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arXiv:2511.01671 · Nature Communications

Spin-adapted neural network wavefunctions in real space

Ruichen Li, Yuzhi Liu A, Jiang Du, Yixiao Chen, Xuelan Wen, Wenrui Li, Di He, Liwei Wang, Ji Chen, Weiluo Ren

Published Oct 24, 2026• 2 citations
arXivHTMLPDF

Abstract

Spin plays a fundamental role in understanding electronic structure, yet many real-space wavefunction methods fail to adequately consider it. We introduce the spin-adapted antisymmetrization method, a general procedure that enforces exact total spin symmetry for antisymmetric many-electron wavefunctions in real space. In the context of neural network-based quantum Monte Carlo, our method leverages the expressiveness of deep neural networks to capture electron correlation while enforcing exact spin adaptation via group representation theory. This framework provides a principled route to embed physical priors into otherwise black-box neural network wavefunctions, yielding a compact representation of correlated systems with neural network orbitals. To demonstrate its effectiveness, we study the spin ladder of iron-sulfur clusters, a long-standing challenge for many-body methods due to their dense spectrum of nearly degenerate spin states. Our results reveal accurate resolution of low-lying spin states and spin gaps in [Fe2S2] and [Fe4S4] clusters, offering insights into their electronic structures. In sum, these findings establish our framework as a practical route toward spin-adapted neural network-based quantum Monte Carlo for strongly correlated systems. The authors introduce a general framework that embeds exact spin symmetry directly into the neural network wavefunction, guaranteeing spin-pure states by construction. This design enables accurate resolution of near-degenerate spin states in challenging chemical systems.

DOI
10.1038/s41467-026-78600-y

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Spin plays a fundamental role in understanding electronic structure, yet many real-space wavefunction methods fail to adequately consider it. We introduce the spin-adapted antisymmetrization method, a general procedure that enforces exact total spin symmetry for antisymmetric many-electron wavefunctions in real space. In the context of neural network-based quantum Monte Carlo, our method leverages the expressiveness of deep neural networks to capture electron correlation while enforcing exact spin adaptation via group representation theory. This framework provides a principled route to embed physical priors into otherwise black-box neural network wavefunctions, yielding a compact representation of correlated systems with neural network orbitals. To demonstrate its effectiveness, we study the spin ladder of iron-sulfur clusters, a long-standing challenge for many-body methods due to their dense spectrum of nearly degenerate spin states. Our results reveal accurate resolution of low-lying spin states and spin gaps in [Fe2S2] and [Fe4S4] clusters, offering insights into their electronic structures. In sum, these findings establish our framework as a practical route toward spin-adapted neural network-based quantum Monte Carlo for strongly correlated systems. The authors introduce a general framework that embeds exact spin symmetry directly into the neural network wavefunction, guaranteeing spin-pure states by construction. This design enables accurate resolution of near-degenerate spin states in challenging chemical systems.
Strongly correlated molecules often contain dense manifolds of low-lying spin states, making total-spin symmetry essential for predictive electronic-structure theory. Neural-network quantum states provide flexible variational wavefunctions, but commonly used fermionic architectures do not enforce this symmetry and can therefore converge to spin-contaminated states with misleading energies and properties. Here we introduce a spin-adapted neural-network backflow (SA-NNBF) ansatz in second quantization, which combines configuration-dependent spatial orbitals with a compressed spin eigenfunction. A projected tensor compression scheme for spin eigenfunctions and a particle-hole representation make variational Monte Carlo calculations with SA-NNBF practical for active spaces containing more than one hundred electrons. Across hydrogen chains and iron-sulfur clusters, SA-NNBF eliminates spin contamination and consistently achieves lower variational energies than standard NNBF with a comparable number of parameters. For the CAS(113e,76o) active-space model of FeMoco, SA-NNBF yields a highly compact spin-adapted variational state, achieving an energy competitive with recent spin-adapted DMRG calculations at bond dimension D=10000D=10000D=10000 while using orders of magnitude fewer parameters. Our work establishes a general framework for developing spin-symmetry-preserving neural-network quantum states for chemically realistic strongly correlated electrons.
The integration of deep neural networks with the Variational Monte Carlo (VMC) method has marked a significant advancement in solving the Schrödinger equation. In this work, we enforce spin symmetry in the neural network-based VMC calculation with modified optimization target. Our method is designed to solve for the ground state and multiple excited states with target spin symmetry at a low computational cost. It predicts accurate energies while maintaining the correct symmetry in strongly correlated systems, even in cases where different spin states are nearly degenerate. Our approach also excels at spin-gap calculations, including the singlet-triplet gap in biradical systems, which is of high interest in photochemistry. Overall, this work establishes a robust framework for efficiently calculating various quantum states with specific spin symmetry in correlated systems, paving the way for novel discoveries in quantum science.
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Directly solving the full electron-nuclear Schrödinger equation remains one of the grand challenges in quantum mechanics. Here, we propose a quantum-chemically motivated orbital-based neural network wavefunction ansatz with explicit many-body electron-electron correlation terms, with parameters depending on the entire electron-nuclear configuration, enabling an accurate description of strongly correlated and/or multi-reference electronic structures by a single Slater-determinant across a broad nuclear configuration space. To train this wavefunction, we further develop an efficient hybrid optimization strategy that combines the variational Monte Carlo technique with local-energy constraints, mitigating the intrinsic sampling bias of the former and considerably reducing the statistical error. These advances are embedded in a unified neural network framework, Schrödinger, enabling the determination of globally accurate potential energy surfaces with a single training for systems ranging from strongly correlated diatomic molecules, triatomic reactions containing conical intersections, to polyatomic multi-reference molecules. This work offers a practical machine-learning tool for solving the electron-nuclear Schrödinger equation and capturing full quantum effects beyond the Born-Oppenheimer approximation.
Simulating strongly correlated fermionic systems remains a fundamental challenge in quantum physics, largely due to the sign problem in quantum Monte Carlo (QMC) methods. We present a neural network-based variational Monte Carlo (NN-VMC) approach, leveraging a flexible neural network ansatz to represent the many-body wavefunction. Focusing on quantum dots with up to 30 electrons, we demonstrate that NN-VMC significantly reduces variational bias and achieves ground-state energies surpassing those of fixed-node diffusion Monte Carlo (DMC). A key feature is that the neural network adaptively learns and optimizes nodal structures during energy minimization. We provide qualitative insights into the nodal structure of fermionic wavefunctions by comparing the nodal structures generated by NN-VMC with those obtained from traditional trial functions. Additionally, we reveal spin-resolved radial distributions and electron density profiles, highlighting the versatility and accuracy of NN-VMC. This work underscores the potential of machine learning to advance quantum simulations and deepen our understanding of strongly correlated systems.