Successful modern generalized gradient approximations (GGA's) are biased toward atomic energies. Restoration of the first-principles gradient expansion for exchange over a wide range of density gradients eliminates this bias. We introduce PBEsol, a revised Perdew-Burke-Ernzerhof GGA that improves equilibrium properties of densely-packed solids and their surfaces.
Table I of this article, and Table SVII of its Supplementary Information, reported imprecise revTPSS meta-GGA cohesive energies for some of the 9 nontransition metals and 6 insulators.The revTPSS cohesive energies in the original Table SVII for the metals were computed from BAND, and those for some insulators were computed from VASP. LiCl, NaCl, and MgO were also calculated from BAND. Converged BAND and VASP calculations agreed rather well, but there were different technical difficulties for each code. For technical reasons, the nonself-consistent revTPSS values from VASP used PBE orbitals, while those from BAND used LSDA orbitals. Recently, we have discovered that the VASP calculations required improved PAWs for meta-GGA calculations and that the BAND free atomic energies were not low enough for many of the solids. These errors produced too-large cohesive energies in both VASP and BAND. Here we report improved BAND results, using LSDA orbitals.The ME and MAE values for revTPSS in Table I change to À0:03 and 0:12 eV=atom, respectively. This makes the cohesive energies slightly less accurate from revTPSS than from PBE, in comparison with the experimental values given in the Supplementary Information. While the present non-self-consistent BAND and self-consistent VASP cohesive energies agree well in most of the cases, relatively large (more than 0.1 eV) discrepancies can be observed for LiCl, NaCl, and MgO. We attribute these discrepancies to the too-high free atomic energies for Cl and O given by BAND. The more precise VASP results show slightly worse ME and MAE values for revTPSS than the BAND results do.
Semilocal density functionals for the exchange-correlation energy are needed for large electronic systems. The Tao-Perdew-Staroverov-Scuseria (TPSS) meta-generalized gradient approximation (meta-GGA) is semilocal and usefully accurate, but predicts too-long lattice constants. Recent "GGA's for solids" yield good lattice constants but poor atomization energies of molecules. We show that the construction principle for one of them (restoring the density gradient expansion for exchange over a wide range of densities) can be used to construct a "revised TPSS" meta-GGA with accurate lattice constants, surface energies, and atomization energies for ordinary matter.
Some fundamental issues in ground-state density functional theory are discussed without equations: (1) The standard Hohenberg-Kohn and Kohn-Sham theorems were proven for a Hamiltonian that is not quite exact for real atoms, molecules, and solids. (2) The density functional for the exchange-correlation energy, which must be approximated, arises from the tendency of electrons to avoid one another as they move through the electron density. (3) In the absence of a magnetic field, either spin densities or total electron density can be used, although the former choice is better for approximations. (4) "Spin contamination" of the determinant of Kohn-Sham orbitals for an open-shell system is not wrong but right. (5) Only to the extent that symmetries of the interacting wave function are reflected in the spin densities should those symmetries be respected by the Kohn-Sham noninteracting or determinantal wave function. Functionals below the highest level of approximations should however sometimes break even those symmetries, for good physical reasons. (6) Simple and commonly used semilocal (lower-level) approximations for the exchange-correlation energy as a functional of the density can be accurate for closed systems near equilibrium and yet fail for open systems of fluctuating electron number. (7) The exact Kohn-Sham noninteracting state need not be a single determinant, but common approximations can fail when it is not. (8) Over an open system of fluctuating electron number, connected to another such system by stretched bonds, semilocal approximations make the exchange-correlation energy and hole-density sum rule too negative. (9) The gap in the exact Kohn-Sham band structure of a crystal underestimates the real fundamental gap but may approximate the first exciton energy in the large-gap limit. (10) Density functional theory is not really a mean-field theory, although it looks like one. The exact functional includes strong correlation, and semilocal approximations often overestimate the strength of static correlation through their semilocal exchange contributions. (11) Only under rare conditions can excited states arise directly from a ground-state theory.
We use the asymptotic expansions of the semiclassical neutral atom as a reference system in density functional theory to construct accurate generalized gradient approximations (GGAs) for the exchange-correlation and kinetic energies without any empiricism. These asymptotic functionals are among the most accurate GGAs for molecular systems, perform well for solid state, and overcome current GGA state of the art in frozen density embedding calculations. Our results also provide evidence for the conjointness conjecture between exchange and kinetic energies of atomic systems.
We construct a Laplacian-level meta-generalized gradient approximation (meta-GGA) for the non-interacting (Kohn-Sham orbital) positive kinetic energy density τ of an electronic ground state of density n. This meta-GGA is designed to recover the fourth-order gradient expansion τ GE4 in the appropiate slowly-varying limit and the von Weizsäcker expression τ W = |∇n| 2 /(8n) in the rapidlyvarying limit. It is constrained to satisfy the rigorous lower bound τ W (r) ≤ τ (r). Our meta-GGA is typically a strong improvement over the gradient expansion of τ for atoms, spherical jellium clusters, jellium surfaces, the Airy gas, Hooke's atom, one-electron Gaussian density, quasi-two dimensional electron gas, and nonuniformly-scaled hydrogen atom. We also construct a Laplacian-level meta-GGA for exchange and correlation by employing our approximate τ in the Tao, Perdew, Staroverov and Scuseria (TPSS) meta-GGA density functional. The Laplacian-level TPSS gives almost the same exchange-correlation enhancement factors and energies as the full TPSS, suggesting that τ and ∇ 2 n carry about the same information beyond that carried by n and ∇n. Our kinetic energy density integrates to an orbital-free kinetic energy functional that is about as accurate as the fourth-order gradient expansion for many real densities (with noticeable improvement in molecular atomization energies) , but considerably more accurate for rapidly-varying ones.
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