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arXiv:cond-mat/0107536 · Physical Review A

Improved tensor-product expansions for the two-particle density matrix

Gábor Cśanyi, Stefan Goedecker, T. A. Arias

Published Feb 20, 2002• 74 citations
arXivHTMLPDF

Abstract

We present a density-matrix functional within the recently introduced framework for tensor-product expansions of the two-particle density matrix. It performs well both for the homogeneous electron gas as well as atoms. For the homogeneous electron gas, it performs significantly better than all previous density-matrix functionals, becoming very accurate for high densities and outperforming the Hartree-Fock method at metallic valence electron densities. For isolated atoms and ions, it is on par with generalized-gradient approximations to the density-functional theory. We also present analytic results for the correlation energy in the low-density limit of the free-electron gas for a broad class of such functionals.

DOI
10.1103/physreva.65.032510

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We present a density-matrix functional within the recently introduced framework for tensor-product expansions of the two-particle density matrix. It performs well both for the homogeneous electron gas as well as atoms. For the homogeneous electron gas, it performs significantly better than all previous density-matrix functionals, becoming very accurate for high densities and outperforming the Hartree-Fock method at metallic valence electron densities. For isolated atoms and ions, it is on par with generalized-gradient approximations to the density-functional theory. We also present analytic results for the correlation energy in the low-density limit of the free-electron gas for a broad class of such functionals.
We explore a class of computationally feasible approximations of the two-body density matrix as a finite sum of tensor products of single-particle operators. Physical symmetries then uniquely determine the two-body matrix in terms of the one-body matrix. Representing dynamical correlation alone as a single tensor product results in a theory that predicts near zero dynamical correlation in the homogeneous electron gas at moderate to high densities. But, representing both dynamical and statistical correlation effects together as a tensor product leads to the recently proposed ``natural orbital functional.'' We find that this latter theory has some asymptotic properties consistent with established many-body theory but is no more accurate than Hartree-Fock in describing the homogeneous electron gas for the range of densities typically found in the valence regions of solids.
We explore a new class of computationally feasible approximations of the two-body density matrix as a finite sum of tensor products of single-particle operators. Physical symmetries then uniquely determine the two-body matrix in terms of the one-body matrix. Representing dynamical correlation alone as a single tensor product results in a theory which predicts near zero dynamical correlation in the homogeneous electron gas at moderate to high densities. But, representing both dynamical and statistical correlation effects together as a tensor product leads to the recently proposed ``natural orbital functional.'' We find that this latter theory has some asymptotic properties consistent with established many-body theory but is no more accurate than Hartee-Fock in describing the homogeneous electron gas for the range of densities typically found in the valence regions of solids. PACS 71.10.-w 71.15.Mb, Accepted for publication in Physical Review B
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.
Accurate treatment of the electronic correlation in inhomogeneous electronic systems, combined with the ability to capture the correlation energy of the homogeneous electron gas, allows to reach high predictive power in the application of density-functional theory. For two-dimensional systems we can achieve this goal by generalizing our previous approximation [Phys. Rev. B 79, 085316 (2009)] to a parameter-free form, which reproduces the correlation energy of the homogeneous gas while preserving the ability to deal with inhomogeneous systems. The resulting functional is shown to be very accurate for finite systems with an arbitrary number of electrons with respect to numerically exact reference data.
The semiclassical $\ensuremath{\hbar}$ expansion of the one-particle density matrix for a two-dimensional Fermi gas is calculated within the Wigner transform method of B. Grammaticos and A. Voros [Ann. Phys. (N.Y.) 123, 359 (1979)], originally developed in the context of nuclear physics. The method of Grammaticos and Voros has the virtue of preserving both the Hermiticity and idempotency of the density matrix to all orders in the $\ensuremath{\hbar}$ expansion. As a topical application, we use our semiclassical expansion to go beyond the local-density approximation for the construction of the total dipole-dipole interaction energy functional of a two-dimensional, spin-polarized dipolar Fermi gas. We find a finite, second-order gradient correction to the Hartree-Fock energy, which takes the form $\ensuremath{\varepsilon}{(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}/\sqrt{\ensuremath{\rho}}$, with $\ensuremath{\varepsilon}$ being small ($|\ensuremath{\varepsilon}|\ensuremath{\ll}1$) and negative. We test the quality of the corrected energy by comparing it with the exact results available for harmonic confinement. Even for small particle numbers, the gradient correction to the dipole-dipole energy provides a significant improvement over the local-density approximation.
The momentum distributions of a homogeneous electron gas, arising from a simple one-matrix functional for the electron-electron repulsion energy that generalizes the Hartree-Fock and Goedecker-Umrigar approximations, are analyzed in detail. Their properties are found to depend strongly on the exponent \ensuremath{\beta} that enters the functional. Smooth momentum distributions that, together with the corresponding energies per particle, exhibit simple scaling behavior with respect to the electron density \ensuremath{\rho}, are obtained only for $\frac{4}{5}<\ensuremath{\beta}<\frac{4}{3}$ and $\ensuremath{\rho}<~{\ensuremath{\rho}}_{\mathrm{crit}}(\ensuremath{\beta}).$