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Learning local equivariant representations for quantum operators
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Proposes SLEM, a strictly local equivariant message-passing network with SO(2) convolution and invariant Slater-Koster overlap parameterization that predicts Hamiltonian, density, and overlap matrices in LCAO DFT with high accuracy and data efficiency.
Problem
Predicting DFT quantum operator matrices (Hamiltonian, density, overlap) for large systems is hindered by the growing receptive field of equivariant message-passing networks and the high cost of tensor products for high-order orbitals (f, g).
Approach
Strictly local node/edge updates that exclude neighbours of neighbours beyond the cutoff, SO(2) convolution to cut tensor-product cost from O(l^6) to O(l^3), and overlap predicted as invariant two-centre Slater-Koster scalars.
Key results
- SLEM achieves lowest Hamiltonian MAE on MoS2 (0.34 meV with 0.7M params vs 0.46 for DeepH-E3), graphene (0.26 meV) and Si (0.10 meV).
- On GaN and HfO2 with f and g orbitals, SLEM reaches 0.21 and 0.28 meV MAE, versus 0.87 meV for DeepH-E3 on GaN; DeepH-E3 ran out of memory on HfO2.
- Time scaling of SO(2)-based tensor product fitted at slope 1.2 versus 3.7 for DeepH-E3 and E3NN, with over two orders of magnitude gains in time and memory.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SLEM instead of DeepH-E3 for Hamiltonian prediction | MoS₂ · MoS₂ | MAE (meV): 0.46 -> 0.34 | Helped |
| Use SLEM instead of DeepH-E3 for Hamiltonian prediction | Graphene · Graphene | MAE (meV): 0.40 -> 0.26 | Helped |
| Use SLEM instead of DeepH-E3 for Hamiltonian prediction | Si (300K) · Si (300K) | MAE (meV): 0.10 -> 0.10 | No clear effect |
| Use SLEM instead of DeepH-E3 for high-orbital Hamiltonian prediction | GaN · GaN | MAE (meV): 0.87 -> 0.21 | Helped |
| Use the semi-local LEM variant instead of QHnet | QH-9 OOD Hamiltonian prediction · QH-9 OOD | Hamiltonian MAE all (10⁻⁶ Eₕ): 83.22 -> 56.57 | Helped |
| Use the semi-local LEM variant instead of QHnet | QH-9 OOD Hamiltonian prediction · QH-9 OOD | Hamiltonian MAE non-diag (10⁻⁶ Eₕ): 80.26 -> 44.86 | Helped |
| Train SLEM rather than DeepH-E3 with reduced training data | Hamiltonian prediction on MoS₂ and Graphene · MoS₂ and Graphene, 20%–80% training data | validation MAE (meV) | Helped |
| Use SO(2) convolution instead of conventional tensor products | SLEM · High-order tensor products | computational complexity: O(l_max^6) to O(l_max^3) | Helped |
Surprising findings
- Strict locality with a 0.7M-parameter model outperforms larger baselines in Hamiltonian accuracy, and the 20% data subset remains accurate (e.g., MoS2 0.37 meV).
Limitations
- Strict locality is less suited to molecules or confined systems lacking screening; a semi-local LEM variant is offered for these.
- Overlap and density results are only on Si, GaN, HfO2 datasets, and QH9 density is inferred from Hamiltonian.
- Cutoff choice (r_cut / r_e) is system dependent and must be tuned.
Experimental setup
Datasets: MoS2 dataset graphene dataset Si MD dataset (generated) GaN MD dataset (generated) HfO2 MD dataset (generated) QH9
Benchmarks: MoS2 Hamiltonian graphene Hamiltonian Si Hamiltonian GaN Hamiltonian HfO2 Hamiltonian Si/GaN/HfO2 density and overlap matrices QH9
Metrics: MAE wall time per iteration GPU memory band structure eigenvalue MAE
Baselines: DeepH-E3 HamGNN QHnet E3NN
Builds on: Allegro DeepH-E3 SO(2) convolution (eSCN) DeePTB Slater-Koster parameterization
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