In Kohn-Sham density functional theory for the ground-state energy and density of a many-electron system, only the exchange-correlation energy as a functional of the density needs to be approximated in practice. The simplest approximation takes the exchange-correlation energy density at position to be a function of just the local electron density (or more generally the local spin densities) there, chosen to make this approximation exact for an electron gas of uniform density. For real atoms, molecules and solids, this looked to its proposers (Kohn and Sham 1965) like a very crude approximation. Yet early tests for bulk solids, surfaces, molecules and atoms showed that it was far more accurate than they expected. Looking for an explanation in the mid 1970’s, Langreth and Perdew as well as Gunnarsson and Lundqvist found that LDA satisfies hidden exact constraints on the exchange-correlation hole that surrounds an electron at . That was the root of the “constraint-satisfaction” approach to the construction of density-functional approximations. This approach will be explained in three lectures, at three levels of functional complexity.
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