Momentum distribution of the uniform electron gas: improved parametrization and exact limits of the cumulant expansion

The momentum distribution of the unpolarized uniform electron gas in its Fermi-liquid regime, n(k,r_s), with the momenta k measured in units of the Fermi wave number k_F and with the density parameter r_s, is constructed with the help of the convex Kulik function G(x). It is assumed that $n(0,r_s), n(1^\pm, r_s)$, the on-top pair density g(0,r_s) and the kinetic energy t(r_s) are known (respectively, from effective-potential calculations, from the solution of the Overhauser model, and from Quantum Monte Carlo calculations via the virial theorem). Information from the high- and the low-density limit, corresponding to the random-phase approximation and to the Wigner crystal limit, is used. The result is an accurate parametrization of n(k,r_s), which fulfills most of the known exact constraints. It is in agreement with the effective-potential calculation of Takada and Yasuhara [1991 {\it Phys. Rev.} B {\bf 44} 7879], is compatible with Quantum Monte Carlo data, and is valid in the density range $r_s \lesssim 12$. The corresponding cumulant expansions of the pair density and of the static structure factor are discussed, and some exact limits are derived

Study of the discontinuity of the exchange-correlation potential in an exactly soluble case

It was found by Perdew, Parr, Levy, and Balduz [Phys. Rev. Lett. {\bf 49}, 1691 (1982)] and by Sham and Schl\"uter [Phys. Rev. Lett. {\bf 51}, 1884 (1983)] that the exact Kohn-Sham exchange-correlation potential of an open system may jump discontinuosly as the particle number crosses an integer, with important physical consequences. Recently, Sagvolden and Perdew [Phys. Rev. A {\bf 77}, 012517 (2008)] have analyzed the discontinuity of the exchange-correlation potential as the particle number crosses one, with an illustration that uses a model density for the H$^-$ ion. In this work, we extend their analysis to the case in which the external potential is the simple harmonic confinement, choosing spring-constant values for which the two-electron hamiltonian has an analytic solution. This way, we can obtain the exact, analytic, exchange and correlation potentials for particle number fluctuating between zero and two, illustrating the discontinuity as the particle number crosses one without introducing any model or approximation. We also discuss exchange and correlation separately.Comment: Submitted to Int. J. Quantum Chem., special issue honoring Prof. Mayer. New version, where an important error has been correcte

Range separation combined with the Overhauser model: Application to the H2_2 molecule along the dissociation curve

The combination of density-functional theory with other approaches to the many-electron problem through the separation of the electron-electron interaction into a short-range and a long-range contribution (range separation) is a successful strategy, which is raising more and more interest in recent years. We focus here on a range-separated method in which only the short-range correlation energy needs to be approximated, and we model it within the "extended Overhauser approach". We consider the paradigmatic case of the H$_2$ molecule along the dissociation curve, finding encouraging results. By means of very accurate variational wavefunctions, we also study how the effective electron-electron interaction appearing in the Overhauser model should be in order to yield the exact correlation energy for standard Kohn-Sham density functional theory.Comment: submitted to Int. J. Quantum Chem., special issue dedicated to Prof. Hira

London dispersion forces without density distortion: a path to first principles inclusion in density functional theory

We analyse a path to construct density functionals for the dispersion interaction energy from an expression in terms of the ground state densities and exchange-correlation holes of the isolated fragments. The expression is based on a constrained search formalism for a supramolecular wavefunction that is forced to leave the diagonal of the many-body density matrix of each fragment unchanged, and is exact for the interaction between one-electron densities. We discuss several aspects: the needed features a density functional approximation for the exchange-correlation holes of the monomers should have, the optimal choice of the one-electron basis needed (named "dispersals"), and the functional derivative with respect to monomer density variations.Comment: 12 pages, 4 figure

Challenging the Lieb-Oxford Bound in a systematic way

The Lieb-Oxford bound, a nontrivial inequality for the indirect part of the many-body Coulomb repulsion in an electronic system, plays an important role in the construction of approximations in density functional theory. Using the wavefunction for strictly-correlated electrons of a given density, we turn the search over wavefunctions appearing in the original bound into a more manageable search over electron densities. This allows us to challenge the bound in a systematic way. We find that a maximizing density for the bound, if it exists, must have compact support. We also find that, at least for particle numbers $N\le 60$, a uniform density profile is not the most challenging for the bound. With our construction we improve the bound for $N=2$ electrons that was originally found by Lieb and Oxford, we give a new lower bound to the constant appearing in the Lieb-Oxford inequality valid for any $N$, and we provide an improved upper bound for the low-density uniform electron gas indirect energy.Comment: accepted in Mol. Phys. in the special issue in honour of Andreas Savin; revised version with new calculation