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

arXiv:1105.2473 · Physical Review B

Constraints of reduced density-matrix functional theory for the two-dimensional homogeneous electron gas

A. Putaja, E. Räsänen

Published Jul 15, 2011• 7 citations
arXivHTMLPDF

Abstract

Reduced density-matrix functional theory (RDMFT) has become an appealing alternative to density-functional theory to describe electronic properties of strongly correlated systems. Here we derive exact conditions for the suitability of RDMFT to describe the two-dimensional homogeneous electron gas, which is the base system for semiconductor quantum dots and quantum Hall devices, for example. Following the method of Cioslowski and Pernal [J. Chem. Phys. 111, 3396 (1999)] we focus on the properties of power functionals of the form $f(n,{n}^{\ensuremath{'}})={(n{n}^{\ensuremath{'}})}^{\ensuremath{\alpha}}$ for the scaling function in the exchange-correlation energy. We show that in order to have stable and analytic solutions, and for $f$ to satisfy the homogeneous scaling constraint, the power is restricted to $1/4\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}3/4$. Applying a reasonable ansatz for the momentum distribution and the lower bound for the exchange-correlation energy tightens the physical regime further to $0.64\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\leqslant}0.75$.

DOI
10.1103/physrevb.84.035104

Related papers

Validity of power functionals for a homogeneous electron gas in reduced-density-matrix-functional theory

A. Putaja, F. G. Eich, Tim Baldsiefen +1 more

Physically valid and numerically efficient approximations for the exchange and correlation energy are critical for reduced-density-matrix-functional theory to become a widely used method in electronic structure calculations. Here we examine the physical limits of power functionals of the form $f(n,{n}^{\ensuremath{'}})={(n{n}^{\ensuremath{'}})}^{\ensuremath{\alpha}}$ for the scaling function in the exchange-correlation energy. To this end we obtain numerically the minimizing momentum distributions for the three- and two-dimensional homogeneous electron gas, respectively. In particular, we examine the limiting values for the power $\ensuremath{\alpha}$ to yield physically sound solutions that satisfy the Lieb-Oxford lower bound for the exchange-correlation energy and exclude pinned states with the condition $n(\mathbf{k})<1$ for all wave vectors $\mathbf{k}$. The results refine the constraints previously obtained from trial momentum distributions. We also compute the values for $\ensuremath{\alpha}$ that yield the exact correlation energy and its kinetic part for both the three- and two-dimensional electron gas. In both systems, narrow regimes of validity and accuracy are found at $\ensuremath{\alpha}\ensuremath{\gtrsim}0.6$ and at ${r}_{s}\ensuremath{\gtrsim}10$ for the density parameter, corresponding to relatively low densities.

arXiv:1602.03624•Physical Review A•Mar 10, 2016•3 citations

Performance of one-body reduced density-matrix functionals for the homogeneous electron gas

Nektarios N. Lathiotakis, N. Helbig, Eberhard K. U. Gross

The subject of this study is the exchange-correlation-energy functional of reduced density-matrix functional theory. Approximations of this functional are tested by applying them to the homogeneous electron gas. We find that two approximations recently proposed by Gritsenko et al., [J. Chem. Phys. 122, 204102 (2005)] yield considerably better correlation energies and momentum distributions than previously known functionals. We introduce modifications to these functionals, which, by construction, reproduce the exact correlation energy of the homogeneous electron gas.

arXiv:cond-mat/0605531•Physical Review B•May 21, 2007•59 citations

Exchange-correlation approximations for reduced-density-matrix-functional theory at finite temperature: Capturing magnetic phase transitions in the homogeneous electron gas

Tim Baldsiefen, Attila Cangi, F. G. Eich +1 more

We derive an intrinsically temperature-dependent approximation to the correlation grand potential for many-electron systems in thermodynamical equilibrium in the context of finite-temperature reduced-density-matrix-functional theory (FT-RDMFT). We demonstrate its accuracy by calculating the magnetic phase diagram of the homogeneous electron gas. We compare it to known limits from highly accurate quantum Monte Carlo calculations as well as to phase diagrams obtained within existing exchange-correlation approximations from density functional theory and zero-temperature RDMFT.

