Machine Learning Hamiltonians are Accurate Energy-Force Predictors

TL;DR

This work proposes QHFlow2, a state-of-the-art Hamiltonian model with an SO(2)-equivariant backbone and a two-stage edge update that achieves lower Hamiltonian error than the previous best model with fewer parameters and demonstrates that improvements in Hamiltonian accuracy effectively translate into more accurate energy and force computations.

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.

Figures (8)

• Figure 1: QHFlow2 overall workflow. Given the molecular structure $\mathcal{M}$, flow time $t$, and an intermediate Hamiltonian state $\mathbf{H}_t$, QHFlow2 applies an SO(2) backbone, a two-stage pairwise update, and construct Hamiltonian via tensor expansion for energy and force evaluation. (a) Pipeline overview. (b) Detailed architecture of the matrix encode, the SO(2) backbone, and the two-stage pair update.
• Figure 2: Energy and force accuracy under direct evaluation on the MD benchmark. We report mean absolute errors (MAE) of total energy (top) and forces (bottom) computed from predicted Hamiltonians on six molecular systems. gray bars denote MLIP baselines, green bars denote prior Hamiltonian predictors, and blue bars denote QHFlow2. All methods use the same data splits and evaluation setup. Overall, the results show that Hamiltonian-based direct evaluation yields competitive energy and force accuracy on MD trajectories, with prior Hamiltonian models narrowing the gap to MLIPs and QHFlow2 providing the strongest improvements. The right-hand panels summarize mean MAE across the six systems, and the dashed line marks the NequIP reference. QHFlow2 attains low energy error and, for the first time among Hamiltonian predictors, reaches NequIP-level force accuracy. Each number is reported in \ref{['tab:0_md17_EF_sub']}.
• Figure 3: Relative performance of MLIP baselines and QHFlow2 on QH9. Relative errors for total energy and orbital quantities are normalized by the best MLIP result of baselines for each quantity. EQ and EQ2 denote Equiformer and EquiformerV2, respectively. MLIP numbers are taken from published results on QM9 as reference points, whereas QHFlow2 is trained and evaluated on QH9-stable-id. All exact values are reported in \ref{['app:experiment']}.
• Figure 4: Relative reduction of SCF cycles on QH9. We report the SCF cycle ratio (lower is better): SCF iterations to converge when initializing with a predicted Hamiltonian $\hat{\mathbf H}$, normalized by the MinAO baseline (100%). All SCF/DFT settings are fixed; only the initialization is changed. Stronger predictors reduce SCF cycles and approach REF, which initializes with the dataset Hamiltonian $\mathbf H$ (data limit), while RC initializes with a converged Hamiltonian recomputed under our evaluation settings (solver limit).
• Figure 5: Data scaling on the MD benchmark.(a) Energy and force MAE versus training set size for QHFlow2-m and MLIP baselines on salicylic acid and aspirin. (b) Hamiltonian MAE versus training set size for QHFlow2-m. (c) Occupied orbital energy MAE versus training set size for QHFlow2-m. Improved Hamiltonian accuracy consistently accompanies improved orbital, energy, and force accuracy.