The Hohenberg-Kohn density functional was long ago shown to reduce to the Thomas-Fermi approximation in the non-relativistic semiclassical (or large-Z) limit for all matter, i.e, the kinetic energy becomes local. Exchange also becomes local in this limit. Numerical data on the correlation energy of atoms supports the conjecture that this is also true for correlation, but much less relevant to atoms. We illustrate how expansions around large particle number are equivalent to local density approximations and their strong relevance to density functional approximations. Analyzing highly accurate atomic correlation energies, we show that E C → −A C Z ln Z + B C Z as Z → ∞, where Z is the atomic number, A C is known, and we estimate B C to be about 37 millihartrees. The local density approximation yields A C exactly, but a very incorrect value for B C , showing that the local approximation is less relevant for correlation alone. This limit is a benchmark for the non-empirical construction of density functional approximations. We conjecture that, beyond atoms, the leading correction to the local density approximation in the large-Z limit generally takes this form, but with B C a functional of the TF density for the system. The implications for construction of approximate density functionals are discussed.

Recent Quantum Monte Carlo data for the exchange‐correlation energy density of pseudopotential systems strongly suggest the value of using the Laplacian of the density as a variable for constructing first‐order corrections to the local density approximation of density functional theory. We report on an exchange functional built on these observations and extended to the all‐electron case. The model keeps the typical properties of constraint‐based generalized gradient approximations (GGAs) and also has a finite‐valued potential at the nucleus, unlike the GGA. Problems with oscillatory behavior in the potential due to higher order derivatives are controlled by a curvature minimization constraint. The results are tested against exact potentials for the He and Ne atom. A combination of gradient and Laplacian as suggested by a gradient expansion of the exchange hole gives the best overall results. © 2012 Wiley Periodicals, Inc.

We consider the implications of the Lieb-Simon limit for correlation in density functional theory. In this limit, exemplified by the scaling of neutral atoms to large atomic number, local density approximation (LDA) becomes relatively exact, and the leading correction to this limit for correlation has recently been determined for neutral atoms. We use the leading correction to the LDA and the properties of the real-space cutoff of the exchange-correlation hole to design, based upon Perdew-Burke-Ernzerhof (PBE) correlation, an asymptotically corrected generalized gradient approximation (acGGA) correlation which becomes more accurate per electron for atoms with increasing atomic number. When paired with a similar correction for exchange, this acGGA satisfies more exact conditions than PBE. Combined with the known -dependence of the gradient expansion for correlation, this correction accurately reproduces correlation energies of closed-shell atoms down to Be. We test this acGGA for atoms and molecules, finding consistent improvement over PBE but also showing that optimal global hybrids of acGGA do not improve upon PBE0 and are similar to meta-GGA values. We discuss the relevance of these results to Jacob's ladder of non-empirical density functional construction.

We visualize the Kohn-Sham kinetic energy density (KED) and the ingredients--the electron density, its gradient, and Laplacian--used to construct orbital-free models of it, for the AE6 test set of molecules. These are compared to related quantities used in metaGGA's, to characterize two important limits--the gradient expansion and the localized-electron limit typified by the covalent bond. We find the second-order gradient expansion of the KED to be a surprisingly successful predictor of the exact KED, particularly at low densities where this approximation fails for exchange. This contradicts the conjointness conjecture that the optimal enhancement factors for orbital-free kinetic and exchange energy functionals are closely similar in form. In addition we find significant problems with a recent metaGGA-level orbital-free KED, especially for regions of strong electron localization. We define an orbital-free description of electron localization and a revised metaGGA that improves upon atomization energies significantly.

We model the exchange-correlation (XC) energy density of the Si crystal and atom as calculated by variational Monte Carlo (VMC) methods with a gradient analysis beyond the local density approximation (LDA). We find the Laplacian of the density to be an excellent predictor of the discrepancy between VMC and LDA energy densities in each system. A simple Laplacian-based correction to the LDA energy density is developed by means of a least square fit to the VMC XC energy density for the crystal, which fits the homogeneous electron gas and Si atom without further effort.