Machine Learning Hamiltonians are Accurate Energy-Force Predictors

TL;DR

This paper introduces QHFlow2, a novel machine learning Hamiltonian model that accurately predicts energies and forces directly from Hamiltonians, outperforming previous models in both accuracy and efficiency.

Contribution

QHFlow2 is the first Hamiltonian model to reach NequIP-level force accuracy and significantly reduce energy errors, with a new benchmark for evaluating energy-force prediction performance.

Findings

QHFlow2 achieves 40% lower Hamiltonian error than previous models.

It reaches NequIP-level force accuracy and up to 20x lower energy MAE on MD17/rMD17.

The model demonstrates consistent scaling and effective translation of Hamiltonian accuracy into energy-force prediction accuracy.

Abstract

Recently, machine learning Hamiltonian (MLH) models have gained traction as fast approximations of electronic structures such as orbitals and electron densities, while also enabling direct evaluation of energies and forces from their predictions. However, despite their physical grounding, existing Hamiltonian models are evaluated mainly by reconstruction metrics, leaving it unclear how well they perform as energy-force predictors. We address this gap with a benchmark that computes energies and forces directly from predicted Hamiltonians. Within this framework, we propose QHFlow2, a state-of-the-art Hamiltonian model with an SO(2)-equivariant backbone and a two-stage edge update. QHFlow2 achieves $40\%$ lower Hamiltonian error than the previous best model with fewer parameters. Under direct evaluation on MD17/rMD17, it is the first Hamiltonian model to reach NequIP-level force accuracy while achieving up to $20\times$ lower energy MAE. On QH9, QHFlow2 reduces energy error by up to $20\times$ compared to MACE. Finally, we demonstrate that QHFlow2 exhibits consistent scaling behavior with respect to model capacity and data, and that improvements in Hamiltonian accuracy effectively translate into more accurate energy and force computations.