Physical Review A•Dec 18, 2017•8 citations

Orbital Functionals in Density-Matrix- and Current-Density-Functional Theory

N. Helbig

Density-Functional Theory (DFT), although widely used and very successful in the calculation of several observables, fails to correctly describe strongly correlated materials. In the first part of this work we, therefore, introduce reduced-density-matrix-functional theory (RDMFT) which is one possible way to treat electron correlation beyond DFT. Within this theory the one-body reduced density matrix (1-RDM) is used as the basic variable. Our main interest is the calculation of the fundamental gap which proves very problematic within DFT. In order to calculate the fundamental gap we generalize RDMFT to fractional particle numbers M by describing the system as an ensemble of an N and an N+1 particle system (with N≤ M≤N+1). For each fixed particle number, M, the total energy is minimized with respect to the natural orbitals and their occupation numbers. This leads to the total energy as a function of M. The derivative of this function with respect to the particle number has a discontinuity at integer particle number which is identical to the gap. In addition, we investigate the necessary and sufficient conditions for the 1-RDM of a system with fractional particle number to be N-representable. Numerical results are presented for alkali atoms, small molecules, and periodic systems. Another problem within DFT is the description of non-relativistic many- electron systems in the presence of magnetic fields. It requires the paramagnetic current density and the spin magnetization to be used as basic variables besides the electron density. However, electron-gas-based functionals of current-spin-density-functional Theory (CSDFT) exhibit derivative discontinuities as a function of the magnetic field whenever a new Landau level is occupied, which makes them difficult to use in practice. Since the appearance of Landau levels is, intrinsically, an orbital effect it is appealing to use orbital-dependent functionals. We have developed a CSDFT version of the optimized effective potential (OEP) method which allows for the use of explicitly orbital-dependent functionals in the context of CSDFT. We present the derivation of corresponding equations and show results for a quantum dot in external magnetic fields in the second part of this thesis.

•Jan 1, 2006•3 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

Reduced density matrix functional theory at finite temperature

Tim Baldsiefen

In this dissertation we present a new ab-initio functional theory (FT-RDMFT) to describe many-particle quantum systems at finite temperature. The work can be divided roughly into three parts. In the first part, we lay the mathematical foundation for FT-RDMFT. In the second part, we develop analytical and numerical methods to approximate and minimize the exact functional. We conclude by applying FT-RDMFT to the important model system of the homogeneous electron gas.

