Asymptotic Behavior of the Exchange-Correlation Energy Density and the Kohn-Sham Potential in Density Functional Theory: Exact Results and Strategy for Approximations

Density functional theory (DFT) is nowadays the leading theoretical framework for quantum description of materials from first principles. Being an exact theory on one hand and computationally efficient on the other hand, DFT allows to address large and complex many-electron systems and accurately predict their properties. The predictive power of DFT critically depends though on an accurate approximation to the generally unknown exchange-correlation (xc) energy functional. Approximations can be constructed from first principles by satisfying known properties of the exact functional. In this work I review two such exact properties: the asymptotic behavior of the xc energy density per particle and the asymptotic behavior of the Kohn-Sham potential, in finite many-electron systems. The derivation of the asymptotic forms for both quantities is reviewed, employing the concepts of the adiabatic connection and of the xc hole with relation to the first quantity and the exact electron factorization approach for the second one. Furthermore, it is shown that the correct asymptotic behavior of one of the aforementioned quantities does not guarantee a correct behavior of the other. These quantities are related via the xc hole response function, which is defined, examined and its exact exchange part is analytically derived. The extent to which existing xc approximations satisfy the named exact properties is reviewed and the relationship between correct asymptotics and freedom from one-electron self-interaction in DFT is discussed. Finally, a strategy for development of advanced approximations for exchange and correlation with a correct asymptotic behavior is suggested.