The long-wavelength behaviour of the exchange-correlation kernel in the Kohn-Sham theory of periodic systems

TL;DR

This paper investigates the divergence behavior of the exchange-correlation kernel in periodic insulators within Kohn-Sham density-functional theory, highlighting differences in dielectric constant definitions and the limitations of common approximations.

Contribution

It reveals the ${ m O}(1/q^2)$ divergence in the XC kernel for small q and compares the dielectric constant definitions, emphasizing the limitations of local and semi-local functionals.

Findings

The XC kernel diverges as ${ m O}(1/q^2)$ for small q in insulators.

Test-charge and electronic dielectric constants differ in exact KS theory.

Common approximations do not capture the divergence behavior.

Abstract

The polarization-dependence of the exchange-correlation (XC) energy functional of periodic insulators within Kohn-Sham (KS) density-functional theory requires a ${\cal O} (1/q^2)$ divergence in the XC kernel for small vectors q. This behaviour, exemplified for a one-dimensional model semiconductor, is also observed when an insulator happens to be described as a KS metal, or vice-versa. Although it can occur in the exchange-only kernel, it is not found in the usual local, semi-local or even non-local approximations to KS theory. We also show that the test-charge and electronic definitions of the macroscopic dielectric constant differ from one another in exact KS theory, but are equivalent in the above-mentioned approximations.