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

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 $\mathbb{R}<sup>3$ 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.

• The paper introduces CliffordSTF, a dual-track architecture that combines Clifford multivectors with rank-2 and rank-3 symmetric-traceless tensors to recover missing angular components through L=3 without Clebsch–Gordan tables.
• CliffordSTF raises rMD17 force-cosine similarity from 0.055 for the base Clifford model to 0.551, while reducing force MAE by 15.8% and energy MAE by 10.9% at a matched approximately 1-million-parameter budget.
• The results show that higher-order features must participate in message passing through bidirectional cross-track coupling, although the model remains less directionally accurate than EquiformerV2 and incurs higher training costs.

The structural gap in Clifford-algebra interatomic potentials

Machine learning interatomic potentials (MLIPs) require features that transform correctly under rotations, and the dominant constructions—Clebsch–Gordan (CG) tensor products over spherical harmonics—carry a combinatorial cost that grows steeply with maximum angular momentum $L_{\max}$. Clifford (geometric) algebra offers an alternative: embed atomic environments into the eight-dimensional algebra $\mathrm{Cl}(3,0)$ and use the closed-form geometric product as the sole bilinear. This paper identifies a precise expressivity deficiency in that lineage and proposes an architecture, CliffordSTF, that repairs it.

The diagnosis is algebraic. Under $\mathrm{SO}(3)$, the graded components of $\mathrm{Cl}(3,0)$ decompose as $2\cdot L{=}0 \oplus 2\cdot L{=}1$: grade 0 and grade 3 carry $L{=}0$, while grade 1 and the Hodge dual of grade 2 carry $L{=}1$. No grade contains the five-dimensional $L{=}2$ irrep. Because the geometric product is closed within the algebra, the per-edge bilinear that drives every message-passing layer can only realise the $L{=}0$ and $L{=}1$ pieces of $\mathrm{Cl}(3,0)$0; the symmetric-traceless rank-2 component is absent regardless of depth. Two clarifications matter here. Higher algebraic grade does not correspond to higher angular momentum—bivectors transform as $\mathrm{Cl}(3,0)$1 pseudovectors—and the same deficit carries over to the projective algebra $\mathrm{Cl}(3,0)$2 used by GATr, whose null basis encodes translational rather than angular content.

The empirical consequence is stark. Across ten rMD17 molecules, every $\mathrm{Cl}(3,0)$3 baseline in the paper's twelve-model study attains aggregate force-cosine similarity below 0.25; the base Clifford model reaches only 0.055. Force cosine similarity isolates directional fidelity from magnitude accuracy, and directional fidelity governs whether molecular dynamics trajectories remain in the correct potential-energy basin even when scalar errors look small.

Constructive CG recovery via STF tracks

CliffordSTF couples the Clifford multivector to closed-form symmetric-traceless (STF) Cartesian tensor tracks at ranks two and three. The theoretical core is a constructive theorem: for vectors $\mathrm{Cl}(3,0)$4 carrying the $\mathrm{Cl}(3,0)$5 irrep, the full Clebsch–Gordan decomposition of $\mathrm{Cl}(3,0)$6 is recovered as

$\mathrm{Cl}(3,0)$7

where $\mathrm{Cl}(3,0)$8. Analogous contractions recover the higher couplings $\mathrm{Cl}(3,0)$9, $\mathrm{SO}(3)$0, and $\mathrm{SO}(3)$1 up to an $\mathrm{SO}(3)$2 cut (the $\mathrm{SO}(3)$3 component of $\mathrm{SO}(3)$4 is omitted). Every map is a polynomial with constant rational coefficients matching the classical detracing rule $\mathrm{SO}(3)$5 in $\mathrm{SO}(3)$6, and each was verified against a Wigner-$\mathrm{SO}(3)$7 reference implementation to below $\mathrm{SO}(3)$8 absolute error. The construction uses one learned bilinear—the geometric product, implemented as an $\mathrm{SO}(3)$9 Cayley table—with no CG tables, no Wigner-$\mathrm{Cl}(3,0)$0 matrices, and no e3nn calls.

Architecture

The model carries two parallel feature substrates per atom: a Clifford multivector ($\mathrm{Cl}(3,0)$1, dimension 8) and STF channels at ranks 2 and 3 (5 and 7 components respectively). Message passing proceeds on both tracks simultaneously. The Clifford track uses the geometric product with grade-sparse dispatch and invariant dot-product attention; the STF tracks are generated from vector features and propagated with radial modulation. The critical element is bidirectional cross-track coupling: grade-1 vectors generate STF$\mathrm{Cl}(3,0)$2 features upward, while STF$\mathrm{Cl}(3,0)$3 features contract back into grade-1 through $\mathrm{Cl}(3,0)$4. Without this coupling, STF parameters receive only weak invariant-norm gradients and training degrades—a point confirmed empirically. An iterated augmented product at the many-body stage provides correlations up to body order 4 through six bilinear terms, each corresponding bijectively to a CG channel. Forces are read out from concatenated polar-vector and Hodge-dual bivector channels plus a scatter-summed $\mathrm{Cl}(3,0)$5 contribution gated by the STF$\mathrm{Cl}(3,0)$6 norm—an explicit $\mathrm{Cl}(3,0)$7 path capturing angular force components inaccessible to $\mathrm{Cl}(3,0)$8 architectures.

All equivariance guarantees are $\mathrm{Cl}(3,0)$9-only. Under $2\cdot L{=}0 \oplus 2\cdot L{=}1$0, polar vectors and axial vectors carry opposite parity, so full inversion equivariance would require explicit parity tracking; the current force readout concatenates them, which is consistent under rotations but conflates parities under reflections.

