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TraceGrad: a Framework Learning Expressive SO(3)-equivariant Non-linear Representations for Electronic-Structure Hamiltonian Prediction

ICML 2025 · Probabilistic and scientific modeling › Molecules and proteins · approach: Equivariant and disentangled latent representations · 2 citations (OpenAlex)

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Proposes TraceGrad, which supervises SO(3)-invariant trace quantities tr(H·H†) to learn non-linear invariant features, then derives SO(3)-equivariant features via gradients of those invariants, improving DeepH-E3 and QHNet Hamiltonian prediction.

Problem

Predicting electronic-structure Hamiltonians with both strict SO(3)-equivariance and strong non-linear expressiveness; applying non-linearities directly to equivariant features (degree l≥1) breaks equivariance.

Approach

Use the trace tr(H·H†) of equivariant Hamiltonian blocks as an invariant supervision label for an invariant non-linear branch, and obtain equivariant features as the gradient of that invariant scalar with respect to the equivariant input feature, which preserves equivariance and dimensional consistency with H.

Key results

• DeepH-E3+TraceGrad lowers MAE_all and challenging-sample MAE by up to 34% and 35% relative to DeepH-E3 across six DeepH databases.
• QHNet+TraceGrad on QH9-dynamic reduces MAE_all from 4.733 to 2.819 meV and orbital energy MAE from 264.483 to 63.375 meV.
• On QH9-stable, QHNet+TraceGrad reduces MAE_all from 1.962 to 1.191 meV and raises wavefunction similarity from 0.937 to 0.948.

Measured effects reported in the paper

Limitations

• Equivariance is proven theoretically but the gradient-based construction relies on differentiable invariant branches; the paper's gated-mechanism comparison is only empirical in an appendix.
• Results on the DeepH series for TraceGrad are averages over 10 runs with a fixed seed, so seed variance is small by design.
• Evaluated only on DeepH-E3 and QHNet backbones.

Experimental setup

Datasets: DeepH benchmark databases (MG, MM, BG, BB, BT, BS) QH9-stable QH9-dynamic

Benchmarks: DeepH series (MG, MM, BG, BB, BT, BS) QH9-stable QH9-dynamic

Metrics: MAE (meV) MAE on challenging samples/blocks orbital energy MAE wavefunction cosine similarity

Baselines: DeepH-E3 QHNet

Builds on: DeepH-E3 QHNet gated activation

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