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. B, Condensed matter

Correlation-energy functional and its high-density limit obtained from a coupling-constant perturbation expansion

Andreas Görling, Mel Levy

Published May 15, 1993• 436 citations
Publisher

Abstract

A perturbation theory is developed for the correlation energy ${\mathit{E}}_{\mathit{c}}$[n], of a finite-density system, with respect to the coupling constant \ensuremath{\alpha} which multiplies the electron-electron repulsion operator in ${\mathit{H}}^{\mathrm{\ensuremath{\alpha}}}$=T^+\ensuremath{\alpha}V${\mathrm{^}}_{\mathit{e}\mathit{e}}$+${\mathit{tsum}}_{\mathit{i}}$${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$(${\mathbf{r}}_{\mathit{i}}$). The external potential ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is constrained to keep the gound-state density n fixed for all \ensuremath{\alpha}\ensuremath{\ge}0. ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is given completely in terms of functional derivatives at full charge (\ensuremath{\alpha}=1), from which ${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]=${\mathit{e}}_{\mathit{c},2}$[n]+ ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$${\mathit{e}}_{\mathit{c},3}$[n]+${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}2}$${\mathit{e}}_{\mathit{c},4}$[n]+..., where each ${\mathit{e}}_{\mathit{c},}$j[n] is expressed in terms of integrals involving Kohn-Sham determinants. Here, ${\mathit{n}}_{\ensuremath{\lambda}}$(x,y,x)=${\ensuremath{\lambda}}^{3}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,\ensuremath{\lambda}z) and \ensuremath{\lambda}=${\mathrm{\ensuremath{\alpha}}}^{\mathrm{\ensuremath{-}}1}$. The identification of ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$], which is a high-density limit, as the second-order energy ${\mathit{e}}_{\mathit{c},2}$[n] allows one to compute bounds upon ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]; the bounds imply that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$]\ensuremath{\simeq}${\mathit{E}}_{\mathit{c}}$[n] for a large class of small atoms and molecules, and suggest that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$] should be of the same order of magnitude as ${\mathit{E}}_{\mathit{c}}$[n] in finite insulators and semiconductors.Approximations to ${\mathit{E}}_{\mathit{c}}$[n] should reflect all this. In contrast, perhaps the well-known overbinding of the local-density approximation (LDA) in molecules and solids is due, in part, to the fact that the LDA correlation energy is too sensitive to a coordinate scaling of n. Indeed, the LDA for ${\mathit{E}}_{\mathit{c}}$[${\mathit{n}}_{\ensuremath{\lambda}}$] diverges when \ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty} because of the presence of the -ln(\ensuremath{\lambda}) term in the Gell-Mann and Brueckner high-density expression for the correlation energy, per particle, of a homogeneous density, which is infinite. In a sense, the derived perturbation expansion transforms the Gell-Mann and Brueckner expression into one that applies specifically to an inhomogeneous density which integrates to a finite number of electrons.

DOI
10.1103/physrevb.47.13105

Related papers

The correlation energy as an explicit functional of the one-particle density matrix from a determinantal reference state

James P. Finley

Using an approach based on many body perturbation theory, the correlation energy $\cEco$ is expressed as an explicit functional of $ρ_1$, $v$, and $v_s$, where $ρ_1$ is the one-particle density matrix from the noninteracting, or reference, determinantal-state; $v$ is the external potential from the interacting, or target, state; $v_s$ is the (kernel of the) external potential from the noninteracting determinantal-state. In other words we have $\cEco[ρ_1,v,v_s]$. Anther possibility is the following explicit functional: $\cEco[ρ_1,v_{\text{co}},v_s]$, where $v_{\text{co}}$ is the (kernel of the) correlation potential from the noninteracting Hamiltonian. The proposed method can, in principle, be used to compute $\cEco$ in a very accurate and efficient manner, since, like the Kohn--Sham approach, there are no virtual orbitals to consider. However, in contrast to the Kohn--Sham approach, $\cEco$ is a known, explicit functional that can be approximated in a systematic manner. For simplicity, we only consider noninteracting closed-shell states and target states that are nondegenerate, singlet ground-states; so, in that case, $ρ_1$ denotes the spin-less one-particle density matrix from the determinantal reference state.

