Meta-local density functionals: a new rung on Jacob's ladder

TL;DR

This paper introduces a new family of density functionals called meta-LDAs, derived from properties of the homogeneous electron gas, which improve upon traditional local density approximations in density-functional theory.

Contribution

The authors propose a novel meta-LDA approach based on a weighted geometric average of densities, forming a new rung on Jacob's ladder, and demonstrate its improved stability and accuracy.

Findings

Meta-LDAs are exact for the homogeneous electron gas.

A specific functional with x=0.50 yields the best atomic energies.

Meta-LDAs significantly reduce LDA overbinding in molecules.

Abstract

The homogeneous electron gas (HEG) is a key ingredient in the construction of most exchange-correlation functionals of density-functional theory. Often, the energy of the HEG is parameterized as a function of its spin density $n$, leading to the local density approximation (LDA) for inhomogeneous systems. However, the connection between the electron density and kinetic energy density of the HEG can be used to generalize the LDA by evaluating it on a weighted geometric average of the local spin density and the spin density of a HEG that has the local kinetic energy density of the inhomogeneous system, with a mixing ratio $x$. This leads to a new family of functionals that we term meta-local density approximations (meta-LDAs), which are still exact for the HEG, which are derived only from properties of the HEG, and which form a new rung of Jacob's ladder of density functionals. The first functional of this ladder, the local $\tau$ approximation (LTA) of Ernzerhof and Scuseria that corresponds to $x=1$ is unfortunately not stable enough to be used in self-consistent field calculations, because it leads to divergent potentials as we show in this work. However, a geometric averaging of the LDA and LTA densities with smaller values of $x$ not only leads to numerical stability of the resulting functional, but also yields more accurate exchange energies in atomic calculations than the LDA, the LTA, or the tLDA functional ($x=1/4$) of Eich and Hellgren. We choose $x=0.50$ as it gives the best total energy in self-consistent exchange-only calculations for the argon atom. Atomization energy benchmarks confirm that the choice $x=0.50$ also yields improved energetics in combination with correlation functionals in molecules, almost eliminating the well-known overbinding of the LDA and reducing its error by two thirds.