The Hartree and exchange-correlation (xc) potential in Kohn-Sham (KS) density functional theory (DFT) is a term that needs to be added to a given external potential in order to force N non-interacting fermions to have the same one-particle density of N interacting electrons in that same external potential. In KS DFT, the xc potential is the functional derivative with respect to the density of the xc energy functional.
For small systems, very accurate one-electron densities can be obtained from wave function methods, and a lot of numerical and theoretical work has been done in the past 20-25 years to compute and rationalize ``exact’’ (in the sense of accurate) xc potentials. In this lecture I will only briefly mention these inversion techniques. Rather, the main focus will be on the exact expression of the xc potential in terms of one- and two-body reduced density matrices originally derived by Baerends and coworkers. These expressions involve both correlated and KS quantities, and are derived from the equation for the square root of the density [Levy, Perdew, and Sahni, Phys. Rev. A 30, 2745 (1984)?]. They allow us to analyze the xc potential in terms of its kinetic, response, and xc hole parts, which have different properties and peculiarities. Very recently, Staroverov and coworkers have generalized these expressions in order to obtain a meaningful xc potential in a given finite basis set via a smart iterative procedure.
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