arXiv:physics/0506186•Jun 24, 2005•0 citations

Strong Correlation in Kohn-Sham Density Functional Theory

Francesc Malet, Paola Gori‐Giorgi

We use the exact strong-interaction limit of the Hohenberg-Kohn energy density functional to approximate the exchange-correlation energy of the restricted Kohn-Sham scheme. Our approximation corresponds to a highly nonlocal density functional whose functional derivative can be easily constructed, thus transforming exactly, in a physically transparent way, an important part of the electron-electron interaction into an effective local one-body potential. We test our approach on quasi-one-dimensional systems, showing that it captures essential features of strong correlation that restricted Kohn-Sham calculations using the currently available approximations cannot describe.

arXiv:1207.2775•Physical Review Letters•Dec 10, 2012•118 citations

Local-density-matrix approximation: Exact asymptotic results for a high-density homogeneous electron gas

Jerzy Ciosłowski, Katarzyna Pernal

Scaling arguments show that the exchange-correlation energy density of a spin-unpolarized homogeneous electron gas equals $A{\ensuremath{\omega}}^{4∕3}$, where $\ensuremath{\omega}$ is given by a simple functional of the momentum distribution $n(k)$ and $A$ is a constant. This functional possesses the property of yielding the true $n(k)$ (including the discontinuity at the Fermi momentum, which is not reproduced by any of the presently known density matrix functionals) upon minimization of the energy density $e$ constrained by the electron density $\ensuremath{\rho}$. At the limit of $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\omega}$ coincides with $\ensuremath{\rho}$ and the dependence of $\ensuremath{\omega}$ on $n(k)$, which can be readily recovered from the actual values of $e$ and $n(k)$, is given by an explicit analytical formula. Application of this formalism to general Coulombic systems yields a local version of the density matrix theory, in which the exchange-correlation energy is given by a spatial integral of an expression that depends on the amplitudes of natural orbitals at a given point and the respective occupation numbers.

Physical Review B•Mar 22, 2005•11 citations

Density-Functional Theory for Strongly Interacting Electrons

Paola Gori‐Giorgi, Michael Seidl, Giovanni Vignale

We present an alternative to the Kohn-Sham formulation of density-functional theory for the ground-state properties of strongly interacting electronic systems. The idea is to start from the limit of zero kinetic energy and systematically expand the universal energy functional of the density in powers of a "coupling constant" that controls the magnitude of the kinetic energy. The problem of minimizing the energy is reduced to the solution of a strictly correlated electron problem in the presence of an effective potential, which in our theory plays the same role as the Kohn-Sham potential plays in the traditional formulation. We discuss several schemes for approximating the energy functional, and report preliminary results for low-density quantum dots.

arXiv:0908.0669•Physical Review Letters•Oct 12, 2009•106 citations

Equation satisfied by the energy-density functional for electron-electron mutual Coulomb repulsion

Daniel P. Joubert

It is shown that the electron-electron mutual Coulomb repulsion energy-density functional ${V}_{ee}^{\ensuremath{\gamma}}[\ensuremath{\rho}]$ satisfies the equation${V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N}^{1}]\ensuremath{-}{V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}]=\ensuremath{\int}{d}^{3}r\frac{\ensuremath{\delta}{V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N}^{1}]}{\ensuremath{\delta}{\ensuremath{\rho}}_{N}^{1}(\mathbf{r})}[{\ensuremath{\rho}}_{N}^{1}(\mathbf{r})\ensuremath{-}{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})],$where ${\ensuremath{\rho}}_{N}^{1}(\mathbf{r})$ and ${\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})$ are $N$-electron and $(N\ensuremath{-}1)$-electron densities determined from the same adiabatic scaled external potential of the $N$-electron system at coupling strength $\ensuremath{\gamma}.$

Physical Review A•Oct 14, 2011•4 citations

Self-consistent density-functional approach to the correlated ground states and an unrestricted many-body perturbation theory

Behnam Farid

We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.

