# Semi-local Exchange Energy Functional For Two-Dimensional Quantum Systems: A Step Beyond Generalized Gradient Approximations

## TL;DR

This paper introduces a new semi-local exchange energy functional for two-dimensional quantum systems, improving accuracy and efficiency over existing density functionals by utilizing a hole based on the density matrix expansion.

## Contribution

It presents a novel semi-local exchange functional for 2D systems that satisfies all relevant constraints and outperforms previous local and non-empirical functionals.

## Key findings

- Achieves high accuracy compared to exact exchange results
- Reduces errors present in previous density functionals for 2D systems
- Demonstrates excellent performance in comprehensive testing

## Abstract

Semi-local density functionals for the exchange-correlation energy of electrons are extensively used as it produce realistic and accurate results for finite and extended systems. The choice of techniques play crucial role in constructing such functionals of improved accuracy and efficiency. An accurate and efficient semi-local exchange energy functional in two dimensions is constructed by making use of the corresponding hole based on the density matrix expansion. The exchange hole involved is localized under the generalized coordinate transformation and satisfies all the relevant constraints. Comprehensive testing and excellent performance of the functional is demonstrated versus exact exchange results. The functional also achieves remarkable accuracy by substantially reducing the errors present in the local and non-empirical density functionals proposed so far for two dimensional systems. The underlying principles involved in the functional construction are physically appealing and practically useful for developing range separated and non-local functionals in two dimensions.