Excited Pfaffians: Generalized Neural Wave Functions Across Structure and State

2026-03-15

AI Summary

This work addresses the prohibitive computational cost in variational Monte Carlo when estimating high-accuracy overlaps among multiple excited states with conventional neural network wave functions, a challenge that escalates sharply with the number of states due to rapidly increasing sampling requirements. The authors propose a multi-state importance sampling (MSIS) method combined with a novel Excited Pfaffians neural network architecture, enabling efficient representation of multiple electronic states within a single model and achieving near-constant sampling complexity for overlap estimation. By integrating Hartree–Fock principles into the neural wave function, the approach yields a unified, transferable representation across molecules and excitation manifolds. Demonstrations include modeling 50% more states in the carbon dimer at over 200× faster training speed while matching natural orbital accuracy, and the first complete neural-network-based solution of all discrete energy levels of the beryllium atom, alongside cross-molecular excited-state representation within a single model.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work
• Social Networks and Social Media: Computational social science

Abstract

Neural-network wave functions in Variational Monte Carlo (VMC) have achieved great success in accurately representing both ground and excited states. However, achieving sufficient numerical accuracy in state overlaps requires increasing the number of Monte Carlo samples, and consequently the computational cost, with the number of states. We present a nearly constant sample-size approach, Multi-State Importance Sampling (MSIS), that leverages samples from all states to estimate pairwise overlap. To efficiently evaluate all states for all samples, we introduce Excited Pfaffians. Inspired by Hartree-Fock, this architecture represents many states within a single neural network. Excited Pfaffians also serve as generalized wave functions, allowing a single model to represent multi-state potential energy surfaces. On the carbon dimer, we match the $O(N_s^4)$-scaling natural excited states while training $>200\times$ faster and modeling 50\% more states. Our favorable scaling enables us to be the first to use neural networks to find all distinct energy levels of the beryllium atom. Finally, we demonstrate that a single wave function can represent excited states across various molecules.

Problem

• neural-network wave functions
• excited states
• state overlaps
• computational cost
• Variational Monte Carlo

Innovation

• Excited Pfaffians
• Multi-State Importance Sampling
• Neural Wave Functions
• Variational Monte Carlo
• Multi-state Potential Energy Surfaces