Designing explicit functionals for the charge density in terms of a potential

TL;DR

This paper explores explicit functionals for charge density based on model data, aiming to improve predictions for real materials without solving the Kohn-Sham equations.

Contribution

It introduces a hierarchy of approximations for charge density functionals derived from the homogeneous electron gas, avoiding direct Schrödinger equation solutions.

Findings

Higher approximation levels systematically improve charge density predictions for cubic helium.

Explicit functionals based on model data can effectively describe inhomogeneous materials.

The approach offers a promising route for simple, analyzable charge density expressions.

Abstract

One of the most powerful strategies to address properties of real many-body systems is to incorporate data obtained for models, for example, to use data of the homogeneous electron gas in order to build the Local Density Approximation for the Kohn-Sham exchange-correlation potential. In the present work, we examine to what extent we can use model data to design functionals directly for observables of materials. In particular, we study different approximations for the charge density of real inhomogeneous materials expressed as a simple, explicit functional of a given Kohn-Sham potential, using as central building block the Lindhard density-density response function of the homogeneous electron gas. Our increasingly realistic set of approximations includes a fully nearsighted expression equivalent to the Thomas-Fermi approximation, functional Taylor expansions, and different approximations to the Connector Theory developed in [Aouina \textit{et al.}, npj Computational Materials {\bf 11}, 242 (2025)]. In all cases, the charge density is obtained without ever solving the Kohn-Sham Schr\"odinger equation. Results for cubic helium, a prototypical strongly inhomogeneous material, systematically improve with higher levels of approximation, indicating that this is a promising route to obtain expressions that are relatively simple to calculate and to analyze.