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Infusing Self-Consistency into Density Functional Theory Hamiltonian Prediction via Deep Equilibrium Models
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Proposes DEQH, a deep-equilibrium Hamiltonian solver (instantiated as DEQHNet on QHNet) that iteratively refines predicted DFT Hamiltonians to a fixed point without DFT calls during training, evaluated on MD17 and QH9.
Problem
Predicting DFT Hamiltonian matrices with machine learning while capturing their self-consistent iterative nature, which prior approaches only incorporate by running costly DFT iterations during training.
Approach
Wrap an equivariant Hamiltonian predictor (QHNet-based) in a deep equilibrium model so the network takes its own Hamiltonian (and overlap matrix) as input and is trained to reach a fixed point H* = f(H*, Z, R, S), making the model a learned Hamiltonian solver rather than a one-shot predictor.
Key results
- On QH9-stable-id, DEQHNet reduces total Hamiltonian MAE from 76.31 to 61.42 (10^-6 Eh) relative to QHNet and raises orbital coefficient similarity from 95.85 to 99.84 (10^-2).
- On QH9-dynamic-mol, DEQHNet Hamiltonian MAE drops from 173.92 to 105.80, nearly halving the error.
- On MD17, DEQHNet beats QHNet on Hamiltonian MAE for ethanol, malonaldehyde and uracil but is worse on water (36.07 vs 10.79) and has much higher orbital energy MAE on water (335.86 vs 33.76).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use DEQHNet instead of QHNet | Hamiltonian prediction · MD17 Water | Hamiltonian MAE: 10.79 -> 36.07 | Hurt |
| Use DEQHNet instead of QHNet | Hamiltonian prediction · MD17 Ethanol | Hamiltonian MAE: 20.91 -> 18.73 | Helped |
| Use DEQHNet instead of QHNet | Hamiltonian prediction · MD17 Malonaldehyde | Hamiltonian MAE: 21.52 -> 17.97 | Helped |
| Use DEQHNet instead of QHNet | Hamiltonian prediction · MD17 Uracil | Hamiltonian MAE: 20.12 -> 15.07 | Helped |
| Use DEQHNet instead of QHNet | Hamiltonian prediction · QH9-stable-id | Hamiltonian MAE, all: 76.31 -> 61.42 | Helped |
| Use DEQHNet instead of QHNet | Hamiltonian prediction · QH9-stable-ood | Hamiltonian MAE, all: 72.11 -> 53.23 | Helped |
| Use DEQHNet instead of QHNet | Hamiltonian prediction · QH9-dynamic-geo | Hamiltonian MAE, all: 96.85 -> 62.14 | Helped |
| Use DEQHNet instead of QHNet | Hamiltonian prediction · QH9-dynamic-mol | Hamiltonian MAE, all: 173.92 -> 105.80 | Helped |
| Use DEQHNet instead of QHNet | Orbital energy prediction · QH9-stable-id | Orbital energy MAE: 798.51 -> 4383.10 | Hurt |
| Use DEQHNet instead of QHNet | Orbital energy prediction · QH9-stable-ood | Orbital energy MAE: 644.17 -> 5657.07 | Hurt |
| Use DEQHNet instead of QHNet | Orbital energy prediction · QH9-dynamic-geo | Orbital energy MAE: 834.47 -> 1864.06 | Hurt |
| Use DEQHNet instead of QHNet | Orbital energy prediction · QH9-dynamic-mol | Orbital energy MAE: 9719.58 -> 4625.88 | Helped |
Surprising findings
- Lower Hamiltonian MAE does not necessarily translate into lower orbital energy MAE; DEQHNet's orbital energy error is higher than QHNet's on QH9-stable-id and on MD17 water.
- DEQHNet converges to a fixed point in about 2-3 iterations, fewer than the 15 SCF iterations PySCF needs for the same configurations.
Limitations
- Orbital energy MAE is worse than QHNet on several settings (e.g., 4383.10 vs 798.51 on QH9-stable-id), so Hamiltonian accuracy gains do not carry over to orbital energies.
- Water on MD17 shows worse Hamiltonian MAE than QHNet, and the authors attribute the MD17 comparisons partly to differing training data sizes.
- Evaluation is limited to small molecules (MD17, QH9); scalability to large systems is claimed but not demonstrated.
Experimental setup
Datasets: MD17 QH9-stable QH9-dynamic
Benchmarks: MD17 QH9-stable-id QH9-stable-ood QH9-dynamic-geo QH9-dynamic-mol
Metrics: Hamiltonian MAE orbital energy MAE orbital coefficient cosine similarity
Baselines: QHNet SchNOrb PhiSNet DeepH PySCF SCF
Builds on: deep equilibrium models QHNet Equiformer OrbNet-Equi
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