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High-order Equivariant Flow Matching for Density Functional Theory Hamiltonian Prediction
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Proposes QHFlow, an SE(3)-equivariant flow-matching model that generates DFT Kohn-Sham Hamiltonians conditioned on molecular geometry, with GOE/tensor-expansion invariant priors and an orbital-energy fine-tuning stage.
Problem
Predicting DFT Hamiltonian matrices directly from molecular geometry to bypass the costly SCF loop; existing equivariant regression models output deterministic pointwise predictions that miss structural correlations in the Hamiltonian.
Approach
Reformulate Hamiltonian prediction as conditional flow matching from an SE(3)-invariant prior (GOE or tensor-expansion-based) to the target Hamiltonian, using a QHNet-based equivariant vector-field network, plus a post-hoc fine-tuning step that aligns predicted orbital energies with DFT references.
Key results
- On MD17, QHFlow reduces Hamiltonian MAE to 4.93 μEh (water) vs 11.70 for QHNet, up to 73% reduction.
- On QH9-stable (id), QHFlow (WA-FT) reaches HOMO error 92.22 μEh and LUMO error 187.48 μEh versus QHNet's 1546.27 and 18257.34.
- When initializing SCF with QHFlow predictions on 300 QH9 id molecules, total runtime falls to 46% of conventional DFT, versus 53% for QHNet initialization.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use QHFlow instead of the strongest reported baseline | Hamiltonian prediction · MD17 Water, Ethanol, Malondialdehyde, and Uracil | Hamiltonian MAE: 11.70 -> 4.93; 20.09 -> 5.33; 20.67 -> 3.80; 19.36 -> 3.68 | Helped |
| use QHFlow instead of the strongest reported baseline | Hamiltonian prediction · QH9-stable id, stable ood, dynamic-300k geo, and dynamic-300k mol | Hamiltonian MAE: 45.48 -> 22.95; 43.33 -> 20.01; 52.18 -> 25.94; 108.19 -> 45.91 | Helped |
| fine-tune with weighted alignment loss (ablation) | QHFlow · QH9-stable id and stable ood | occupied orbital energy MAE: 119.67 -> 101.92; 84.54 -> 72.64 | Helped |
| fine-tune with weighted alignment loss (ablation) | QHFlow · QH9-dynamic-300k geo and mol | occupied orbital energy MAE: 103.11 -> 89.03; 442.56 -> 424.75 | Helped |
| fine-tune with weighted alignment loss (ablation) | QHFlow · QH9-stable id, stable ood, dynamic-300k geo, and dynamic-300k mol | Hamiltonian MAE: 22.95 -> 23.85; 20.01 -> 20.55; 25.94 -> 27.12; 45.91 -> 46.60 | Hurt |
| replace GOE prior with TE prior (ablation) | QHFlow · QH9-stable id, stable ood, and dynamic-300k geo | Hamiltonian MAE: 25.93 -> 22.95; 20.41 -> 20.01; 29.39 -> 25.94 | Helped |
| replace GOE prior with TE prior (ablation) | QHFlow · QH9-dynamic-300k mol | Hamiltonian MAE: 46.78 -> 45.91 | Helped |
| initialize SCF with QHFlow-predicted Hamiltonians instead of conventional initialization | DFT SCF · QH9 id benchmark | Total runtime ratio: 46%; 54% relative speedup | Helped |
| initialize SCF with QHFlow-predicted Hamiltonians instead of QHNet initialization | DFT SCF · QH9 id benchmark | Total runtime ratio: 53% -> 46%; 7% points | Helped |
Surprising findings
- The fine-tuned WA-FT variant improves orbital energies but slightly worsens Hamiltonian MAE on several QH9 splits.
- The stochastic flow sampling gives stable predictions across seeds (mean absolute deviation about 0.03% and energy std below 0.3 μEh).
Limitations
- Flow-based sampling adds ODE-solving inference overhead.
- Experiments limited to gas-phase molecules up to 29 atoms and only PBE and B3LYP functionals.
- Generalization to WANet, SPHNet, and PubChemQH not verified due to lack of code and data.
Experimental setup
Datasets: MD17 QH9
Benchmarks: MD17 QH9-stable QH9-dynamic-300k
Metrics: Hamiltonian MAE occupied orbital energy MAE orbital coefficient similarity LUMO error HOMO error HOMO-LUMO gap error SCF iteration ratio SCF/total runtime ratio
Baselines: SchNOrb PhiSNet QHNet SPHNet WANet
Builds on: QHNet WANet Equiformer Tensor Field Networks flow matching
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