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arXiv:2405.14762 · neural information processing systems

Neural Pfaffians: Solving Many Many-Electron Schrödinger Equations

Nicholas Gao, Stephan Günnemann

Published Jan 1, 2024• 3 citations
arXivHTMLPDF

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.

DOI
10.52202/079017-3981

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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 1.9mEhE_hEh​ and reduce energy errors compared to previous generalized neural wave functions by up to an order of magnitude.
Recent neural network-based wave functions have achieved state-of-the-art accuracies in modeling ab-initio ground-state potential energy surface. However, these networks can only solve different spatial arrangements of the same set of atoms. To overcome this limitation, we present Graph-learned orbital embeddings (Globe), a neural network-based reparametrization method that can adapt neural wave functions to different molecules. Globe learns representations of local electronic structures that generalize across molecules via spatial message passing by connecting molecular orbitals to covalent bonds. Further, we propose a size-consistent wave function Ansatz, the Molecular orbital network (Moon), tailored to jointly solve Schrödinger equations of different molecules. In our experiments, we find Moon converging in 4.5 times fewer steps to similar accuracy as previous methods or to lower energies given the same time. Further, our analysis shows that Moon's energy estimate scales additively with increased system sizes, unlike previous work where we observe divergence. In both computational chemistry and machine learning, we are the first to demonstrate that a single wave function can solve the Schrödinger equation of molecules with different atoms jointly.
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Given access to accurate solutions of the many-electron Schrödinger equation, nearly all chemistry could be derived from first principles.Exact wave functions of interesting chemical systems are out of reach because they are NP-hard to compute in general, but approximations can be found using polynomially scaling algorithms.The key challenge for many of these algorithms is the choice of wave function approximation, or Ansatz, which must trade off between efficiency and accuracy.Neural networks have shown impressive power as accurate practical function approximators and promise as a compact wave-function Ansatz for spin systems, but problems in electronic structure require wave functions that obey Fermi-Dirac statistics.Here we introduce a novel deep learning architecture, the Fermionic neural network, as a powerful wave-function Ansatz for many-electron systems.The Fermionic neural network is able to achieve accuracy beyond other variational quantum Monte Carlo Ansatz on a variety of atoms and small molecules.Using no data other than atomic positions and charges, we predict the dissociation curves of the nitrogen molecule and hydrogen chain, two challenging strongly correlated systems, to significantly higher accuracy than the coupled cluster method, widely considered the most accurate scalable method for quantum chemistry at equilibrium geometry.This demonstrates that deep neural networks can improve the accuracy of variational quantum Monte Carlo to the point where it outperforms other ab initio quantum chemistry methods, opening the possibility of accurate direct optimization of wave functions for previously intractable many-electron systems.
Recent advances in machine learning have facilitated numerically accurate solution of the electronic Schrödinger equation (SE) by integrating various neural network (NN)-based wave function ansatzes with variational Monte Carlo methods. Nevertheless, such NN-based methods are all based on the Born-Oppenheimer approximation (BOA) and require computationally expensive training for each nuclear configuration. In this work, we propose a novel NN architecture, SchrödingerNet, to solve the full electronic-nuclear SE by defining a loss function designed to equalize local energies across the system. This approach is based on a translationally, rotationally and permutationally symmetry-adapted total wave function ansatz that includes both nuclear and electronic coordinates. This strategy not only allows for an efficient and accurate generation of a continuous potential energy surface at any geometry within the well-sampled nuclear configuration space, but also incorporates non-BOA corrections, through a single training process. Comparison with benchmarks of atomic and small molecular systems demonstrates its accuracy and efficiency.