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Neural Pfaffians: Solving Many Many-Electron Schrödinger Equations
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Proposes Neural Pfaffian (NeurPf), a generalized neural wave function that replaces Slater determinants with overparametrized Pfaffians, removing discrete orbital selection and enabling learnable generalization across molecules, atoms, ions and structures.
Problem
Generalized neural wave functions for many-electron Schrödinger equations need permutation antisymmetry, but existing Slater-determinant approaches require exact orbital counts and hand-crafted or Hartree-Fock-based discrete orbital selection, limiting generalization and accuracy.
Approach
Use a Pfaffian of a learnable skew-symmetric matrix built from overparametrized (N_orb >= max(N_up, N_down)) permutation-equivariant orbitals, which is a superset of Slater determinants and needs no orbital selection; orbitals are predicted per nucleus by a MetaGNN, with new Hartree-Fock pretraining via rotation/antisymmetric orthogonal matrices and memory-efficient envelopes.
Key results
- On TinyMol small structures, NeurPf reaches 1.9 mHa lower energy than CCSD(T) CBS after 32k steps and 5.9 mHa lower than TAO and 11.3 mHa lower than Globe.
- On TinyMol large structures, NeurPf reaches 21.1 mHa lower energies than TAO after 32k steps, about 6 times lower error than the CCSD(T) baseline.
- On the nitrogen PES, NeurPf keeps about 2 mHa average error with and without ethene, while Globe rises from 2.7 to 5.3 mHa when ethene is added.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace Globe with NeurPf | Nitrogen potential energy surface trained only on nitrogen structures · Nitrogen potential energy surface | average error (milliHartree): 2.7 -> 2 | Helped |
| add ethene structures to training | Globe · Nitrogen potential energy surface | average error (milliHartree): 2.7 -> 5.3 | Hurt |
| add ethene structures to training | NeurPf · Nitrogen potential energy surface | average error (milliHartree): 2 -> 2 | No clear effect |
| use NeurPf instead of TAO | TinyMol small structures after 32k steps · TinyMol small | energy relative to CCSD(T) CBS: 1.9 milliHartree lower | Helped |
| use NeurPf instead of TAO | TinyMol small structures · TinyMol small | energy: 5.9 milliHartree lower | Helped |
| use NeurPf instead of Globe | TinyMol small structures · TinyMol small | energy: 11.3 milliHartree lower | Helped |
| use NeurPf instead of TAO | TinyMol large structures after 32k steps · TinyMol large | energy error: 21.1 milliHartree lower | Helped |
| use NeurPf instead of the CCSD(T) CBS baseline | TinyMol large structures after 128k steps · TinyMol large | energy: 4.4 milliHartree lower | Helped |
| replace full envelopes with memory-efficient envelopes (ablation) | NeurPf · molecular systems tested | performance | Helped |
| replace bottleneck envelopes with the proposed efficient envelopes (ablation) | NeurPf · systems tested | convergence | Helped |
Surprising findings
- NeurPf's variational energies on small TinyMol structures fall below the CCSD(T) CBS reference, which is not variational.
- The Pfaffian wave function is slower than a comparably sized Slater determinant due to lack of hardware support and more orbitals.
Limitations
- Pfaffian is slower than Slater determinants because of limited hardware/software support and more orbitals.
- Wave functions are not built to respect symmetries of the wave function itself.
- Transfer to periodic systems is left for future work; in single-structure calculations NeurPf may not improve accuracy.
Experimental setup
Datasets: TinyMol nitrogen potential energy surface ethene structures hydrogen chains second-row elements (ionization/affinity)
Benchmarks: TinyMol nitrogen dimer potential energy surface second-row atoms ionization and electron affinity hydrogen chains
Metrics: variational energy error (mHa) mean energy difference to CCSD(T) CBS chemical accuracy threshold
Baselines: Globe TAO FermiNet PESNet CCSD(T) CBS
Builds on: FermiNet PauliNet PESNet Globe Moon Spring
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