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The local-density approximation (LDA) for the exchange-correlation potential underestimates the fundamental energy gaps of insulators by about 40%. When a simplified self-interaction correction (SSIC) is applied to the band structures of Ne and NaCl, vast improvements over LDA are found in the gaps, with little change in the valence-band widths or conduction-band structures. Because it is applied directly to the Bloch orbital representation, SSIC is very easy to incorporate into LDA computer codes. Results are also reported using the Langreth-Mehl (LM) generalized gradient correction to LDA. The LM potential, which we regard as a close approximation to the exact Kohn-Sham potential, yields band structures very close to those of LDA.
We assess the performance of recent density functionals for the exchange-correlation energy of a nonmolecular solid, by applying accurate calculations with the GAUSSIAN, BAND, and VASP codes to a test set of 24 solid metals and nonmetals. The functionals tested are the modified Perdew-Burke-Ernzerhof generalized gradient approximation (PBEsol GGA), the second-order GGA (SOGGA), and the Armiento-Mattsson 2005 (AM05) GGA. For completeness, we also test more standard functionals: the local density approximation, the original PBE GGA, and the Tao-Perdew-Staroverov-Scuseria meta-GGA. We find that the recent density functionals for solids reach a high accuracy for bulk properties (lattice constant and bulk modulus). For the cohesive energy, PBE is better than PBEsol overall, as expected, but PBEsol is actually better for the alkali metals and alkali halides. For fair comparison of calculated and experimental results, we consider the zero-point phonon and finite-temperature effects ignored by many workers. We show how GAUSSIAN basis sets and inaccurate experimental reference data may affect the rating of the quality of the functionals. The results show that PBEsol and AM05 perform somewhat differently from each other for alkali metal, alkaline-earth metal, and alkali halide crystals (where the maximum value of the reduced density gradient is about 2), but perform very similarly for most of the other solids (where it is often about 1). Our explanation for this is consistent with the importance of exchange-correlation nonlocality in regions of core-valence overlap.
Because of its useful accuracy and efficiency, density functional theory (DFT) is one of the most widely used electronic structure theories in physics, materials science, and chemistry. Only the exchange-correlation energy is unknown, and needs to be approximated in practice. Exact constraints provide useful information about this functional. The local spin-density approximation (LSDA) was the first constraint-based density functional. The Lieb-Oxford lower bound on the exchange-correlation energy for any density is another constraint that plays an important role in the development of generalized gradient approximations (GGAs) and meta-GGAs. Recently, a strongly and optimally tightened lower bound on the exchange energy was proved for one- and two-electron densities, and conjectured for all densities. In this article, we present a realistic "meta-GGA made very simple" (MGGA-MVS) for exchange that respects this optimal bound, which no previous beyond-LSDA approximation satisfies. This constraint might have been expected to worsen predicted thermochemical properties, but in fact they are improved over those of the Perdew-Burke-Ernzerhof GGA, which has nearly the same correlation part. MVS exchange is however radically different from that of other GGAs and meta-GGAs. Its exchange enhancement factor has a very strong dependence upon the orbital kinetic energy density, which permits accurate energies even with the drastically tightened bound. When this nonempirical MVS meta-GGA is hybridized with 25% of exact exchange, the resulting global hybrid gives excellent predictions for atomization energies, reaction barriers, and weak interactions of molecules.
Within density functional theory, a coordinate-scaling relation for the coupling-constant dependence of the exchange-correlation kernel ${f}_{\mathrm{xc}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\omega})$ is utilized to express the correlation energy of a many-electron system in terms of ${f}_{\mathrm{xc}}.$ As a test of several of the available approximations for the exchange-correlation kernel, or equivalently the local-field factor, we calculate the uniform-gas correlation energy. While the random phase approximation ${(f}_{\mathrm{xc}}$ $=$ 0) makes the correlation energy per electron too negative by about 0.5 eV, the adiabatic local-density approximation $[{f}_{\mathrm{xc}}$ $=$ ${f}_{\mathrm{xc}}(q$ $=$ $0,\ensuremath{\omega}$ $=$ 0)] makes a comparable error in the opposite direction. The adiabatic nonlocal approximation $[{f}_{\mathrm{xc}}$ $=$ ${f}_{\mathrm{xc}}(q,\ensuremath{\omega}$ $=$ 0)] reduces this error to about 0.1 eV, and inclusion of the full frequency dependence $[{f}_{\mathrm{xc}}$ $=$ ${f}_{\mathrm{xc}}(q,\ensuremath{\omega})]$ in an approximate parametrization reduces it further to less than 0.02 eV. We also report the wave-vector analysis and the imaginary-frequency analysis of the correlation energy for each choice of kernel.
Abstract A set of exact conditions is compiled for the purpose of developing and testing approximations for the exchange‐correlation energy as a functional of the electron density. Special emphasis is placed upon recently developed density‐scaling relationships. Commonly used generalized gradient approximations are compared against several of these conditions. A direct tabular comparison of these functionals (not of calculated properties) with one another is also made. © 1994 John Wiley & Sons, Inc.