John P. Perdew

Restoring the density-gradient expansion for exchange in solids and surfaces.

Popular modern generalized gradient approximations are biased toward the description of free-atom energies. Restoration of the first-principles gradient expansion for exchange over a wide range of density gradients eliminates this bias. We introduce a revised Perdew-Burke-Ernzerhof generalized gradient approximation that improves equilibrium properties of densely packed solids and their surfaces.Successful modern generalized gradient approximations (GGAs) are biased toward atomic energies. Restoration of the first-principles gradient expansion for the exchange energy over a wide range of density gradients eliminates this bias. With many collaborators, I introduce PBEsol, a revised Perdew-Burke-Ernzerhof GGA that improves equilibrium properties of densely-packed solids and their surfaces.

Rationale for mixing exact exchange with density functional approximations

Density functional approximations for the exchange‐correlation energy EDFAxc of an electronic system are often improved by admixing some exact exchange Ex: Exc≊EDFAxc+(1/n)(Ex−EDFAx). This procedure is justified when the error in EDFAxc arises from the λ=0 or exchange end of the coupling‐constant integral ∫10 dλ EDFAxc,λ. We argue that the optimum integer n is approximately the lowest order of Gorling–Levy perturbation theory which provides a realistic description of the coupling‐constant dependence Exc,λ in the range 0≤λ≤1, whence n≊4 for atomization energies of typical molecules. We also propose a continuous generalization of n as an index of correlation strength, and a possible mixing of second‐order perturbation theory with the generalized gradient approximation.

Comparative assessment of a new nonempirical density functional: Molecules and hydrogen-bonded complexes

A comprehensive study is undertaken to assess the nonempirical meta-generalized gradient approximation (MGGA) of Tao, Perdew, Staroverov, and Scuseria (TPSS) against 14 common exchange-correlation energy functionals. Principal results are presented in the form of statistical summaries of deviations from experiment for the G3/99 test set (223 enthalpies of formation, 86 ionization potentials, 58 electron affinities, 8 proton affinities) and three additional test sets involving 96 bond lengths, 82 harmonic vibrational frequencies, and 10 hydrogen-bonded complexes, all computed using the 6-311++G(3df,3pd) basis. The TPSS functional matches, or exceeds in accuracy all prior nonempirical constructions and, unlike semiempirical functionals, consistently provides a high-quality description of diverse systems and properties. The computational cost of self-consistent MGGA is comparable to that of ordinary GGA, and exact exchange (unavailable in some codes) is not required. A one-parameter global hybrid version of ...

The exchange-correlation energy of a metallic surface

Abstract The exchange-correlation energy of a metal surface is analyzed in terms of the wavelength of the fluctuations which contribute to it. A scheme is proposed to interpolate between the short-wavelength region properly described by the local density functional approximation and the long-wavelength region for which an exact limiting form is established. This interpolation scheme is tested and verified for the soluble infinite barrier model and then applied to realistic density profiles.

Tests of functionals for systems with fractional electron number

In the exact theory, the ground state energy of an open system varies linearly when the electron number is changed between two adjacent integers. This linear dependence is not reproduced by common approximate density functionals. Deviation from linearity in this dependence has been suggested as a basis for the concept of many-electron self-interaction error (SIE). In this paper, we quantify many-electron SIE of a number of approximations by performing calculations on fractionally charged atoms. We demonstrate the direct relevance of these studies to such problems of common approximate functionals as instabilities of anions, spurious fractional charges on dissociated atoms, and poor description of charge transfer. Semilocal approximations have the largest many-electron SIE, which is only slightly reduced in typical global hybrids. In these approximations the energy versus fractional electron number curves upward, while in Hartree-Fock theory the energy curves downward. Perdew-Zunger self-interaction correction [Phys. Rev. B 23, 5048 (1981)] significantly reduces the many-electron SIE of semilocal functionals but impairs their accuracy for equilibrium properties. In contrast, a long-range corrected hybrid functional can be nearly many-electron SIE-free in many cases (for reasons we discuss) and at the same time performs remarkably well for many molecular properties.

Jacob’s ladder of density functional approximations for the exchange-correlation energy

The ground-state energy and density of a many-electron system are often calculated by Kohn-Sham density functional theory. We describe a ladder of approximations for the exchange-correlation energy as a functional of the electron density. At the lowest rung of this ladder, the contribution to the energy from a volume element of 3-dimensional space is determined by the local density there. Higher rungs or levels incorporate increasingly complex ingredients constructed from the density or the Kohn-Sham orbitals in or around this volume element. We identify which additional exact conditions can be satisfied at each level, and discuss the extent to which the functionals at each level may be constructed without empirical input. We also discuss the research that remains to be done at the exact-exchange level, and present our “dreams of a final theory.” “Jacob left Beer-sheba and went toward Haran. He came to a certain place and stayed there for the night, because the sun had set. Taking one of the stones of the...

Meta-generalized gradient approximation: Explanation of a realistic nonempirical density functional

Tao, Perdew, Staroverov, and Scuseria (TPSS) have constructed a nonempirical meta-generalized gradient approximation (meta-GGA) [Phys. Rev. Lett. 91, 146401 (2003)] for the exchange-correlation energy, imposing exact constraints relevant to the paradigm densities of condensed matter physics and quantum chemistry. Results of their extensive tests on molecules, solids, and solid surfaces are encouraging, suggesting that this density functional achieves uniform accuracy for diverse properties and systems. In the present work, this functional is explained and details of its construction are presented. In particular, the functional is constructed to yield accurate energies under uniform coordinate scaling to the low-density or strong-interaction limit. Its nonlocality is displayed by plotting the factor F(xc) that gives the enhancement relative to the local density approximation for exchange. We also discuss an apparently harmless order-of-limits problem in the meta-GGA. The performance of this functional is investigated for exchange and correlation energies and shell-removal energies of atoms and ions. Non-self-consistent molecular atomization energies and bond lengths of the TPSS meta-GGA, calculated with GGA orbitals and densities, agree well with those calculated self-consistently. We suggest that satisfaction of additional exact constraints on higher rungs of a ladder of density functional approximations can lead to further progress.

Assessing the performance of recent density functionals for bulk solids

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 zeropoint 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.

Generalized gradient approximation to the angle- and system-averaged exchange hole

A simple analytic model is proposed for the angle- and system-averaged exchange hole of a many-electron system. The model hole depends on the local density and density gradient. It recovers a nonoscillatory local-spin density (LSD) approximation to the exchange hole for a vanishing density gradient. The model hole reproduces the exchange energy density of the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) for exchange, and facilitates a detailed understanding of the PBE GGA. The hole model is applied to atoms and molecules, and a comparison is made to exact and LSD angle- and system-averaged exchange holes. We find that the GGA hole model significantly improves upon the LSD model. Furthermore, the GGA hole model accurately describes the change in the exchange hole upon the formation of single bonds, but is less accurate for the formation of multiple bonds, where it misses the appearance of a long-range tail.

Generalized gradient approximations for exchange and correlation : a look backward and forward

Abstract Ground-state properties of many-electron systems are calculable from the exchange-correlation energy as a functional of the electron spin densities n↑ (r), n↓ (r). After a review of the local spin density (LSD) approximation, the second-order density-gradient expansion (GEA), and the generalized gradient approximation (GGA) for exchange, prospects are discussed for a simple, first-principles GGA for correlation. The new GGA for correlation, like that for exchange, is based upon a real-space cut-off of the spurious long-range part of the gradient expansion for the fluctuation hole around an electron. Scaling relations, which lead to simple functionals, are stressed.