Force directionality on rMD17

The headline diagnostic is aggregate force-cosine similarity on rMD17 at a matched $2\cdot L{=}0 \oplus 2\cdot L{=}1$1-parameter budget:

Model	Aggregate $2\cdot L{=}0 \oplus 2\cdot L{=}1$2
EquiformerV2 $2\cdot L{=}0 \oplus 2\cdot L{=}1$3–$2\cdot L{=}0 \oplus 2\cdot L{=}1$4	0.956–0.961
CliffordSTF	0.551 ± 0.064
MACE $2\cdot L{=}0 \oplus 2\cdot L{=}1$5	0.228
ICTP $2\cdot L{=}0 \oplus 2\cdot L{=}1$6	0.174
Base Clifford	0.055
PaiNN / ViSNet / NequIP	≤ 0.014

The decisive comparison is Clifford versus CliffordSTF, which share infrastructure, protocol, and budget and differ only by the STF tracks. Force-cosine similarity rises from 0.055 to 0.551—an order-of-magnitude relative gain—while force MAE improves from 20.9 to 17.6 meV/Å (15.8% lower) and energy MAE from 34.1 to 30.4 meV (10.9% lower). Directional and magnitude accuracy improve together rather than trading off. CliffordSTF also outperforms every CG-free or body-ordered baseline evaluated (all ≤ 0.17), though it remains well short of fully spherical-harmonic EquiformerV2 at ~0.96. Four baselines carry caveats: SchNet and DimeNet++ are strictly invariant, FAENet uses stochastic frame averaging, GotenNet did not converge under the fixed-budget protocol, and EquiformerV2's energy MAE reflects unreferenced absolute energies.

Catalysis results

On OC22, CliffordSTF attains the best out-of-distribution S2EF energy MAE in the study (5.083 eV) and the best in-distribution IS2RE MAE among $2\cdot L{=}0 \oplus 2\cdot L{=}1$7 methods (4.089 eV, versus GotenNet's 4.318 eV). On OC20 it stays competitive across splits, though base Clifford holds the best S2EF energy MAE there (1.889 eV) and EquiformerV2 leads on force cosine similarity. A scaling check to $2\cdot L{=}0 \oplus 2\cdot L{=}1$8 parameters shows stable training: the best CliffordSTF seed reaches 0.0162 eV/atom on OC22 IS2RE, roughly a threefold gain, although EquiformerV2-L2 scales more steeply (0.0094 eV/atom) and overtakes it at this capacity. On scalar benchmarks the extension does not degrade performance: CliffordSTF ranks fifth of ten on QM9 aggregate MAE and mid-pack on Molecule3D.

Ablation: complementarity of the two tracks

An eleven-variant ablation on four rMD17 molecules separates three tiers. A vanilla $2\cdot L{=}0 \oplus 2\cdot L{=}1$9 Clifford configuration reaches $L{=}0$0; an STF-readout-only configuration (no STF inside message passing) reaches 0.097; only the full hybrid scaffold reaches $L{=}0$1. The roughly fivefold separation between the readout-only and scaffold tiers rules out the explanation that an $L{=}0$2 readout alone suffices—the $L{=}0$3 representations must participate throughout message passing. Within the scaffold, individual flag toggles produce effects often within seed variance on headline $L{=}0$4, but cross-track coupling matters substantially for stability: removing it raises mean energy MAE from 46.0 to 70.4 meV and inflates its standard deviation. The paper accordingly reads cross-track coupling as a training-stability lever rather than a raw-accuracy driver.

Cost profile

At matched budget, base Clifford infers at 0.128 s/step, faster than every $L{=}0$5 or Cartesian-tensor baseline including EquiformerV2 $L{=}0$6 (0.142 s). CliffordSTF infers at 0.352 s/step, competitive with slower spherical-harmonic baselines. Training is the weak point: CliffordSTF trains at 7.77 s/step, a two-to-threefold overhead against the fastest spherical-harmonic models, with forward/backward profiling attributing the residual to the autograd graph through the dual-track scaffold rather than the geometric-product dispatch itself. The authors estimate 40–60% further speedup is tractable via full torch.compile engagement (currently blocked upstream by torch_scatter custom-op tracing) and fused Triton kernels.

Limitations

The paper concedes several constraints plainly. Absolute directional fidelity remains below spherical-harmonic architectures—EquiformerV2's ~0.96 versus 0.551—so closing that gap without CG machinery is left open. The fixed-budget protocol deliberately targets cross-architecture comparability rather than per-architecture tuning, so MACE, ICTP, and GotenNet underconverge relative to their published recipes, and the catalysis results characterise the sub-10M-parameter regime rather than the leaderboard frontier. QM9 and Molecule3D required gradient clipping to prevent early-training divergence from the cross-track bilinear. Finally, the directional diagnostic is restricted to small organics in vacuum; whether the $L{=}0$7 clustering at poor cosine alignment persists on periodic or solvated systems remains untested.

Conclusion

This paper makes a precise structural claim—that the $L{=}0$8 geometric product cannot reach $L{=}0$9 content at the per-edge bilinear, and that this, not capacity or depth, drives the poor force-directional fidelity of Clifford-based potentials—and substantiates it with a controlled ablation showing an order-of-magnitude directional gain from adding closed-form STF tracks. The constructive theorem shows the hybrid spans the same SO(3)-irrep content as spherical harmonics through $L{=}1$0 without any learned angular bookkeeping. The broader lesson offered is that representational choice in equivariant architectures should follow the angular content actually required, not allegiance to a particular algebraic formalism. The remaining gap to fully spherical-harmonic models in absolute directional fidelity, and the training-cost overhead of the dual-track scaffold, define the concrete open problems this work leaves unresolved.