Communication: Near-locality of exchange and correlation density functionals for 1- and 2-electron systems

Abstract

The uniform electron gas and the hydrogen atom play fundamental roles in condensed matter physics and quantum chemistry. The former has an infinite number of electrons uniformly distributed over the neutralizing positively-charged background, and the latter only one electron bound to the proton. The uniform electron gas was used to derive the local spin density approximation (LSDA) to the exchange-correlation functional that undergirds the development of the Kohn-Sham density functional theory. We show here that the ground-state exchange-correlation energies of the hydrogen atom and many other 1-and 2-electron systems are modeled surprisingly well by a different local spin density approximation (LSDA0). Our LSDA0 is constructed to satisfy exact constraints, but agrees surprisingly well with the exact results for a uniform two-electron density in a finite, curved threedimensional space. We also apply LSDA0 to excited or noded 1-electron densities.Furthermore, we show that the locality of an orbital can be measured by the ratio between the exact exchange energy and its optimal lower bound.The uniform electron gas (UEG) and the hydrogen atom are two of the most important models in condensed matter physics and quantum chemistry. These models represent two opposite limits from several perspectives (e.g., extended vs. confined, and ∞ vs. 1 in electron number). When the density functional theory (DFT) 1-3 was developed, the UEG was first used to derive the local spin density approximation (LSDA) 4-6 to its exchange-correlation energy, the only part that needs to be approximated in DFT. LSDA was at first believed to be too crude for any practical applications, but it performed

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