The local spin-density (LSD) functional and Perdew–Wang 91 (PW91) generalized gradient approximations to atomization energies of molecules are investigated. We discuss the coupling-constant dependence of the atomization energy and why exchange errors of the functionals are greater than exchange–correlation errors. This fact helps to justify hybrid schemes which mix some exact exchange with density functional approximations for exchange and correlation. It is shown that the biggest errors in the atomization energies occur when there is a strong interaction between different electron pairs, which vanishes upon atomization. We argue that the amount of exchange character of a molecular property, such as the atomization energy, depends on the property itself. We define an exact mixing coefficient b, which measures this exchange character, and show that both LSD and PW91 typically overestimate this quantity. Thus, nonempirical hybrid schemes which approximate this quantity by its LSD or PW91 value typically do not improve the exchange–correlation energy. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64: 285–295, 1997
The ground-state energy, electron density, and related properties of ordinary matter can be computed efficiently when the exchange-correlation energy as a functional of the density is approximated semilocally. We propose the first meta-generalized-gradient approximation (meta-GGA) that is fully constrained, obeying all 17 known exact constraints that a meta-GGA can. It is also exact or nearly exact for a set of "appropriate norms," including rare-gas atoms and nonbonded interactions. This strongly constrained and appropriately normed meta-GGA achieves remarkable accuracy for systems where the exact exchange-correlation hole is localized near its electron, and especially for lattice constants and weak interactions.
We present a self-consistent calculation of the ground state of the metallic planar surface in which the discrete-lattice perturbation $\ensuremath{\delta}v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is treated variationally, with use of a single variational parameter, rather than perturbatively. Our calculation, which reduces to the perturbation theory of Lang and Kohn in the limit of weak $\ensuremath{\delta}v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ and retains most of the simplicity and broad utility of their approach, shows that for many metal surfaces $\ensuremath{\delta}v(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is not a weak perturbation: The electron density profiles at real metal surfaces are often unlike those of the jellium model and in fact show a strong dependence on the choice of exposed crystallographic face. This face dependence, which is rather simply related to the average value ${〈\ensuremath{\delta}v〉}_{\mathrm{av}}$ of the discrete-lattice perturbation over the volume of the semi-infinite crystal may have important consequences in the calculation of many surface-related properties, including chemisorption, to which our method is easily applicable. We calculate the face-dependent surface energies, density profiles, and work functions of nine simple metals. Calculated surface energies, including the correction to the local-density approximation for exchange and correlation given by the method of wave-vector analysis, are in good agreement with measured surface tensions for those seven of the nine metals in which the ionic pseudopotential gives a good account of the bulk binding energy. Problems still to be considered include improvements in the pseudopotential, lattice relaxation at the surface, and variation of the electron density over planes parallel to the surface.
First principles predictions of lattice dynamics are of vital importance for a broad range of topics in materials science and condensed matter physics. The large-scale nature of lattice dynamics calculations and the desire to design novel materials with distinct properties demands that first principles predictions are accurate, transferable, efficient, and reliable for a wide variety of materials. In this work, we demonstrate that the recently constructed r2SCAN density functional approximation meets this need for general systems by demonstrating phonon dispersions for typical 12 systems with distinct chemical characteristics. The approximation’s performance opens a door for phonon-mediated materials discovery from first principles calculations.
We propose a model for the angle- and system-averaged exchange-correlation hole of a many-electron system. This hole analyzes the exchange-correlation energy into contributions of various distances $u$ from an electron. The model is ``reverse-engineered'' (derived from and not used to derive a density functional). It satisfies known exact hole constraints, including ones that can only be satisfied by a meta-generalized gradient approximation or meta-GGA. It incorporates the exchange-correlation energy density of the Tao-Perdew-Staroverov-Scuseria (TPSS) nonempirical meta-GGA. The hole model is tested for atoms and applied to jellium surfaces. The Fourier transform $(u\ensuremath{\rightarrow}k)$ of the hole is needed for wave-vector interpolation of the jellium surface energy from an exact small-$k$ or large-$u$ asymptote. We find essentially the same surface energies (close to the uncorrected TPSS values) whether we apply the wave-vector interpolation correction to the local spin density approximation, the GGA or the meta-GGA. These and other considerations suggest that these surface energies are accurate. Moreover, we find that the uncorrected TPSS surface energies have a realistic wave-vector analysis. Our TPSS hole model can be used to build the hole model for a TPSS-based global hybrid functional, or for a hyper-GGA that uses full exact exchange.
