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

NeurIPS 2026 · Architectures and efficiency › RNNs and normalization · approach: Scalable approximation and equivariant compute

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Proposes SpinGTP, a Gaunt-style tensor product over real spin-weighted spherical harmonics that recovers antisymmetric (odd l-sum) coupling paths missing from scalar GTP while keeping O(L^4 log^2 L)-type efficiency, and evaluates on Tetris, 3BPA, SPICE chiral subsets, and OC20 IS2RE.

Problem

Scalar Gaunt Tensor Product reduces CG tensor product cost but removes odd l-sum paths (e.g., vector cross product), making E(3)-equivariant networks incomplete and weak on chiral/parity-sensitive 3D atomistic tasks.

Approach

Generalize the Gaunt coefficient from scalar spherical harmonics to spin-weighted spherical harmonics with signed spin weights satisfying s3 = s1 + s2, so the spin-weighted 3j factor is nonzero for odd l-sums; use real, parity-labeled SWSH bases and pre-contracted kernels to keep FFT-style efficiency.

Key results

• On 3BPA at 1200 K, SpinGTP reports energy RMSE 29.0 meV and force RMSE 61.4 meV/Å, best among compared models (MACE 29.8 / 62.0).
• On chirality parity generalization (train R, test S), SpinGTP reaches 94.1% accuracy on unseen S structures versus 0% for MACE-Gaunt and 88.6% for MACE.
• On OC20 IS2RE direct, SpinGTP achieves ID energy MAE 0.5066 eV, better than Equiformer's 0.5088, but a lower average (0.5648 vs 0.5489 for Equiformer).

Surprising findings

• Scalar GTP (MACE-Gaunt) reaches 0% accuracy on unseen S-chirality structures after training only on R, indicating complete failure of parity generalization.
• The authors report no clear speedup from the spherical transform itself, attributing this to large multiplicity relative to small L.

Limitations

• Nonzero spin features require consistent gauge/frame construction; ambiguity remains for highly symmetric point clouds.
• No clear acceleration observed from the spherical transform at small L with large multiplicity.
• On OC20 the model is only comparable to Equiformer, not clearly better on average.

Experimental setup

Datasets: Tetris (synthetic chiral) 3BPA SPICE-MACE-OFF chiral subset OC20 IS2RE

Benchmarks: Chiral Tetris 3BPA SPICE-MACE-OFF chiral subset OC20 IS2RE direct

Metrics: accuracy energy MAE force MAE EwT runtime (ms)

Baselines: Allegro NequIP BOTNet MACE MACE-Gaunt Equiformer EquiformerV2 SchNet DimeNet++ GemNet-dT SphereNet ComENet

Builds on: Gaunt Tensor Product MACE Equiformer e3nn Clebsch-Gordan tensor product

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