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 A

Exact differential equation for the density and ionization energy of a many-particle system

Mel Levy, John P. Perdew, Viraht Sahni

Published Nov 1, 1984• 905 citations
Publisher

Abstract

The ground-state density $n$ of a many-electron system obeys a Schr\"odinger-like differential equation for ${n}^{\frac{1}{2}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, which may be solved by standard Kohn-Sham programs. The exact local effective (nonexternal) potential, ${v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is displayed explicitly in terms of wave-function expectation values, from which ${v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})>~0$ for all $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$. A derivation for $n$ as $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|\ensuremath{\rightarrow}\ensuremath{\infty}$ implies that this new effective potential tends asymptotically to zero, as does the exact Kohn-Sham potential, with the highest occupied eigenvalue as the exact ionization energy. A new exact expression is also presented for the exchange-correlation hole density ${\ensuremath{\rho}}_{\mathrm{xc}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ about an electron at $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$, as $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|\ensuremath{\rightarrow}\ensuremath{\infty}$.

DOI
10.1103/physreva.30.2745

Related papers

Expressions for the Exchange Correlation Potential and Exchange Correlation Functional of Kohn--Sham Density Functional Theory

James P. Finley

The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential $\vxc$ and exchange-correlation energy $\Eco$ as explicit functionals of $v_s$ and $ϕ_1$, where $v_s$ is the local, one-body potential from the Kohn--Sham equations, and $ϕ_1$ is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say $|ϕ_1\ran$. In other words, $|ϕ_1\ran$ is the ground state eigenfunction of the noninteracting Schrödinger equation with the one-body potential $v_s$. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.

arXiv:physics/0506109•Jun 12, 2005•0 citations

A Rigorous Extension of the Kohn-Sham Equation for Strongly Correlated Electron Systems

Koichi Kusakabe

By introducing a set of auxiliary equations representing a many-body system, we have derived an extension of the Kohn-Sham scheme for the density functional theory. These equations consist of a Kohn-Sham-type equation determining single-particle orbitals and an eigen-value equation for an effective many-body problem. A variational method similar to the Kohn-Sham technique was utilized to derive effective interactions as well as effective potentials without artificial substitution of a Hubbard-type interaction and a mean-field correction in the energy functional. The second equation is described by an effective many-body Hamiltonian with both 2-body interactions and mean-field terms. Rigorous formulation of the extended Kohn-Sham equation is also given in accordance with the Hadjisavvas-Theophilou formulation. Our formulation can be interpreted as a way to define models of the strongly correlated electron systems, e.g. the Hubbard model

arXiv:cond-mat/0103094•Journal of the Physical Society of Japan•Jul 15, 2001•23 citations

Ab initioapproach for many-electron systems without invoking orbitals: An integral formulation of density-functional theory

Weitao Yang

A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.

Physical Review A•Dec 1, 1988•19 citations

TREATMENTS OF THE EXCHANGE ENERGY IN DENSITY-FUNCTIONAL THEORY

TAMÁS GÁL

Following a recent work [Gál, Phys. Rev. A64, 062503 (2001)], a simple derivation of the density-functional correction of the Hartree–Fock equations, the Hartree–Fock–Kohn–Sham equations, is presented, completing an integrated view of quantum mechanical theories, in which the Kohn–Sham equations, the Hartree–Fock–Kohn–Sham equations and the ground-state Schrödinger equation formally stem from a common ground: density-functional theory, through its Euler equation for the ground-state density. Along similar lines, the Kohn–Sham formulation of the Hartree–Fock approach is also considered. Further, it is pointed out that the exchange energy of density-functional theory built from the Kohn–Sham orbitals can be given by degree-two homogeneous N-particle density functionals (N = 1, 2, …), forming a sequence of degree-two homogeneous exchange-energy density functionals, the first element of which is minus the classical Coulomb-repulsion energy functional.

