Relevance of coordinate and particle-number scaling in density functional theory

TL;DR

This paper explores a family of density scalings in density functional theory, highlighting their importance in semiclassical and quantum electronic systems, and emphasizing their role as constraints in developing new functionals.

Contribution

It introduces and analyzes a family of density scalings parameterized by $eta$ and $\lambda$, demonstrating their significance in various physical regimes and in the construction of density functionals.

Findings

Different scalings correspond to important physical limits like semiclassical atoms and metallic clusters.

Curvature energy of metallic clusters relates to second-order gradient expansion.

Scaling properties serve as constraints for developing improved density functionals.

Abstract

We discuss a $\beta$-dependent family of electronic density scalings of the form $n_\lambda(\R)=\lambda^{3\beta+1}\; n(\lambda^\beta \R)$ in the context of density functional theory. In particular, we consider the following special cases: the Thomas-Fermi scaling ($\beta=1/3$ and $\lambda \gg 1$), which is crucial for the semiclassical theory of neutral atoms; the uniform-electron-gas scaling ($\beta=-1/3$ and $\lambda\gg 1$), that is important in the semiclassical theory of metallic clusters; the homogeneous density scaling ($\beta=0$) which can be related to the self-interaction problem in density functional theory when $\lambda \leq 1$; the fractional scaling ($\beta=1$ and $\lambda\leq 1$), that is important for atom and molecule fragmentation; and the strong-correlation scaling ($\beta=-1$ and $\lambda \gg 1$) that is important to describe the strong correlation limit. The results of our work provide evidence for the importance of this family of scalings in semiclassical and quantum theory of electronic systems, and indicate that these scaling properties must be considered as important constraints in the construction of new approximate density functionals. We also show, using the uniform-electron-gas scaling, that the curvature energy of metallic clusters is related to the second-order gradient expansion of kinetic and exchange-correlation energies.