Stepwise Refinement and Label-Free Error Ranking for Kohn–Sham Hamiltonians

Machine-learned Kohn–Sham Hamiltonians can bypass costly self-consistent density functional theory (DFT) calculations, but their practical use requires identifying predictions with large errors that should be recomputed with DFT. In this work, we introduce HIVE (Hamiltonian Iterative Velocity Estimator), an equivariant Hamiltonian prediction model that also provides an error ranking at no additional cost. During training, the model sees one randomly sampled point on a deterministic path from zero to the target Hamiltonian. At inference, we repeatedly apply the same field to its evolving Hamiltonian prediction, a process we call the rollout. With this strategy, averaging the stepwise estimates gives the final Hamiltonian, while their dispersion provides a label-free score for ranking likely failures without additional evaluations beyond the rollout. We test HIVE across four periodic benchmarks with PBE and HSE06 labels, and the results improve the best reported Hamiltonian mean absolute error on three and match it on the fourth. More steps improve accuracy without retraining, mainly because the finer rollout produces better estimates rather than because averaging cancels noise. On newly DFT-labelled structural shifts, the dispersion ranks per-structure error with Spearman correlations of 0.98–0.99 across all three PBE systems. Overall, HIVE provides both an accurate Hamiltonian prediction and a practical uncertainty signal without training a separate model.