# Nonlocal Machine-Learned Exchange Functional for Molecules and Solids

## TL;DR

This paper introduces a machine-learned, nonlocal exchange functional for DFT that achieves hybrid-DFT accuracy at a computational cost comparable to semilocal functionals, benefiting both molecules and solids.

## Contribution

It develops a novel machine learning-based exchange functional that is orbital-dependent and nonlocal, offering high accuracy with low computational cost for diverse systems.

## Key findings

- Achieves hybrid-DFT accuracy on thermochemical benchmarks.
- Improves band gap predictions over semilocal DFT.
- Demonstrates scalability in large supercell calculations.

## Abstract

The design of better exchange-correlation functionals for Density Functional Theory (DFT) is a central challenge of modern electronic structure theory. However, current developments are limited by the mathematical form of the functional, with efficient semilocal functionals being inaccurate for many technologically important systems and the more accurate hybrid functionals being too expensive for large solid-state systems due to the use of the exact exchange operator. In this work, we use machine learning combined with exact physical constraints to design an exchange functional that is both orbital-dependent and nonlocal, but which can be evaluated at roughly the cost of semilocal functionals and is significantly faster than hybrid DFT in plane-wave codes. By training functionals with several different feature sets, we elucidate the roles of orbital-dependent and nonlocal features in learning the exchange energy and determine that both types of features provide vital and independently important information to the model. Having trained our new exchange functional with an expressive, nonlocal feature set, we substitute it into existing hybrid functionals to achieve hybrid-DFT accuracy on thermochemical benchmark sets and improve the accuracy of band gap predictions over semilocal DFT. To demonstrate the scalability of our approach as well as the practical benefits of improved band gap prediction, we compute charged defect transition levels in silicon using large supercells. Due to its transferability and computational efficiency for both molecular and extended systems, our model overcomes the cost-accuracy trade-off between semilocal and hybrid DFT, and our general approach provides a feasible path toward a universal exchange-correlation functional with post-hybrid DFT accuracy and semilocal DFT cost.