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Learning the Electronic Hamiltonian of Large Atomic Structures
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Presents a strictly local equivariant GNN (eSCN/EquiformerV2-style) that predicts electronic Hamiltonians of large amorphous materials (up to 3,000 atoms) by training on augmented graph partitions with virtual nodes/edges.
Problem
Predicting ground-state electronic Hamiltonians of large, disordered materials (thousands of atoms, 10+ Å cells) is infeasible with DFT and existing GNN Hamiltonian models were limited to small molecules or periodic crystals with small unit cells.
Approach
A strictly local SO(2)-convolution equivariant GNN with multi-head attention predicts diagonal and off-diagonal Hamiltonian blocks, trained on augmented partitions where virtual nodes/edges preserve exact 1-hop connectivity so arbitrarily large graphs can be split across memory-limited batches without accuracy loss.
Key results
- Best achievable Hamiltonian prediction error of 2.17 to 2.58 meV on unseen amorphous test structures of 1,000-3,000 atoms (a-GST, a-PtGe, a-HfO2).
- Predicted Hamiltonian eigenvalues match DFT reference within 0.530% relative L1 error on a 3,000-atom a-HfO2 structure.
- Augmented partitioning with 8 slices gives a 6.5x per-epoch speedup (0.38 vs 2.5 s) and 7.2x lower memory per rank (8.59 vs 61.68 GiB) relative to full-batch training.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add virtual nodes and edges to preserve connectivity across partition boundaries (ablation) | GNN trained on a-HfO2 partitions · a-HfO2 test structure 3 | ϵn and ϵe: 5.39 -> 2.16 mEh; 0.55 -> 0.16 mEh | Helped |
| vary partition size from tslice ∼2 Å to tslice ∼52 Å (ablation) | Augmented-partition GNN trained on a-HfO2 · a-HfO2 test structure 3 | ϵn and ϵe: 2.43, 2.16, 2.24, 2.64, 2.75, 2.50 mEh; 0.23, 0.16, 0.16, 0.20, 0.19, 0.17 mEh | No clear effect |
| increase the interaction cutoff from 8 Å to 12 Å (ablation) | Augmented-partition GNN on a-HfO2 · a-HfO2 test structure 3 | ϵtot [meV]: 4.84 -> 2.58 | Helped |
| train on a-HfO1.8 and test across oxygen-vacancy stoichiometries a-HfO1.7, a-HfO1.8, and a-HfO1.9 | Augmented-partition GNN · a-HfOx | ϵn and ϵe: 2.34-2.45 mEh; 0.15-0.16 mEh | No clear effect |
| predict the Hamiltonian with the learned model instead of DFT | Device transport modeling · Devices 0–2 | current [A]: 8.00 ×10−9 -> 5.54×10−9; 1.48×10−5 -> 1.06×10−5; 6.99×10−6 -> 4.89×10−6 | Mixed |
| use augmented partitions rather than full-batch graph training | GNN training on large materials · a-HfO2 training | speedup per epoch: 6.5× speedup per epoch (0.38 vs. 2.5 s) | Helped |
| use augmented partitions rather than full-batch graph training | GNN training on large materials · a-HfO2 training | memory consumption per rank: 7.2× decrease (8.59 vs. 61.68 GiB) | Helped |
Surprising findings
- Virtual nodes/edges reduce node and edge prediction errors by about 60% and 70%, versus training on raw partitions without boundary connectivity.
Limitations
- Quantitative agreement with DFT in device transport simulations is imperfect, though currents are within the right order of magnitude.
- Model is strictly local with a 1-hop receptive field, so long-range interactions beyond the cutoff are not captured explicitly.
- Training data limited to three amorphous materials with few structures each.
Experimental setup
Datasets: custom amorphous HfO2 dataset custom amorphous GST dataset custom amorphous PtGe dataset
Benchmarks: a-HfO2 test structures a-GST test structure a-PtGe test structure
Metrics: node error (mEh) edge error (mEh) total Hamiltonian error (meV) relative L1 eigenvalue error (%) wall-clock time per epoch memory per rank
Baselines: Wang et al. 2024a (Hamiltonian GNN on 2D materials) Wang et al. 2024b (equivariant GNN Hamiltonian) DFT
Builds on: eSCN EquiformerV2 Tensor Field Networks Allegro
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