Neural Pfaffians: Solving Many Many-Electron Schrödinger Equations

TL;DR

This work defines overparametrized, fully learnable neural wave functions suitable for generalization across molecules by relying on Pfaffians rather than Slater determinants and finds that a single neural Pfaffian calculates the ground state and ionization energies with chemical accuracy across various systems.

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.

Figures (16)

• Figure 1: Schematic of the Slater determinant (\ref{['fig:schematic-slater']}) and our NeurPf (\ref{['fig:schematic-pfaffian']}). Where the Slater formulation requires exactly ${N_\mathrm{e}}$ orbital functions, the Pfaffian formulation works for any number ${N_\mathrm{o}}\geq\max\{{N_\uparrow},{N_\downarrow}\}$ of orbital functions, indicated by the rectangular orbital blocks.
• Figure 2: Orbital parametrization per nucleus. , /indicate electrons and nuclei, respectively.
• Figure 3: Ground state, electron affinity, and ionization potential errors of second-row elements during training. A single NeurPf has been trained on all systems jointly while references pfauInitioSolutionManyelectron2020 were calculated separately for each system. Energies are averaged over the last 10% of steps.
• Figure 4: Potential energy surface of nitrogen. Energies are relative to leroyAccurateAnalyticPotential2006.
• Figure 5: Convergence of mean energy difference on the TinyMol dataset from scherbelaTransferableFermionicNeural2024. The y-axis is linear $<1$ and logarithmic $\geq 1$. Due to the variational principle, NeurPf is better than the reference CCSD(T) on the small molecules.