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Self-Consistency Training for Density-Functional-Theory Hamiltonian Prediction
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Proposes self-consistency training, which trains a DFT Hamiltonian predictor by minimizing the residual of the Kohn-Sham equation on unlabeled molecular structures, making it more efficient than DFT labeling and extending Hamiltonian prediction to 56-atom molecules.
Problem
Hamiltonian prediction for DFT is limited by scarce labeled Hamiltonian data, restricting applicability to small molecules and making generalization hard.
Approach
Exploit the self-consistency principle of the Kohn-Sham equation: the loss penalizes the mismatch between the predicted Hamiltonian and the Hamiltonian reconstructed from its own eigenvectors, so no Hamiltonian labels are needed; gradients are back-propagated through the eigensolver and the reconstruction, and this amortizes DFT's SCF cost over many molecules.
Key results
- On MD17 data-scarce setting, adding self-consistency loss cut Hamiltonian MAE by 14.4% to 52.8% and reduced HOMO/LUMO/gap MAE by several folds.
- On the QH9 OOD split, self-consistency fine-tuning reduced LUMO and gap MAE by an order of magnitude versus zero-shot.
- On MD22 molecules ALA3 (42 atoms) and DHA (56 atoms), self-consistency fine-tuning gave 1-2 orders lower property MAE than e2e ET and Equiformer predictors.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add self-consistency training on unlabeled structures | QHNet trained with 100 labels · MD17 ethanol, malondialdehyde, and uracil test sets | Hamiltonian MAE: 160.36 -> 75.65; 101.19 -> 86.60; 88.26 -> 63.82 | Helped |
| Add self-consistency training on unlabeled structures | QHNet trained with 100 labels · MD17 ethanol, malondialdehyde, and uracil test sets | HOMO MAE: 911.64 -> 336.97; 471.92 -> 274.45; 1217.17 -> 359.98 | Helped |
| Add self-consistency training on unlabeled structures | QHNet trained with 100 labels · MD17 ethanol, malondialdehyde, and uracil test sets | LUMO MAE: 6800.84 -> 1203.60; 1093.22 -> 279.14; 12496.1 -> 369.67 | Helped |
| Add self-consistency training on unlabeled structures | QHNet trained with 100 labels · MD17 ethanol, malondialdehyde, and uracil test sets | HOMO-LUMO gap MAE: 6643.11 -> 1224.86; 1115.94 -> 324.37; 11850.56 -> 388.30 | Helped |
| Fine-tune all parameters with self-consistency on unlabeled QH9-large structures | QHNet pretrained on QH9-small · QH9-large test | HOMO-LUMO gap MAE: 12203.12 -> 1316.96 | Helped |
| Fine-tune an adapter only with self-consistency on unlabeled QH9-large structures | QHNet pretrained on QH9-small · QH9-large test | Hamiltonian MAE: 69.67 -> 64.48 | Helped |
| Use adapter-only rather than all-parameter self-consistency fine-tuning (ablation) | QHNet pretrained on QH9-small · QH9-large test | Hamiltonian MAE: 65.74 -> 64.48 | Helped |
| Use adapter-only rather than all-parameter self-consistency fine-tuning (ablation) | QHNet pretrained on QH9-small · QH9-large test | HOMO-LUMO gap MAE: 1316.96 -> 1394.29 | Hurt |
| Use self-consistency fine-tuning on unlabeled structures | QHNet pretrained on QH9-full · MD22 ALA3 and DHA | Hamiltonian MAE: 237.71 -> 52.49; 397.87 -> 56.12 | Helped |
| Use self-consistency training instead of extended-label supervised training | QHNet fine-tuned on QH9-large · QH9-large test | Hamiltonian MAE: 62.13 -> 65.74; 59.67 -> 64.48 | Hurt |
| Use self-consistency training instead of extended-label supervised training | QHNet fine-tuned on QH9-large · QH9-large test | HOMO-LUMO gap MAE: 6137.66 -> 1316.96; 6445.33 -> 1394.29 | Helped |
| Replace conventional DFT with self-consistency training to solve the structures | MD17 molecules · MD17 unlabeled structures | Computation time (s): 4.50×10^4 vs 6.40×10^4; 4.81×10^4 vs 1.05×10^5; 1.23×10^5 vs 2.15×10^5 | Helped |
Surprising findings
- Supervised training with full extended labels reached lower Hamiltonian MAE than self-consistency training in the OOD scenario, though self-consistency still won on derived properties like HOMO/LUMO.
Limitations
- Numerically unstable eigen-decomposition gradients when eigenvalues are close, needing truncation and update skipping.
- Self-consistency loss can be costlier than supervised training per step because it requires eigensolving and Hamiltonian reconstruction.
- Evaluated on relatively small molecules from MD17, QH9 and MD22; scale tests use only 500 labeled evaluation structures.
Experimental setup
Datasets: MD17 QH9 MD22
Benchmarks: MD17 Hamiltonian QH9 MD22
Metrics: Hamiltonian MAE orbital energy MAE orbital coefficient cosine similarity HOMO MAE LUMO MAE HOMO-LUMO gap MAE SCF acceleration ratio
Baselines: supervised label training extended-label extended-label-online zero-shot ET Equiformer
Builds on: QHNet Kohn-Sham DFT DIIS
Closest papers by meaning
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