Philosophical Magazine B•Aug 1, 1997•18 citations
A perturbation theory is developed for the correlation energy Ec{\mathit{E}}_{\mathit{c}}Ec​[n], of a finite-density system, with respect to the coupling constant \ensuremath{\alpha} which multiplies the electron-electron repulsion operator in H\ensuremathα{\mathit{H}}^{\mathrm{\ensuremath{\alpha}}}H\ensuremathα=T^+\ensuremath{\alpha}V{\mathrm{^}}_{\mathit{e}\mathit{e}}+{\mathit{tsum}}_{\mathit{i}}$${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}(ri{\mathbf{r}}_{\mathit{i}}ri​). The external potential v\ensuremathα{\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}v\ensuremathα​ is constrained to keep the gound-state density n fixed for all \ensuremath{\alpha}\ensuremath{\ge}0. v\ensuremathα{\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}v\ensuremathα​ is given completely in terms of functional derivatives at full charge (\ensuremath{\alpha}=1), from which Ec{\mathit{E}}_{\mathit{c}}Ec​[n\ensuremathλ{\mathit{n}}_{\ensuremath{\lambda}}n\ensuremathλ​]=ec,2{\mathit{e}}_{\mathit{c},2}ec,2​[n]+ {\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$${\mathit{e}}_{\mathit{c},3}[n]+{\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}2}$${\mathit{e}}_{\mathit{c},4}[n]+..., where each ec,{\mathit{e}}_{\mathit{c},}ec,​j[n] is expressed in terms of integrals involving Kohn-Sham determinants. Here, n\ensuremathλ{\mathit{n}}_{\ensuremath{\lambda}}n\ensuremathλ​(x,y,x)=\ensuremathλ3{\ensuremath{\lambda}}^{3}\ensuremathλ3n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,\ensuremath{\lambda}z) and \ensuremath{\lambda}=\ensuremathα\ensuremath−1{\mathrm{\ensuremath{\alpha}}}^{\mathrm{\ensuremath{-}}1}\ensuremathα\ensuremath−1. The identification of {\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}[n\ensuremathλ{\mathit{n}}_{\ensuremath{\lambda}}n\ensuremathλ​], which is a high-density limit, as the second-order energy ec,2{\mathit{e}}_{\mathit{c},2}ec,2​[n] allows one to compute bounds upon {\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}[n\ensuremathλ{\mathit{n}}_{\ensuremath{\lambda}}n\ensuremathλ​]; the bounds imply that {\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}[n\ensuremathλ{\mathit{n}}_{\ensuremath{\lambda}}n\ensuremathλ​]\ensuremath{\simeq}Ec{\mathit{E}}_{\mathit{c}}Ec​[n] for a large class of small atoms and molecules, and suggest that {\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathit{E}}_{\mathit{c}}[n\ensuremathλ{\mathit{n}}_{\ensuremath{\lambda}}n\ensuremathλ​] should be of the same order of magnitude as Ec{\mathit{E}}_{\mathit{c}}Ec​[n] in finite insulators and semiconductors.Approximations to Ec{\mathit{E}}_{\mathit{c}}Ec​[n] should reflect all this. In contrast, perhaps the well-known overbinding of the local-density approximation (LDA) in molecules and solids is due, in part, to the fact that the LDA correlation energy is too sensitive to a coordinate scaling of n. Indeed, the LDA for Ec{\mathit{E}}_{\mathit{c}}Ec​[n\ensuremathλ{\mathit{n}}_{\ensuremath{\lambda}}n\ensuremathλ​] diverges when \ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty} because of the presence of the -ln(\ensuremath{\lambda}) term in the Gell-Mann and Brueckner high-density expression for the correlation energy, per particle, of a homogeneous density, which is infinite. In a sense, the derived perturbation expansion transforms the Gell-Mann and Brueckner expression into one that applies specifically to an inhomogeneous density which integrates to a finite number of electrons.
Using an approach based on many body perturbation theory, the correlation energy $\cEco$ is expressed as an explicit functional of ρ1ρ_1ρ1​, vvv, and vsv_svs​, where ρ1ρ_1ρ1​ is the one-particle density matrix from the noninteracting, or reference, determinantal-state; vvv is the external potential from the interacting, or target, state; vsv_svs​ is the (kernel of the) external potential from the noninteracting determinantal-state. In other words we have $\cEco[ρ_1,v,v_s]$. Anther possibility is the following explicit functional: $\cEco[ρ_1,v_{\text{co}},v_s]$, where vcov_{\text{co}}vco​ is the (kernel of the) correlation potential from the noninteracting Hamiltonian. The proposed method can, in principle, be used to compute $\cEco$ in a very accurate and efficient manner, since, like the Kohn--Sham approach, there are no virtual orbitals to consider. However, in contrast to the Kohn--Sham approach, $\cEco$ is a known, explicit functional that can be approximated in a systematic manner. For simplicity, we only consider noninteracting closed-shell states and target states that are nondegenerate, singlet ground-states; so, in that case, ρ1ρ_1ρ1​ denotes the spin-less one-particle density matrix from the determinantal reference state.
