Exchange functionals based on finite uniform electron gases

We show how one can construct \alert{a simple} exchange functional by extending the well-know local-density approximation (LDA) to finite uniform electron gases. This new generalized local-density approximation (GLDA) functional uses only two quantities: the electron density $\rho$ and the curvature of the Fermi hole $\alpha$. This alternative "rung 2" functional can be easily coupled with generalized-gradient approximation (GGA) functionals to form a new family of "rung 3" meta-GGA (MGGA) functionals that we have named factorizable MGGAs (FMGGAs). Comparisons are made with various LDA, GGA and MGGA functionals for atoms and molecules.Comment: 20 pages, 5 figures and 2 table

Nodal surfaces and interdimensional degeneracies

The aim of this paper is to shed light on the topology and properties of the nodes (i.e. the zeros of the wave function) in electronic systems. Using the "electrons on a sphere" model, we study the nodes of two-, three- and four-electron systems in various ferromagnetic configurations ($sp$, $p^2$, $sd$, $pd$, $p^3$, $sp^2$ and $sp^3$). In some particular cases ($sp$, $p^2$, $sd$, $pd$ and $p^3$), we rigorously prove that the non-interacting wave function has the same nodes as the exact (yet unknown) wave function. The number of atomic and molecular systems for which the exact nodes are known analytically is very limited and we show here that this peculiar feature can be attributed to interdimensional degeneracies. Although we have not been able to prove it rigorously, we conjecture that the nodes of the non-interacting wave function for the $sp^3$ configuration are exact.Comment: 7 pages, 3 figures, accepted for publication in the Journal of Chemical Physic

Leading-order behavior of the correlation energy in the uniform electron gas

We show that, in the high-density limit, restricted M{\o}ller-Plesset (RMP) perturbation theory yields $E_{\text{RMP}}^{(2)} = \pi^{-2}(1-\ln 2) \ln r_s + O(r_s^0)$ for the correlation energy per electron in the uniform electron gas, where $r_s$ is the Seitz radius. This contradicts an earlier derivation which yielded $E_{\text{RMP}}^{(2)} = O(\ln|\ln r_s|)$. The reason for the discrepancy is explained.Comment: 4 pages, accepted for publication in Int. J. Quantum Che

Exact energy of the spin-polarized two-dimensional electron gas at high density

We derive the exact expansion, to $O(r_s)$, of the energy of the high-density spin-polarized two-dimensional uniform electron gas, where $r_s$ is the Seitz radius.Comment: 7 pages, 1 figure and 1 table, submitted to Phys. Rev.

The uniform electron gas

The uniform electron gas or UEG (also known as jellium) is one of the most fundamental models in condensed-matter physics and the cornerstone of the most popular approximation --- the local-density approximation --- within density-functional theory. In this article, we provide a detailed review on the energetics of the UEG at high, intermediate and low densities, and in one, two and three dimensions. We also report the best quantum Monte Carlo and symmetry-broken Hartree-Fock calculations available in the literature for the UEG and discuss the phase diagrams of jellium.Comment: 37 pages, 8 figures, 8 tables, accepted for publication in WIRES Computational Molecular Scienc

Correlation energy of two electrons in a ball

We study the ground-state correlation energy $E_{\rm c}$ of two electrons of opposite spin confined within a $D$-dimensional ball ($D \ge 2$) of radius $R$. In the high-density regime, we report accurate results for the exact and restricted Hartree-Fock energy, using a Hylleraas-type expansion for the former and a simple polynomial basis set for the latter. By investigating the exact limiting correlation energy E_{\rm c}^{(0)} = \lim_{R \to 0} \Ec for various values of $D$, we test our recent conjecture [J. Chem. Phys. {\bf 131} (2009) 241101] that, in the large-$D$ limit, $E_{\rm c}^{(0)} \sim -\delta^2/8$ for any spherically-symmetric confining external potential, where $\delta=1/(D-1)$.Comment: 6 pages, 2 figure