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

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.

Correlation energy of two electrons in the high-density limit

We consider the high-density-limit correlation energy \Ec in $D \ge 2$ dimensions for the $^1S$ ground states of three two-electron systems: helium (in which the electrons move in a Coulombic field), spherium (in which they move on the surface of a sphere), and hookium (in which they move in a quadratic potential). We find that the \Ec values are strikingly similar, depending strongly on $D$ but only weakly on the external potential. We conjecture that, for large $D$, the limiting correlation energy \Ec \sim -\delta^2/8 in any confining external potential, where $\delta = 1/(D-1)$.Comment: 4 pages, 0 figur

Chemistry in One Dimension

We report benchmark results for one-dimensional (1D) atomic and molecular systems interacting via the Coulomb operator $|x|^{-1}$. Using various wavefunction-type approaches, such as Hartree-Fock theory, second- and third-order M{\o}ller-Plesset perturbation theory and explicitly correlated calculations, we study the ground state of atoms with up to ten electrons as well as small diatomic and triatomic molecules containing up to two electrons. A detailed analysis of the 1D helium-like ions is given and the expression of the high-density correlation energy is reported. We report the total energies, ionization energies, electron affinities and other interesting properties of the many-electron 1D atoms and, based on these results, we construct the 1D analog of Mendeleev's periodic table. We find that the 1D periodic table contains only two groups: the alkali metals and the noble gases. We also calculate the dissociation curves of various 1D diatomics and study the chemical bond in H$_2^+$, HeH$^{2+}$, He$_2^{3+}$, H$_2$, HeH$^+$ and He$_2^{2+}$. We find that, unlike their 3D counterparts, 1D molecules are primarily bound by one-electron bonds. Finally, we study the chemistry of H$_3^+$ and we discuss the stability of the 1D polymer resulting from an infinite chain of hydrogen atoms.Comment: 27 pages, 7 figure

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

Many-Electron Integrals over Gaussian Basis Functions. I. Recurrence Relations for Three-Electron Integrals

Explicitly correlated F12 methods are becoming the first choice for high-accuracy molecular orbital calculations and can often achieve chemical accuracy with relatively small Gaussian basis sets. In most calculations, the many three- and four-electron integrals that formally appear in the theory are avoided through judicious use of resolutions of the identity (RI). However, for the intrinsic accuracy of the F12 wave function to not be jeopardized, the associated RI auxiliary basis set must be large. Here, inspired by the Head-Gordon-Pople and PRISM algorithms for two-electron integrals, we present an algorithm to directly compute three-electron integrals over Gaussian basis functions and a very general class of three-electron operators without invoking RI approximations. A general methodology to derive vertical, transfer, and horizontal recurrence relations is also presented

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

High-density correlation energy expansion of the one-dimensional uniform electron gas

We show that the expression of the high-density (i.e small-$r_s$) correlation energy per electron for the one-dimensional uniform electron gas can be obtained by conventional perturbation theory and is of the form \Ec(r_s) = -\pi^2/360 + 0.00845 r_s + ..., where $r_s$ is the average radius of an electron. Combining these new results with the low-density correlation energy expansion, we propose a local-density approximation correlation functional, which deviates by a maximum of 0.1 millihartree compared to the benchmark DMC calculations.Comment: 7 pages, 2 figures, 3 tables, accepted for publication in J. Chem. Phy