•Jan 1, 2012•5 citations
Reduced density-matrix functional theory (RDMFT) has become an appealing alternative to density-functional theory to describe electronic properties of strongly correlated systems. Here we derive exact conditions for the suitability of RDMFT to describe the two-dimensional homogeneous electron gas, which is the base system for semiconductor quantum dots and quantum Hall devices, for example. Following the method of Cioslowski and Pernal [J. Chem. Phys. 111, 3396 (1999)] we focus on the properties of power functionals of the form f(n,n\ensuremath′)=(nn\ensuremath′)\ensuremathαf(n,{n}^{\ensuremath{'}})={(n{n}^{\ensuremath{'}})}^{\ensuremath{\alpha}}f(n,n\ensuremath′)=(nn\ensuremath′)\ensuremathα for the scaling function in the exchange-correlation energy. We show that in order to have stable and analytic solutions, and for fff to satisfy the homogeneous scaling constraint, the power is restricted to 1/4\ensuremath⩽\ensuremathα\ensuremath⩽3/41/4\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}3/41/4\ensuremath⩽\ensuremathα\ensuremath⩽3/4. Applying a reasonable ansatz for the momentum distribution and the lower bound for the exchange-correlation energy tightens the physical regime further to 0.64\ensuremath≲\ensuremathα\ensuremath⩽0.750.64\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\leqslant}0.750.64\ensuremath≲\ensuremathα\ensuremath⩽0.75.
Physically valid and numerically efficient approximations for the exchange and correlation energy are critical for reduced-density-matrix-functional theory to become a widely used method in electronic structure calculations. Here we examine the physical limits of power functionals of the form $f(n,{n}^{\ensuremath{'}})={(n{n}^{\ensuremath{'}})}^{\ensuremath{\alpha}}$ for the scaling function in the exchange-correlation energy. To this end we obtain numerically the minimizing momentum distributions for the three- and two-dimensional homogeneous electron gas, respectively. In particular, we examine the limiting values for the power $\ensuremath{\alpha}$ to yield physically sound solutions that satisfy the Lieb-Oxford lower bound for the exchange-correlation energy and exclude pinned states with the condition n(k)<1n(\mathbf{k})<1n(k)<1 for all wave vectors k\mathbf{k}k. The results refine the constraints previously obtained from trial momentum distributions. We also compute the values for $\ensuremath{\alpha}$ that yield the exact correlation energy and its kinetic part for both the three- and two-dimensional electron gas. In both systems, narrow regimes of validity and accuracy are found at $\ensuremath{\alpha}\ensuremath{\gtrsim}0.6$ and at ${r}_{s}\ensuremath{\gtrsim}10$ for the density parameter, corresponding to relatively low densities.
The subject of this study is the exchange-correlation-energy functional of reduced density-matrix functional theory. Approximations of this functional are tested by applying them to the homogeneous electron gas. We find that two approximations recently proposed by Gritsenko et al., [J. Chem. Phys. 122, 204102 (2005)] yield considerably better correlation energies and momentum distributions than previously known functionals. We introduce modifications to these functionals, which, by construction, reproduce the exact correlation energy of the homogeneous electron gas.
We derive an intrinsically temperature-dependent approximation to the correlation grand potential for many-electron systems in thermodynamical equilibrium in the context of finite-temperature reduced-density-matrix-functional theory (FT-RDMFT). We demonstrate its accuracy by calculating the magnetic phase diagram of the homogeneous electron gas. We compare it to known limits from highly accurate quantum Monte Carlo calculations as well as to phase diagrams obtained within existing exchange-correlation approximations from density functional theory and zero-temperature RDMFT.
Density-Functional Theory (DFT), although widely used and very successful in the calculation of several observables, fails to correctly describe strongly correlated materials. In the first part of this work we, therefore, introduce reduced-density-matrix-functional theory (RDMFT) which is one possible way to treat electron correlation beyond DFT. Within this theory the one-body reduced density matrix (1-RDM) is used as the basic variable. Our main interest is the calculation of the fundamental gap which proves very problematic within DFT. In order to calculate the fundamental gap we generalize RDMFT to fractional particle numbers M by describing the system as an ensemble of an N and an N+1 particle system (with N≤ M≤N+1). For each fixed particle number, M, the total energy is minimized with respect to the natural orbitals and their occupation numbers. This leads to the total energy as a function of M. The derivative of this function with respect to the particle number has a discontinuity at integer particle number which is identical to the gap. In addition, we investigate the necessary and sufficient conditions for the 1-RDM of a system with fractional particle number to be N-representable. Numerical results are presented for alkali atoms, small molecules, and periodic systems. Another problem within DFT is the description of non-relativistic many- electron systems in the presence of magnetic fields. It requires the paramagnetic current density and the spin magnetization to be used as basic variables besides the electron density. However, electron-gas-based functionals of current-spin-density-functional Theory (CSDFT) exhibit derivative discontinuities as a function of the magnetic field whenever a new Landau level is occupied, which makes them difficult to use in practice. Since the appearance of Landau levels is, intrinsically, an orbital effect it is appealing to use orbital-dependent functionals. We have developed a CSDFT version of the optimized effective potential (OEP) method which allows for the use of explicitly orbital-dependent functionals in the context of CSDFT. We present the derivation of corresponding equations and show results for a quantum dot in external magnetic fields in the second part of this thesis.
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.