On the Kirzhnits gradient expansion in two dimensions

TL;DR

This paper derives the semiclassical Kirzhnits expansion in two dimensions, revealing that all gradient corrections to the density and kinetic energy vanish, but the exchange energy correction diverges, aligning with prior linear-response findings.

Contribution

The paper provides the first derivation of the 2D Kirzhnits expansion and demonstrates the vanishing of gradient corrections in 2D, while confirming divergence of exchange energy corrections.

Findings

Gradient corrections to density and kinetic energy vanish in 2D.

The Kirzhnits expansion satisfies the Gross and Proetto consistency criterion.

Gradient correction to exchange energy diverges in 2D.

Abstract

We derive the semiclassical Kirzhnits expansion of the D-dimensional one-particle density matrix up to the second order in $\hbar$. We focus on the two-dimensional (2D) case and show that all the gradient corrections both to the 2D one-particle density and to the kinetic energy density vanish. However, the 2D Kirzhnits expansion satisfies the consistency criterion of Gross and Proetto [J. Chem. Theory Comput. 5, 844 (2009)] for the functional derivatives of the density and the noninteracting kinetic energy with respect to the Kohn-Sham potential. Finally we show that the gradient correction to the exchange energy diverges in agreement with the previous linear-response study.