Simple meta-generalization of local density functionals

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 density, 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 including a fraction $x$ of the kinetic energy density. This leads to a new family of functionals that we term meta-local density approximations (meta-LDAs), which are still exact for the HEG. The first functional of this kind, the local $\tau$ approximation (LTA) of Ernzerhof and Scuseria [J. Chem. Phys. 111, 911 (1999)] is unfortunately not stable enough to be used in self-consistent field calculations. However, we show in this work that geometric averaging of the LDA and LTA densities with $x=1/2$ 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 $\tau$-LDA ($x=1/4$) of Eich and Hellgren [J. Chem. Phys. 141, 224107 (2014)]. Furthermore, atomization energy benchmarks confirm that the choice $x=1/2$ 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. Our functional form can also be used as a starting point to construct new meta-generalized gradient functionals by including further dependence on the gradient of the density.