Antonio Cancio

Beyond the local approximation to exchange and correlation : The role of the laplacian of the density in the energy density of Si

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.

Visualization and analysis of the Kohn-Sham kinetic energy density and its orbital-free description in molecules.

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 metaGGAs, 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.

Locality of correlation in density functional theory

The Hohenberg-Kohn density functional was long ago shown to reduce to the Thomas-Fermi (TF) 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 support the conjecture that this is also true for correlation, but much less relevant to atoms. We illustrate how expansions around a 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 EC → -AC ZlnZ + BCZ as Z → ∞, where Z is the atomic number, AC is known, and we estimate BC to be about 37 mhartree. The local density approximation yields AC exactly, but a very incorrect value for BC, showing that the local approximation is less relevant for the 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 BC a functional of the TF density for the system. The implications for the construction of approximate density functionals are discussed.

Visualisation and orbital-free parametrisation of the large-Z scaling of the kinetic energy density of atoms

ABSTRACT The scaling of neutral atoms to large Z, combining periodicity with a gradual trend to homogeneity, is a fundamental probe of density functional theory, one that has driven recent advances in understanding both the kinetic and exchange-correlation energies. Although research focus is normally upon the scaling of integrated energies, insights can also be gained from energy densities. We visualise the scaling of the positive-definite kinetic energy density (KED) in closed-shell atoms, in comparison to invariant quantities based upon the gradient and Laplacian of the density. We notice a striking fit of the KED within the core of any atom to a gradient expansion using both the gradient and the Laplacian, appearing as an asymptotic limit around which the KED oscillates. The gradient expansion is qualitatively different from that derived from first principles for a slowly varying electron gas and is correlated with a nonzero Pauli contribution to the KED near the nucleus. We propose and explore orbital-free meta-GGA models for the kinetic energy to describe these features, with some success, but the effects of quantum oscillations in the inner shells of atoms make a complete parametrisation difficult. We discuss implications for improved orbital-free description of molecular properties. GRAPHICAL ABSTRACT

Scaling properties of exchange and correlation holes of the valence shell of second-row atoms

We study the exchange and correlation hole of the valence shell of second-row atoms using variational Monte Carlo techniques, especially correlated estimates, and norm-conserving pseudopotentials. The well-known scaling of the valence shell provides a tool to probe the behavior of exchange and correlation as a functional of the density and thus test models of density functional theory. The exchange hole shows an interesting competition between two scaling forms\char22{}one caused by self-interaction and another that is approximately invariant under changes in the particle number, related to the known invariance of exchange under uniform scaling to high density with the particle number held constant. The correlation hole shows a scaling trend that is marked by the finite size of the atom relative to the radius of the hole. Both trends are well captured in the main by the Perdew-Burke-Ernzerhof generalized-gradient approximation model for the exchange-correlation hole and energy.

Fitting a round peg into a round hole: Asymptotically correcting the generalized gradient approximation for correlation

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 rs -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 Jacobs ladder of non-empirical density functional construction.