We use the exact strong-interaction limit of the Hohenberg-Kohn energy density functional to approximate the exchange-correlation energy of the restricted Kohn-Sham scheme. Our approximation corresponds to a highly nonlocal density functional whose functional derivative can be easily constructed, thus transforming exactly, in a physically transparent way, an important part of the electron-electron interaction into an effective local one-body potential. We test our approach on quasi-one-dimensional systems, showing that it captures essential features of strong correlation that restricted Kohn-Sham calculations using the currently available approximations cannot describe.
Scaling arguments show that the exchange-correlation energy density of a spin-unpolarized homogeneous electron gas equals $A{\ensuremath{\omega}}^{4∕3}$, where $\ensuremath{\omega}$ is given by a simple functional of the momentum distribution n(k)n(k)n(k) and AAA is a constant. This functional possesses the property of yielding the true n(k)n(k)n(k) (including the discontinuity at the Fermi momentum, which is not reproduced by any of the presently known density matrix functionals) upon minimization of the energy density eee constrained by the electron density $\ensuremath{\rho}$. At the limit of $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\omega}$ coincides with $\ensuremath{\rho}$ and the dependence of $\ensuremath{\omega}$ on n(k)n(k)n(k), which can be readily recovered from the actual values of eee and n(k)n(k)n(k), is given by an explicit analytical formula. Application of this formalism to general Coulombic systems yields a local version of the density matrix theory, in which the exchange-correlation energy is given by a spatial integral of an expression that depends on the amplitudes of natural orbitals at a given point and the respective occupation numbers.
We present an alternative to the Kohn-Sham formulation of density-functional theory for the ground-state properties of strongly interacting electronic systems. The idea is to start from the limit of zero kinetic energy and systematically expand the universal energy functional of the density in powers of a "coupling constant" that controls the magnitude of the kinetic energy. The problem of minimizing the energy is reduced to the solution of a strictly correlated electron problem in the presence of an effective potential, which in our theory plays the same role as the Kohn-Sham potential plays in the traditional formulation. We discuss several schemes for approximating the energy functional, and report preliminary results for low-density quantum dots.
It is shown that the electron-electron mutual Coulomb repulsion energy-density functional ${V}_{ee}^{\ensuremath{\gamma}}[\ensuremath{\rho}]$ satisfies the equation${V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N}^{1}]\ensuremath{-}{V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}]=\ensuremath{\int}{d}^{3}r\frac{\ensuremath{\delta}{V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N}^{1}]}{\ensuremath{\delta}{\ensuremath{\rho}}_{N}^{1}(\mathbf{r})}[{\ensuremath{\rho}}_{N}^{1}(\mathbf{r})\ensuremath{-}{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})],$where ${\ensuremath{\rho}}_{N}^{1}(\mathbf{r})$ and ${\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})$ are NNN-electron and $(N\ensuremath{-}1)$-electron densities determined from the same adiabatic scaled external potential of the NNN-electron system at coupling strength $\ensuremath{\gamma}.$
We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.