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  • Open Access

Particle View of Many-Body Electronic Structure with Neural Network Wave Function

Zichen Wang1,*, Weizhong Fu1,2,*, Zhe Li2, Weiluo Ren2,†, and Ji Chen1,3,4,‡

  • 1School of Physics, Peking University, Beijing 100871, People’s Republic of China
  • 2ByteDance Seed, Beijing, People’s Republic of China
  • 3Interdisciplinary Institute of Light-Element Quantum Materials and Research Center for Light-Element Advanced Materials, Peking University, Beijing 100871, People’s Republic of China
  • 4State Key Laboratory of Artificial Microstructure and Mesoscopic Physics and Frontiers Science Center for Nano-Optoelectronics, Peking University, Beijing 100871, People’s Republic of China

  • *These authors contributed equally to this work.
  • †Contact author: renweiluo@bytedance.com
  • ‡Contact author: ji.chen@pku.edu.cn
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Phys. Rev. X 16, 031048 – Published 24 August, 2026

DOI: https://doi.org/10.1103/g6v2-grnl

Abstract

In the study of electronic structure, the wave function view dominates the current research landscape and forms the theoretical foundation of modern quantum mechanics. In contrast, valence bond (VB) theory represents chemical bonds as shared electron pairs and can provide an intuitive, particle-based insight into chemical bonding. In this work, using a newly developed periodic dynamic Voronoi Metropolis sampling (PDVMS) method, we project classical many-body electronic configurations from the neural network wave function and apply VB theory to construct a complementary particle-view paradigm. The powerful neural network wave function can achieve near-exact ab initio solutions for the ground state of both molecular and solid systems. It allows us to definitively characterize the ground state of benzene by reassessing the competition between its two VB structures. Extending PDVMS to solids, we also predict a spin-staggered VB structure in graphene, which explains emergent magnetic properties in graphene-based nanostructures. Furthermore, this particle view provides insight into the optimization process of the neural network wave function itself. Our work thus introduces a framework for analyzing many-body electronic structure in molecules and solids, opening new avenues for investigating this complex problem and its associated exotic phenomena.

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Popular Summary

Quantum mechanics can predict electron behavior, but the concept of the wave function is challenging for humans to interpret. The mathematical complexity obscures the intuitive, particle-based picture of chemical bonds used in chemistry. We introduce a framework that translates the complex quantum wave function into a clear, classical picture of electron positions. We combine a neural network that calculates a near-exact ground-state wave function with a recently developed method called periodic dynamic Voronoi Metropolis sampling that we extended to periodic solids. This combination pins down the locations of electrons from the wave function’s probability cloud. We applied this framework to benzene and graphene. We found graphene’s ground state features a “spin-staggered” structure, in contrast to the expected network of alternating single and double bonds, providing an intuitive explanation for the emergent magnetic properties observed in similar nanostructures. Our framework provides a complementary lens to visualize and understand the complex electronic choreography ties to exotic phenomena in molecules and solids.

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Phys. Rev. X 16, 031048– Published 24 August, 2026
  • Received 3 October 2025
  • Accepted 1 July 2026
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