Pairformer-H: An E(3)-Equivariant Pairformer for Electronic Hamiltonian Prediction

Learning electronic Hamiltonians from atomic structures offers a route to electronic-property prediction without costly self-consistent density functional theory (DFT) calculations. However, accurately capturing how the surrounding atomic environment shapes orbital couplings between atom pairs remains a central challenge. We introduce Pairformer-H, a sparse E(3)-equivariant architecture that incorporates the Hamiltonian’s orbital-block structure into pairwise information exchange through learned orbital-operator compositions. Along each path connecting a target pair through an intermediate atom, the model composes two pair operators with an environment-dependent operator acting on the intermediate orbital space. This composition contracts over intermediate orbital indices, producing a contribution with the orbital dimensions of the target pair. Adaptive aggregation combines contributions across periodicity-consistent paths to update the target-pair representation. Across seven datasets without spin–orbit coupling and two with spin–orbit coupling, Pairformer-H achieves Hamiltonian matrix-element mean absolute errors of 0.166–0.356 meV and the lowest reported errors among the evaluated baselines on five without spin–orbit coupling datasets and both with spin–orbit coupling datasets. Relative to the best baseline for each dataset, errors decrease by 51.7% on bilayer Bi$\_2$Te$\_3$ without spin–orbit coupling and 35.4% on bulk $T\_\mathrm{d}$\-WTe$\_2$. Representative band structures, densities of states, and optical responses closely agree with DFT references. These results show that aligning a network’s internal interactions with the orbital-block structure of the Hamiltonian yields accurate predictions across materials and spin settings.