Expander attention as exchange-correlation

TL;DR

This paper introduces a linearly scaling non-local exchange-correlation functional using an expander graph transformer, enabling accurate modeling of strongly correlated systems with reduced computational cost.

Contribution

It proposes a novel expander graph transformer-based XC functional that scales linearly, improving efficiency while maintaining accuracy for strongly correlated quantum systems.

Findings

Recovers the H2 dissociation curve in the strongly correlated regime.

Shows promising results on planar H4, where high-level methods fail.

Achieves a balance between accuracy and computational efficiency.

Abstract

Kohn-Sham density functional theory (DFT) is the workhorse of quantum chemistry, offering an attractive balance between accuracy and computational cost. Although exact in principle, DFT in practice relies on an approximation to the unknown exchange-correlation (XC) functional, which encodes the many-body quantum effects beyond the mean-field treatment. Many such approximations exist, and machine-learned XC functionals have proliferated in recent years. A persistent challenge in this area is the trade-off between accuracy and computational cost: while high-accuracy ML functionals have shown success on strongly correlated systems that are notoriously difficult for conventional approximations, their unfavorable scaling has limited broader adoption. Here, we propose a linearly scaling non-local XC approximation based on an expander graph transformer ansatz, improving the scaling of $O(N^2)$ or worse for previous ML functionals capable of reliably capturing strongly correlated systems. We show that it recovers the correct $\mathrm{H_2}$ dissociation curve in the strongly correlated regime, with promising results on planar $\mathrm{H_4}$, a system where even high-level coupled cluster methods break down. Our approach thus charts a path toward ML functionals that are both accurate on strongly correlated systems and cheap enough to deploy at scale.