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

2026-06-28

AI Summary

This work addresses the limitation of existing Clifford algebra–based interatomic potential models, which struggle to accurately predict force directions due to the geometric product’s inability to represent symmetric traceless second-order tensor components. The authors propose CliffordSTF, a novel approach that couples Clifford multivectors with explicit symmetric traceless tensor channels (second- and third-order) within message-passing layers and achieves higher-order equivariance through cross-channel bilinear contractions. Notably, this method integrates geometric algebra with high-order Cartesian tensors without relying on Clebsch–Gordan coefficients, Wigner-D matrices, or the e3nn library, yielding an efficient and complementary equivariant representation. On the rMD17 benchmark, it improves the cosine similarity of force directions from 0.055 to 0.551 and reduces force and energy MAE by 15.8% and 10.9%, respectively, while also achieving state-of-the-art energy prediction performance on the OC22 catalysis benchmark.

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.

Problem

• Geometric Algebra
• Interatomic Potentials
• Equivariance
• Symmetric-Traceless Tensors
• Force Direction

Innovation

• Geometric Algebra
• Symmetric-Traceless Tensors
• Equivariant Interatomic Potentials
• CliffordSTF
• Higher-Order Representations