Spin-Weighted Spherical Harmonics Enable Complete and Scalable E(3)-Equivariant Networks - read online free

Sep 21, 2026

E(3)-equivariant networks are promising for modeling 3D atomistic systems, but their scalability is constrained by the O(L^6) complexity of the Clebsch-Gordan tensor product (CGTP). The Gaunt tensor product (GTP) lowers complexity yet misses antisymmetric interaction paths, leading to incomplete expressivity. This work introduces SpinGTP, which generalizes from scalar functions to spin-weighted spherical harmonics (SWSH) to recover antisymmetric interactions while retaining GTP’s efficiency and enabling a more expressive parity-aware basis. Benchmarks across multiple chiral and non-centrosymmetric tasks show accuracy comparable to full CGTP, with improved performance for chiral materials and providing a complete, scalable route to high-order equivariance. Code is publicly available.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages29

Reading Time1 h 13 min

AccessFree to download

Introduction

Related Work

Gaunt Tensor Product of Spin-Weighted Spherical Harmonics

Gaunt Tensor Product Efficiency and the Antisymmetry Gap

Spin-weighted Spherical Harmonics for Restoring Expressivity

Real-Basis Spin-Weighted Spherical Harmonics

Parity-Equivariant Spin-Weighted Spherical Harmonics

High-Performance Implementation of SpinGTP

Specialized SWSH Equivariant Layers

Experiments

Chiral Tetris Classification

3BPA Performance

SPICE-MACE-OFF Chiral Subset Performance

OC20 IS2RE Direct

Limitations and Summary

Spin-Weighted Spherical Harmonics (SWSH) Appendix

arXiv :2607 .0 1408v2 [ cs .LG] 3 Jul 2026 July 7, 2026 Spin-Weighted Spherical Harmonics Enable Complete and Scalable E(3)-Equivariant Networks Chenxing Liang1∗, Yuchao Lin1∗, Andrii Kryvenko1∗, Wendi Yu1 , Chuan Li2 , Jianwen Xie2 , Xiaofeng Qian3,4,5 , Shuiwang Ji1,3,6 1 Department of Computer Science and Engineering, Texas A&M University 2 Lambda, Inc. 3 Department of Materials Science and Engineering, Texas A&M University 4 Department of Electrical and Computer Engineering, Texas A&M University 5 Department of Physics and Astronomy, Texas A&M University 6 Department of Mechanical Engineering, Texas A&M University Abstract: E (3)-equivariant networks are promising for 3D atomistic system modeling, yet their scalability is limited by the O(L6 ) complexity of the Clebsch-Gordan Tensor Product (CGTP) . The recently proposed Gaunt Tensor Product (GTP) reduces the complexity but is unable to capture the antisymmetric paths, resulting in incomplete expressivity. In this work, we present SpinGTP, an approach to overcome the GTP incompleteness by generalizing from scalar functions to Spin-Weighted Spherical Harmonics (SWSH) . By relying on the algebraic properties of SWSH, SpinGTP recovers the missing antisymmetric interactions while maintaining the asymptotic efficiency of GTP. It also allows for a more expressive equivariant basis that naturally accounts for the parity-odd components of tensor products. We evaluate SpinGTP across diverse benchmarks, including Tetris, 3BPA, SPICE-MACE-OFF, and OC20 . Our results show that SpinGTP achieves accuracies comparable to full CGTP. Notably, by explicitly capturing antisymmetric paths, SpinGTP exhibits superior performance in tasks involving chiral materials and non-centrosymmetric geometries. This work provides a complete, scalable, and mathematically rigorous path toward high-order equivariance in large-scale 3D atomistic system simulations. Our code is publicly available at [https://github.com/divelab/SpinGTP/](https://github.com/divelab/SpinGTP/) . Keywords: AI for Science, Equivariant Networks, Spin-Weighted Spherical Harmonics, 3D Atomistic Simulations * These authors contributed equally Github Repository: [https://github.com/divelab/SpinGTP/](https://github.com/divelab/SpinGTP/) Contents 1 Introduction 4 2 Related Work 5 3 Gaunt Tensor Product of Spin-Weighted Spherical Harmonics 5 3.1 Gaunt Tensor Product Efficiency and the Antisymmetry Gap .................. 5 3.2 Spin-weighted Spherical Harmonics for Restoring Expressivity ................. 6 3.3 Real-Basis Spin-Weighted Spherical Harmonics .......................... 7 3.4 Parity-Equivariant Spin-Weighted Spherical Harmonics ..................... 8 3.5 High-Performance Implementation of SpinGTP .......................... 8 3.6 Specialized SWSH Equivariant Layers ............................... 9 4 Experiments 9 4. 1 Chiral Tetris Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 3BPA Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.3 SPICE-MACE-OFF Chiral Subset Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.4 OC20 IS2RE Direct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Limitations and Summary 13 A Spin-Weighted Spherical Harmonics (SWSH) 17 A.1 Explicit Formula .......................................... 17 A.2 Equivariance and Spin-Weighted Transformations ........................ 17 A.3 Nonzero Spin-Weighted Gaunt Paths ............................... 18 B Best Asymptotic Runtime Cost 19 C Real Basis Spin-Weighted Spherical Harmonics 20 C.1 The Unitary Q Matrix ....................................... 20 C.2 Example: The Q Matrix for ℓ = 1 ................................. 21 D Derivation of the Parity-Equivariant Basis 21 E SpinGTP Implementation and Time Comparison 22 E.1 Tensor Prod

FAQs

What limitation of CGTP affects scalability in E(3)-equivariant networks?

CGTP has computational complexity of O(L^6) with the maximum angular degree L, which often forces using small L and limits high-order geometric information.

Why does GTP lead to incomplete expressivity?

GTP reduces complexity but cannot capture antisymmetric paths, which creates an antisymmetry gap and prevents full expressivity.

How does SpinGTP address GTP’s incompleteness while keeping efficiency?

SpinGTP generalizes from scalar functions to spin-weighted spherical harmonics (SWSH), using their algebraic properties to recover missing antisymmetric interactions while maintaining GTP’s asymptotic efficiency and improving parity-aware expressivity.

Sep 21, 2026

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