Exchange and correlation near the nucleus in density functional theory

TL;DR

This paper analyzes the behavior of the exchange-correlation potential near the nucleus in density functional theory, revealing a nonzero linear term arising from kinetic energy density differences, with implications for Kohn-Sham system construction.

Contribution

It derives an analytical expression for the linear term of the exchange-correlation potential near the nucleus and discusses its implications for density functional theory.

Findings

The linear term of the exchange-correlation potential near the nucleus is nonzero.

The linear term mainly arises from the correlation potential.

Near the nucleus, the potential is nearly quadratic due to the small linear term.

Abstract

The near nucleus behavior of the exchange-correlation potential $v_{xc}({\bf r})$ in Hohenberg-Kohn-Sham density functional theory is investigated. It is shown that near the nucleus the linear term of $O(r)$ of the spherically averaged exchange-correlation potential ${\bar v}_{xc}(r)$ is nonzero, and that it arises purely from the difference between the kinetic energy density at the nucleus of the interacting system and the noninteracting Kohn-Sham system. An analytical expression for the linear term is derived. Similar results for the exchange $v_{x}({\bf r})$ and correlation $v_{c}({\bf r})$ potentials are also obtained separately. It is further pointed out that the linear term in $v_{xc}({\bf r})$ arising mainly from $v_{c}({\bf r})$ is rather small, and $v_{xc}({\bf r})$ therefore has a nearly quadratic structure near the nucleus. Implications of the results for the construction of the Kohn-Sham system are discussed with examples.