Spin-Weighted Spherical Harmonics Enable Complete and Scalable $\mathrm{E}(3)$-Equivariant Networks

• URL: http://arxiv.org/abs/2607.01408v2
• Date: Fri, 03 Jul 2026 12:45:35 GMT
• Title: Spin-Weighted Spherical Harmonics Enable Complete and Scalable $\mathrm{E}(3)$-Equivariant Networks
• Abstract summary: We present SpinGTP, an approach to overcome the GTP incompleteness.<n>We evaluate SpinGTP across diverse benchmarks, including Tetris, 3BPA, SPICE-MACE-OFF, and OC20.<n>This work provides a complete, scalable, and mathematically rigorous path toward high-order equivariant in large-scale 3D atomistic system simulations.
• Score: 46.25293959572394
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: $\mathrm{E}(3)$-equivariant networks are promising for 3D atomistic system modeling, yet their scalability is limited by the $O(L^6)$ 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.

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