Descriptions of exchange and correlation effects in inhomogeneous electron systems

TLDR

The paper discusses approximate descriptions of exchange‑correlation effects in inhomogeneous electron systems, building on Kohn‑Sham density‑functional theory and highlighting limitations of earlier gradient‑expansion approaches. The authors aim to introduce two explicit XC energy functionals that are exact in specific density regimes—one for nearly constant densities with rapid variations and another for simple limits. They construct the functionals by deriving two explicit forms from the XC‑hole formula, one exact for almost constant densities with rapid variations and another exact in simple limits. The new functionals retain the computational simplicity of the local‑density approximation, yield correct asymptotic r⁻¹ and z⁻¹ potentials, and reduce total‑energy errors by roughly an order of magnitude compared with the LD approximation, though they still exhibit shortcomings in surface image‑potential modeling.

Abstract

Starting from a formula relating the exchange-correlation (XC) energy of the Kohn-Sham density-functional formalism to the XC hole, we discuss some general but approximate descriptions of XC effects in inhomogeneous electron systems, in particular valence electrons, using homogeneous-electron-gas data as input. The new descriptions have all the virtues of the local-density (LD) approximation, including the computational simplicity of a local XC potential, and it reduces to the latter in the proper limit. In addition, they have a physically motivated nonlocal dependence on the electron density, which results in such desirable features as an asymptotical ${r}^{\ensuremath{-}1}$ behavior far away from, e.g., atoms and a ${z}^{\ensuremath{-}1}$ behavior of the potential outside solid surfaces. We present two explicit forms of the XC energy functional, one which is exact for a system with almost constant density but with possibly spatially rapid variations, and another which is exact in some simple limits. Illustrations on atoms show them to reduce the error in the total energy by about one order of magnitude compared with the LD approximation. Applications to surfaces show a reasonable modeling of the image-potential effect but also illustrate shortcomings of the approximations. We also point out shortcomings of two earlier methods to extend the LD approximation, the gradient expansion, and the expansion to second order in the density variations, when they are applied to inhomogeneous systems.