Exchange-correlation energy: From LDA to GGA and beyond

Abstract

The local (spin) density approximation (LDA) [3], having long been the standard choice, provides a realistic description of the atomic structure, elastic and vibrational properties for a wide range of systems. Yet the LDA is not reliable enough to describe the energetics of chemical reactions, say, reaction enthalpies and activation energy barriers. In particular, it leads to binding energies of molecules and solids which are overestimated on the order of $\approx 1$~eV/atom. Also several cases are documented where the LDA puts molecular conformations or crystal bulk phases in an even qualitatively wrong energetic order.

The more recent generalized gradient approximations (GGA's) [4] have overcome such deficiencies to a large extent, giving, for instance, a more realistic account of energy barriers and adsorption energies for molecules on metal or semiconductor surfaces. At the same time GGA's are computationally as simple to use as the LDA. On the other hand, the results of many applications as well as formal analysis suggest [5] that GGA functionals are still too limited to yield a fully consistent improvement over the LDA and to describe binding energies within the desired "chemical accuracy" (better than 1 kcal/mol or 50 meV/atom) in general.

In this lecture we review present formulations of the GGA and compare them with the LDA, examine general trends and specific examples to illustrate what should or should not be expected from them. To complement this "phenomenological" view, we consider the exact exchange-correlation functional, in terms of the exchange-correlation hole and the adiabatic connection, and discuss how important aspects, for instance sum rules, are retained in approximate functionals like the LDA and GGA. In addition we look at orbital-dependent exchange-correlation functionals (such as exact Kohn-Sham exchange [6], hybrid functionals [7], Meta GGA's [8], and functionals based on the fluctuation-dissipation theorem [9]) that are being explored in the quest for ever more accurate and more widely applicable exchange-correlation functionals.

Literature

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