Atomic and molecular intracules for excited states

Intracules in position space, momentum space and phase space have been calculated for low-lying excited states of the He atom, Be atom, formaldehyde and butadiene. The phase-space intracules (Wigner intracules) provide significantly more information than the position- and momentum-space intracules, particularly for the Be atom. Exchange effects are investigated through the differences between corresponding singlet and triplet states.This work was supported by the Engineering and Physical Sciences Research Council through the award of an Advanced Research Fellowship (GR/R77636) to NAB and a Joint Research Equipment Initiative grant (GR/R62052)

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.

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

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

Resolutions of the Coulomb operator: VI. Computation of auxiliary integrals

We discuss the efficient computation of the auxiliary integrals that arise when resolutions of two-electron operators (specifically, the Coulomb and long-range Ewald operators) are employed in quantum chemical calculations. We derive a recurrence relation that facilitates the generation of auxiliary integrals for Gaussian basis functions of arbitrary angular momentum and propose a near-optimal algorithm for its use

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

A Remarkable Identity Involving Bessel Functions

We consider a new identity involving integrals and sums of Bessel functions. The identity provides new ways to evaluate integrals of products of two Bessel functions. The identity is remarkably simple and powerful since the summand and integrand are of exactly the same form and the sum converges to the integral relatively fast for most cases. A proof and numerical examples of the identity are discussed.Comment: 10 pages, 2 figure