Neural Pfaffians: Solving Many Many-Electron Schrödinger Equations (2405.14762v3)

Abstract: Neural wave functions accomplished unprecedented accuracies in approximating the ground state of many-electron systems, though at a high computational cost. Recent works proposed amortizing the cost by learning generalized wave functions across different structures and compounds instead of solving each problem independently. Enforcing the permutation antisymmetry of electrons in such generalized neural wave functions remained challenging as existing methods require discrete orbital selection via non-learnable hand-crafted algorithms. This work tackles the problem by defining overparametrized, fully learnable neural wave functions suitable for generalization across molecules. We achieve this by relying on Pfaffians rather than Slater determinants. The Pfaffian allows us to enforce the antisymmetry on arbitrary electronic systems without any constraint on electronic spin configurations or molecular structure. Our empirical evaluation finds that a single neural Pfaffian calculates the ground state and ionization energies with chemical accuracy across various systems. On the TinyMol dataset, we outperform the `gold-standard' CCSD(T) CBS reference energies by 1.9m$E_h$ and reduce energy errors compared to previous generalized neural wave functions by up to an order of magnitude.

• The paper introduces Neural Pfaffians as an overparametrized neural wave function approach that overcomes the rigidity of Slater determinant methods.
• It employs Pfaffian-based antisymmetry and memory-efficient exponential envelopes, paired with a novel pretraining scheme aligned to Hartree-Fock results.
• Empirical evaluations demonstrate that NeurPf achieves chemical accuracy and outperforms CCSD(T), reducing energy errors by up to an order of magnitude.

Neural Pfaffians: Solving Many Many-Electron Schrödinger Equations

The paper "Neural Pfaffians: Solving Many Many-Electron Schrödinger Equations" by Nicholas Gao and Stephan Günnemann presents an intriguing advancement in the domain of quantum chemistry, particularly concerning the highly accurate approximation of the ground states of many-electron systems via neural wave functions. This paper identifies critical limitations of existing methods and introduces a new approach using Neural Pfaffians (NeurPf) to overcome these barriers.

Summary of Contributions

1. Introduction of Neural Pfaffians:
   • The authors propose overparametrized, fully learnable neural wave functions that generalize well across different molecular systems. Unlike traditional models which rely on Slater determinants, NeurPf utilizes Pfaffians. This choice lifts the rigid constraints on the electronic spin configurations and molecular structures, thus enhancing the flexibility and generalization of the wave functions.
2. Pfaffian Wave Function:
   • By defining wave functions through Pfaffians, the authors enforce antisymmetry without mandating a fixed number of orbitals, as required by Slater determinants. This introduces overparametrization with $N_\geq\max\{N_\uparrow,N_\downarrow\}$. Consequently, it avoids the non-learnable discrete orbital selection procedures and achieves better flexibility and accuracy in modeling complex systems.
3. Memory-Efficient Envelopes:
   • A notable innovation is the use of memory-efficient exponential envelopes which decay to zero at large distances, reducing computational overhead while maintaining the required normalization properties of wave functions.
4. Pretraining Scheme:
   • The paper proposes novel pretraining strategies to align the neural Pfaffian wave functions with Hartree-Fock results. These strategies include matching single-electron orbitals and geminals, thereby stabilizing the optimization process and ensuring the wave function starts close to the ground state.
5. Empirical Evaluation:
   • Extensive empirical evaluations highlight the effectiveness of NeurPf. For instance, on the TinyMol dataset, NeurPf not only surpasses the chemical accuracy benchmark of CCSD(T) for small molecules but also achieves significantly lower error rates compared to previous generalized neural wave functions like Globe.

Detailed Insights and Results

• Ground State and Ionization Energies:
  • The paper demonstrates that a single NeurPf can compute ground state and ionization energies with chemical accuracy across various systems. Notably, it outperforms gold-standard CCSD(T) CBS reference energies by \SI{1.9}{\milli\hartree}, reducing energy errors by up to an order of magnitude compared to previous methods.
• Generalization Across Spin Configurations and Systems:
  • NeurPf shows exceptional performance on second-row elements and their ionization potentials and electron affinities, including systems with different spin configurations. It maintains high accuracy where previous methods like Globe would degrade performance due to their restrictions to singlet state systems.
• Potential Energy Surface of Nitrogen:
  • In modeling the nitrogen potential energy surface, NeurPf maintains an average error of \SI{2}{\milli\hartree}. Unlike Globe, which suffers from accuracy deprecation when trained with additional structures (e.g., ethene), NeurPf's accuracy remains stable.
• Dataset Generalization:
  • On the more comprehensive TinyMol dataset, NeurPf sustains its superior performance, even surpassing the accuracy of CCSD(T) by \SI{1.9}{\milli\hartree} for small structures. It also reveals a significant reduction in energy errors on larger molecules, further establishing its robustness and reliability.

Implications and Future Work

• Practical Implications:
  • The development of NeurPf heralds a new phase in computational chemistry, allowing for more accurate and generalizable neural wave functions. This advancement could significantly impact the fields of drug discovery and materials science, where understanding molecular interactions at quantum levels is crucial.
• Theoretical Implications:
  • The proposed methodology provides a new perspective on dealing with the antisymmetry constraints of many-electron systems. By leveraging Pfaffians, this work paves the way for future explorations into overparametrized models and their applications in diverse quantum systems.
• Future Developments:
  • Beyond current implementations, future research could explore extending NeurPf to periodic systems and integrating symmetry-aware neural network architectures. These enhancements could further propel the utility and applicability of neural network VMC in both academic and industrial settings.

In conclusion, the introduction of Neural Pfaffians as proposed by this paper showcases a significant advancement in accurately and flexibly solving many-electron Schrödinger equations. The meticulous empirical verifications underpin a framework that combines theoretical soundness with practical efficiency, suggesting numerous avenues for future advancements in quantum chemistry and artificial intelligence.