arXiv:0708.0576•International Journal of Modern Physics B•Jun 9, 2008•0 citations

Density-functional theory for excited states in a quasi-local-density approximation

Walter Kohn

The starting point of this paper is a recent extension by Theophilou of the Hohenberg-Kohn-Sham (HKS) density-functional theory to ensembles of systems consisting of the M lowest eigenstates, equally weighted. As in the HKS theory the key quantities are the exchange-correlation energy, ${E}_{\mathrm{xc}{}^{M}[\mathrm{n}(\mathrm{r})]}$, and potential, ${v}_{\mathrm{xc}{}^{M}(\mathit{r};[\mathit{n}(\mathit{r}\mathcal{'})])}$. The present paper provides expressions for these quantities, valid for systems of slowly varying density. Even for such systems, however, there are essential nonlocal effects. Nevertheless both ${E}_{\mathrm{xc}{}^{M}}$ and ${v}_{\mathrm{xc}{}^{M}}$ can be calculated in terms of quantities characteristic of appropriate uniform thermal ensembles. This theory is the analog of the ground-state local-density approximation and allows calculation of excited-state energies and densities.

Physical Review A•Aug 1, 1986•125 citations

New Theoretical Support for Density-Functional Theory as Commonly Applied

David C. Langreth

The Kohn-Sham equations are derived from the diagrammatic structure of many-body perturbation theory for the exact exchange-correlation energy, ${E}_{\mathrm{xc}}$, without explicit recourse to a minimum or extremal principle, thus providing new support for this implementation of density-functional theory in a most common situation. Furthermore, it is shown that for a large class of approximate expressions for ${E}_{\mathrm{xc}}$, there exist approximate Kohn-Sham equations which give the same density and energy as other means of calculation within the same diagrammatic approximation.

