Meta-Local Density Functionals: A New Rung on Jacob’s Ladder

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 σ , 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 geometric average n σ avg ( r ) = n σ 1– x ( r ) ñ σ x ( r ) of the local spin density n σ ( r ) and the spin density ñ σ ( r ) of a HEG that has the local kinetic energy density τ σ ( r ) of the inhomogeneous system. 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 [ AIP Conf. Proc. 2001 577 1 ]. The first functional of this ladder, the local τ approximation (LTA) of Ernzerhof and Scuseria [ J. Chem. Phys. 1999 111 911 ] 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 [ 25494732 J. Chem. Phys. 2014 141 224107 ]. 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.

Big data benchmarking: how do DFT methods across the rungs of Jacob's ladder perform for a dataset of 122k CCSD(T) total atomization energies?

“…Let us begin by examining the performance of the GGA methods BLYP, PBE, and B97-D. Of these, the moderately parameterized, dispersion-corrected B97-D functional gives the best performance with an RMSD of 9.05 and a MAD of 8.03 kcal mol À1 . Consistent with previous benchmark studies for TAEs, 20,21,75,76 the nonempirical PBE functional shows exceptionally poor performance with RMSD E MAD E 80 kcal mol À1 . Table 2 lists the total number of positive and negative deviations -for PBE, all deviations but one are positive.…”

Atomic Electronic Structure Calculations with Hermite Interpolating Polynomials

“…For this reason, we will only discuss τ-dependent functionals in this work. We have previously described successful calculations with τ-dependent meta-GGA functionals with LIP basis sets, ,, and we will critically study such calculations here.…”

“…Although LIPs are only functions and thereby do not guarantee derivatives to be continuous across element boundaries, we have demonstrated in a variety of studies performed at the density functional and HF levels of theory 13 , 15 − 18 , 20 − 22 that the total energy converges smoothly to the complete basis set limit when more elements are added in the calculation. The rationale for this behavior is that the kinetic energy term in the Hamiltonian imposes penalties on discontinuous derivatives across boundaries.…”

“…We have shown that the FEM approach used in HelFEM routinely affords sub- μE h total energies for Hartree–Fock (HF) and density functional calculations with local density approximations (LDAs), generalized gradient approximations (GGAs) as well as meta-GGA functionals. ,− ,− We have also shown that range-separated functionals can be implemented within the same scheme …”