Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentials

• URL: http://arxiv.org/abs/2606.29584v1
• Date: Sun, 28 Jun 2026 19:57:50 GMT
• Title: Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentials
• Abstract summary: We show that CliffordSTF couples the Clifford multivector to closed-form symmetrictraceless tensor tracks at ranks two and three.<n>On rMD17, CliffordSTF raises aggregate force-cosine similarity from $0.055$ (base Clifford) to $0.551$, an order-of-magnitude relative directional gain, alongside improved magnitude accuracy (force MAE $15.8% lower; energy MAE $10.9% lower)<n>On benchmarks, CliffordSTF achieves best out-of-distribution S2EF energy MAE on OC22 in our experiments, and the best in-distribution energy MAE
• Score: 2.3982445219832678
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: $\mathrm{Cl}(3,0)$ interatomic potentials, despite their algebraic elegance, predict force magnitudes accurately but force directions poorly. Across ten rMD17 molecules, every $L \leq 1$ baseline in our twelve-model study attains aggregate force-cosine similarity below $0.25$. The cause is structural. The geometric product of two vectors in $\mathbb{R}^3$ realises only the $L=0$ and $L=1$ components of its irreducible representation content, leaving the symmetric-traceless rank-2 component absent from the per-edge bilinear that drives each message-passing layer. We address this with CliffordSTF, which couples the Clifford multivector to closed-form symmetric-traceless tensor tracks at ranks two and three through bilinear cross-track contractions, using a single learned bilinear and no Clebsch--Gordan tables, Wigner-$D$ matrices, or e3nn calls. On rMD17, CliffordSTF raises aggregate force-cosine similarity from $0.055$ (base Clifford) to $0.551$, an order-of-magnitude relative directional gain, alongside improved magnitude accuracy (force MAE $15.8\%$ lower; energy MAE $10.9\%$ lower). It outperforms all CG-free or body-ordered baselines in our study (all $\leq 0.17$). On catalysis benchmarks, CliffordSTF achieves the best out-of-distribution S2EF energy MAE on OC22 in our experiments, and the best in-distribution energy MAE among $L \geq 2$ methods on OC22 IS2RE. An eleven-variant ablation shows the two tracks are complementary: neither alone matches the combined model.

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