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

Abstract

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.

Constraint-based wave vector and frequency dependent exchange-correlation kernel of the uniform electron gas

“…(36) of Ref. [31], in which C xc decreases very slowly to zero as r s increases, taking the values 0.00185, 0.00122, and 0.00015 at r s = 0, 4, and 70, respectively. This means that, at very low densities with r s 70, D will be close to 0 and the LDA kernel will be nearly correct through order ( q 2k F ) 2 .…”

Simple hydrogenic estimates for the exchange and correlation energies of atoms and atomic ions, with implications for density functional theory

“…Interestingly, a much better coefficient of Z ln Z for neutral atoms of large Z can be found by approaching from the fixed N , large Z direction, as shown in Table I and Appendix B. For the series of neutral atoms, one needs to extrapolate carefully [17,18,20] to much larger Z to approach the asymptotic limit, but that limit is clearly pre-figured in the energies of real atoms. This suggests that widely-predictive approximate functionals should be constrained to recover the correct large-Z asymptotics.…”

“…Dirac [11] added LDA exchange to the Thomas-Fermi model, and Schwinger [12] may have been the first to realize that LDA exchange becomes relatively exact for neutral atoms in the limit of large atomic number. In this limit, the bulk of the density becomes Thomas-Fermi like, with a locally-slow spatial variation [13,14], and the exact energies have large-Z asymptotic expansions [15][16][17][18]…”

“…The leading coefficients (A x = 0.221 hartree, A c = 0.021 hartree) are known [17][18][19] to be those from the LDA evaluated on the self-consistent Kohn-Sham (or Thomas-Fermi) density for large N = Z, and corrections to LDA (e.g., Refs. 9 and 10) can give higher-order coefficients.…”

“…The Perdew-Burke-Ernzerhof (PBE [9]) GGA, while it is more accurate than B88 for solids, is less accurate for molecular atomization energies and for B x . But the acGGA [18] revision of PBE, and the SCAN meta-GGA [10], both constructed to satisfy Eqs. 2 and 3 (as well as other exact constraints) without fitting to molecules, are both more accurate for atomization energies than is PBE (and SCAN is remarkably more accurate).…”