Construction of model dielectric constants for two and three dimensional electron liquids from density functionals

TL;DR

This paper demonstrates a general approach to constructing model dielectric constants for electron liquids from energy functionals, extending beyond three dimensions and including exchange and correlation effects, with implications for understanding dielectric behavior.

Contribution

It introduces a unified paradigm linking energy functionals to model dielectric constants, applicable to various dimensions and including exchange and correlation corrections.

Findings

Derived dielectric constants from exchange and correlation corrections in 2D and 3D.

Extended the paradigm beyond kinetic-energy functionals.

Identified inaccuracies in textbook statements about the long-wavelength limit.

Abstract

The Thomas-Fermi (TF) approximation for the static dielectric constant of a three-dimensional electron liquid can be derived from minimizing the TF local-density approximation for the kinetic-energy functional. Here we show that this connection between energy functionals and model dielectric constants is not an artifact of three-dimensionality or of the kinetic-energy functional, but a general paradigm that can be exploited to build specific physical phenomena into model dielectric constants by deriving them from suitable energy functionals. Four examples are worked out in detail, by calculating the dielectric constants that follow, respectively, from (i) exchange corrections to TF theory in three dimensions, i.e., TF-Dirac theory, (ii) further correlation corrections to TF-Dirac theory in three dimensions, (iii) TF theory in two dimensions, and (iv) exchange corrections to TF theory in two dimensions. Each of these cases has certain interesting features that are discussed in some detail. As a byproduct of these investigations we also find that a common textbook statement about the long-wavelength ($k\to0$) limit of the random-phase approximation is not fully correct.