Physical Review Letters•Jun 25, 1984•17 citations
The ground-state density nnn of a many-electron system obeys a Schr"odinger-like differential equation for n12(r\ensuremath→){n}^{\frac{1}{2}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})n21​(r\ensuremath→), which may be solved by standard Kohn-Sham programs. The exact local effective (nonexternal) potential, veff(r\ensuremath→){v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})veff​(r\ensuremath→), is displayed explicitly in terms of wave-function expectation values, from which veff(r\ensuremath→)> 0{v}_{\mathrm{eff}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})>~0veff​(r\ensuremath→)> 0 for all r\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}r\ensuremath→. A derivation for nnn as ∣r\ensuremath→∣\ensuremath→\ensuremath∞|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|\ensuremath{\rightarrow}\ensuremath{\infty}∣r\ensuremath→∣\ensuremath→\ensuremath∞ implies that this new effective potential tends asymptotically to zero, as does the exact Kohn-Sham potential, with the highest occupied eigenvalue as the exact ionization energy. A new exact expression is also presented for the exchange-correlation hole density \ensuremathρxc(r\ensuremath→,r\ensuremath→\ensuremath′){\ensuremath{\rho}}_{\mathrm{xc}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})\ensuremathρxc​(r\ensuremath→,r\ensuremath→\ensuremath′) about an electron at r\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}r\ensuremath→, as ∣r\ensuremath→∣\ensuremath→\ensuremath∞|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|\ensuremath{\rightarrow}\ensuremath{\infty}∣r\ensuremath→∣\ensuremath→\ensuremath∞.
The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential $\vxc$ and exchange-correlation energy $\Eco$ as explicit functionals of vsv_svs​ and ϕ1ϕ_1ϕ1​, where vsv_svs​ is the local, one-body potential from the Kohn--Sham equations, and ϕ1ϕ_1ϕ1​ is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say $|ϕ_1\ran$. In other words, $|ϕ_1\ran$ is the ground state eigenfunction of the noninteracting Schrödinger equation with the one-body potential vsv_svs​. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.
By introducing a set of auxiliary equations representing a many-body system, we have derived an extension of the Kohn-Sham scheme for the density functional theory. These equations consist of a Kohn-Sham-type equation determining single-particle orbitals and an eigen-value equation for an effective many-body problem. A variational method similar to the Kohn-Sham technique was utilized to derive effective interactions as well as effective potentials without artificial substitution of a Hubbard-type interaction and a mean-field correction in the energy functional. The second equation is described by an effective many-body Hamiltonian with both 2-body interactions and mean-field terms. Rigorous formulation of the extended Kohn-Sham equation is also given in accordance with the Hadjisavvas-Theophilou formulation. Our formulation can be interpreted as a way to define models of the strongly correlated electron systems, e.g. the Hubbard model
A new approach for the calculation of ground states of many-electron systems is proposed via an integral formulation of the Hohenberg-Kohn-Sham density-functional theory. Only equations for the total electron density are involved; orbitals are not employed. Exchange and correlation effects are incorporated. In place of the set of single-electron equations, the total electron density is explicitly expressed in terms of the Kohn-Sham effective local potential through multidimensional integrations. The development is based on the first-order density matrix as obtained from the one-body Green's function in polygonal and Fourier path-integral representations. This might open up the possibility of ab initio calculations for molecules with very many electrons. It also provides explicit solutions to two long-standing problems: electron kinetic energy and momentum density as functionals of the total electron density. The formulation can also be used in calculations for general fermions.
Following a recent work [Gál, Phys. Rev. A64, 062503 (2001)], a simple derivation of the density-functional correction of the Hartree–Fock equations, the Hartree–Fock–Kohn–Sham equations, is presented, completing an integrated view of quantum mechanical theories, in which the Kohn–Sham equations, the Hartree–Fock–Kohn–Sham equations and the ground-state Schrödinger equation formally stem from a common ground: density-functional theory, through its Euler equation for the ground-state density. Along similar lines, the Kohn–Sham formulation of the Hartree–Fock approach is also considered. Further, it is pointed out that the exchange energy of density-functional theory built from the Kohn–Sham orbitals can be given by degree-two homogeneous N-particle density functionals (N = 1, 2, …), forming a sequence of degree-two homogeneous exchange-energy density functionals, the first element of which is minus the classical Coulomb-repulsion energy functional.
The starting point of this paper is a recent extension by Theophilou of the Hohenberg-Kohn-Sham (HKS) density-functional theory to ensembles of systems consisting of the M lowest eigenstates, equally weighted. As in the HKS theory the key quantities are the exchange-correlation energy, ExcM[n(r)]{E}_{\mathrm{xc}{}^{M}[\mathrm{n}(\mathrm{r})]}ExcM[n(r)]​, and potential, vxcM(r;[n(r′)]){v}_{\mathrm{xc}{}^{M}(\mathit{r};[\mathit{n}(\mathit{r}\mathcal{'})])}vxcM(r;[n(r′)])​. The present paper provides expressions for these quantities, valid for systems of slowly varying density. Even for such systems, however, there are essential nonlocal effects. Nevertheless both ExcM{E}_{\mathrm{xc}{}^{M}}ExcM​ and vxcM{v}_{\mathrm{xc}{}^{M}}vxcM​ can be calculated in terms of quantities characteristic of appropriate uniform thermal ensembles. This theory is the analog of the ground-state local-density approximation and allows calculation of excited-state energies and densities.
The Kohn-Sham equations are derived from the diagrammatic structure of many-body perturbation theory for the exact exchange-correlation energy, Exc{E}_{\mathrm{xc}}Exc​, without explicit recourse to a minimum or extremal principle, thus providing new support for this implementation of density-functional theory in a most common situation. Furthermore, it is shown that for a large class of approximate expressions for Exc{E}_{\mathrm{xc}}Exc​, there exist approximate Kohn-Sham equations which give the same density and energy as other means of calculation within the same diagrammatic approximation.