# Simple exchange hole models for long-range-corrected density functionals

## TL;DR

This paper introduces simple, closed-form exchange hole models for long-range-corrected density functionals, improving the conversion of semilocal functionals into their short-range analogs with explicit formulas.

## Contribution

It develops new exchange hole models using Hermite functions for better short-range functional approximations in density functional theory.

## Key findings

- Models match the uniform electron gas limit
- Energy densities are within 5% of each other
- New models are non-oscillatory and simple to implement

## Abstract

Density functionals with a range-separated treatment of the exchange energy are known to improve upon their semilocal forerunners and fixed-fraction hybrids. The conversion of a given semilocal functional into its short-range analog is not straightforward, however, and not even unique, because the latter has a higher information content that has to be recovered in some way. Simple models of the spherically-averaged exchange hole as an interpolation between the uniform electron gas limit and a few-term Hermite function are developed here for use with generalized-gradient approximations, so that the energy density of the error-function-weighted Coulomb interaction is given by explicit closed-form expressions in terms of elementary and error functions. For comparison, some new non-oscillatory models in the spirit of earlier works are also built and studied, their energy densities match rather closely (within less than 5%) but do lack the exact uniform electron gas limit.