Spin-Weighted Spherical Harmonics Enable Complete and Scalable $\mathrm{E}(3)$-Equivariant Networks
For researchers in 3D atomistic modeling, this provides a scalable and complete equivariant network that captures antisymmetric interactions, addressing a key limitation of prior efficient approaches.
The authors propose SpinGTP, a method that uses Spin-Weighted Spherical Harmonics to achieve complete E(3)-equivariance with O(L^6) complexity, matching full CGTP accuracy while improving scalability. It outperforms prior methods on chiral and non-centrosymmetric tasks.
$\mathrm{E}(3)$-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.