuremath{\rho}(r) can be used to estimate the potential field v(r) that will give rise to a given \ensuremath{\rho}(r), or the \ensuremath{\rho}(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizs\"acker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizs\"acker type can never predict multiple maxima in \ensuremath{\rho} if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.

Physical Review A•Jan 1, 1988•54 citations

Exact expressions for energy functional in the time‐dependent density functional theory

Ágnes Nagy

Abstract Exact equations for correlation energy and kinetic energy correction functionals are derived in the time‐dependent density functional theory. Hierarchies of equations are also obtained and approximate functionals are proposed by truncation of the hierarchies at different orders. Explicit relations for the correlation energy and its kinetic component along the adiabatic path are given in terms of powers of the moments of the electron density. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

International Journal of Quantum Chemistry•Jan 1, 2003•2 citations
Both model results and some exact relations concerning kinetic, exchange, and correlation energy density functionals will be presented. As to single-particle kinetic energy, a model of harmonic confinement, motivated by recent experimental activity on Fermion vapors, of an arbitrary number of Fermions will be used to construct exact nonlocal kinetic energy density functionals in both one and three dimensions. Then, some further exact results concerning nuclear cusp conditions for exchange only, and for exchange plus correlation, will be discussed. Finally, in Appendices, the one-shell harmonic confinement is generalized to include Coulomb repulsion in the two-electron Hookean atom and some results on the Pauli potentials for many-Fermion systems are obtained. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
In earlier work, expressions have been constructed for the single-particle kinetic-energy functional ${T}_{s}[\ensuremath{\rho}]$ for independent fermions subject to harmonic confinement in low dimensions, with $\ensuremath{\rho}$ the particle density. Here, the differential equation for $\ensuremath{\rho}$ is first obtained in d dimensions for an arbitrary number of closed shells. Then, by using the known Euler-Lagrange equation, the functional derivative $\ensuremath{\delta}{T}_{s}/\ensuremath{\delta}\ensuremath{\rho}(\mathbf{r})$ is constructed. ${T}_{s}[\ensuremath{\rho}]$ itself is proved to take the form of a linear combination of three pieces: (i) a von Weizs\"acker inhomogeneity kinetic energy, but with the original coefficient reduced by a dimensionality factor 1/d,1/d,1/d, (ii) a Thomas-Fermi kinetic energy, and (iii) a truly nonlocal contribution that, however, is shown to involve only the density $\ensuremath{\rho}$ itself and its first derivative. Thus, for this model, which is currently highly relevant to the interpretation of experiments on the evaporative cooling of dilute, and hence almost noninteracting, fermions, a complete density-functional theory now exists.
Using the known perturbation series for the idempotent Dirac density matrix in powers of a given one-body potential V(r)V(\mathbf{r})V(r), Stoddart and March (SM) generated a corresponding series for the kinetic energy density. While the general term of the SM series is known, a summation has not been achieved to date. A contribution to solve this problem is made here by exhibiting an explicit form of the above kinetic energy density for Fermions filling an arbitrary number of closed shells, when the confinement is harmonic. This example is of considerable current interest because of ongoing experiments on ultracold atomic gases of Fermions.
Abstract Determination of the kinetic energy functional is one of the central problems in the density functional theory of a many‐fermion system. We consider several many‐fermion systems to be characterized by simple confining potentials: the harmonic oscillator potential in one and three dimensions, the Coulomb potential, and the potential V(x) = −Uosech2(x/L). We discuss for each of these cases the differential equation that determines the fermion density in the given potential, and the corresponding kinetic energy density. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density \ensuremath{\rho}(r) can be used to estimate the potential field v(r) that will give rise to a given \ensuremath{\rho}(r), or the \ensuremath{\rho}(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizs\"acker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizs\"acker type can never predict multiple maxima in \ensuremath{\rho} if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.
Abstract Exact equations for correlation energy and kinetic energy correction functionals are derived in the time‐dependent density functional theory. Hierarchies of equations are also obtained and approximate functionals are proposed by truncation of the hierarchies at different orders. Explicit relations for the correlation energy and its kinetic component along the adiabatic path are given in terms of powers of the moments of the electron density. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003