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SU(2)-Equivariant Spinor Message Passing for Molecular Magnetism

Erh-Wei Sheng, Beom Seok Kang, Yuchao Lin, Danish Khan, Anima Anandkumar

arXiv:2610.29111v1physics.chem-phcs.LG

TL;DR

Repeated SOC-dependent molecular-property calculations are costly, and equivariant surrogates typically use integer-rank features despite the spinor nature of electronic states. This paper introduces eSpinor, which propagates Hamiltonian-derived spinors with gauge-covariant message passing and an eigenbasis-independent readout. It reports improved cobalt ZFS prediction and Dy dynamical predictions, while emphasizing that static tensor errors need not predict dynamical accuracy.

  • Problem

    SOC-dependent molecular magnetic properties can require costly electronic-structure calculations, while equivariant models typically use integer-rank features despite electronic states transforming as spinors.

    Source: §I · p. 1

  • Method

    eSpinor initializes spinor features from an approximate SOC Hamiltonian and propagates them alongside integer-rank features using gauge-covariant message passing and a gauge-invariant readout.

    Source: §I · pp. 1–2

  • Results

    eSpinor lowers Co(II) ZFS tensor MAE versus NequIP by 29% in distribution and 23% for unseen ligands, and improves Dy electron-spin correlations while using 15.5% fewer parameters than NequIP+H.

    Source: §I · p. 2, §VI · p. 8

  • Takeaways & Limitations

    Similar static tensor errors can yield different dynamical errors, so model selection should reflect accuracy on the physical observables relevant to the intended application.

    Source: §I · p. 2

    The benchmarks are limited to mononuclear complexes because no suitable labeled multinuclear dataset was identified.

    Source: §VI · p. 9

Abstract

from arXiv · show

Predicting molecular magnetism often requires electronic-structure calculations with spin-orbit coupling (SOC), which are costly to repeat across many molecules. The underlying electronic states can transform as spinors, yet equivariant surrogates typically use only integer-rank features. Inspired by the spinor nature of these electronic states, we develop eSpinor, an SU(2)-equivariant graph neural network that propagates spinor features. We initialize spinor features from eigenvectors of an approximate SOC Hamiltonian and use gauge-covariant message passing with a gauge-invariant readout to make predictions independent of the arbitrary eigenbasis choice. On cobalt single-molecule magnet candidates, eSpinor improves zero-field-splitting predictions relative to NequIP, including for unseen ligands. For a dysprosium magnetic resonance imaging probe, eSpinor improves real electron-spin correlations over a Hamiltonian-informed baseline despite similar static crystal-field errors and fewer parameters, while accurately predicting low-field nuclear relaxation in water and after transfer to methanol without retraining. Overall, these results demonstrate that SU(2)-equivariant spinor message passing is a useful inductive bias for molecular magnetism. We also find that similar static tensor errors can yield different dynamical errors, motivating model selection based on accuracy for the intended physical observables.

I. INTRODUCTION

CASSCF+SOC calculations make repeated magnetic-property prediction costly, while existing equivariant models use integer-rank internal features despite SOC-coupled electronic states transforming as spinors. eSpinor addresses this gap with SU(2)-equivariant spinor message passing initialized from an approximate SOC Hamiltonian.

I. INTRODUCTION

  • Repeated CASSCF+SOC calculations can take hours per geometry, making screening or dynamics expensive.The cost reflects combinatorial growth of active-space configurations and the additional cost of averaging over low-lying states.Source: §I · p. 1
  • Existing models use integer-rank internal features even though SOC-coupled electronic states can transform as SU(2) spinors.This motivates testing whether spinor message passing helps predict SOC-dependent molecular properties.Source: §I · p. 1

II. SU(2)-EQUIVARIANT SPINOR MESSAGE PASSING

  • eSpinor propagates spinor features alongside integer-rank tensors, with gauge-covariant updates and a gauge-invariant readout for eigenbasis-independent predictions.The architecture updates both feature types and contracts spinors to supply additional integer tensors.Source: §I · p. 2, §II · p. 2

A. Spinor initialization

  • At magnetic metal sites, eigenvectors of an approximate ligand-field-plus-SOC Hamiltonian initialize spinors, while atomic-species embeddings initialize integer tensors.The approximate Hamiltonian reflects benchmark-specific metal–ligand bonding; its eigenvectors are expressed in coupled angular-momentum bases.Source: §II · p. 2, A. Spinor initialization · pp. 2–3

B. Addressing eigenbasis gauge freedom

Kramers-pair eigenvectors have a unitary basis freedom, so spinor features must transform covariantly while predictions remain independent of that choice. The message-passing design preserves this transformation law through shared operations and spinor contractions.

B. Addressing eigenbasis gauge freedom

  • For energetically isolated Kramers pairs, eigenvectors are defined only up to U(2) transformations, including phase changes and mixing of the pair.Predictions must therefore be independent of this eigenbasis choice.Source: B. Addressing eigenbasis gauge freedom · p. 3
  • Spinor features remain basis-dependent, but contracting their eigenbasis index produces gauge-invariant products that can be projected onto integer-rank tensors.Weights and gates act identically on both columns of each Kramers pair to preserve covariance.Source: B. Addressing eigenbasis gauge freedom · p. 3

C. Gauge-covariant message passing

  • The message-passing convolution combines integer–integer, spinor–integer, and conjugated spinor–spinor couplings with Clebsch–Gordan projections.The mixed coupling updates spinors without changing their eigenbasis index; contraction of the conjugated spinor product cancels the unitary basis transformation.Source: C. Gauge-covariant message passing · p. 4
  • Shared channel maps, angular projections, and invariant gates preserve the spinor transformation law at each layer, while contractions yield gauge-invariant integer tensors.In the mononuclear benchmarks, only the metal carries spinors, and ligand neighbors condition updates to that metal’s spinor features.Source: C. Gauge-covariant message passing · p. 4

D. Tensor readout

The tensor readout combines integer-rank features with gauge-invariant spinor contractions to produce predictions independent of the initial eigenbasis. The cobalt benchmark evaluates these predictions on in-distribution and unseen-ligand splits using several model families and initialization choices.

D. Tensor readout

  • The readout combines final integer-rank features with gauge-invariant spinor contractions projected onto the same output rank.Under the stated initialization assumptions, including separated Kramers pairs, the resulting tensors are gauge-invariant and obey the required SO(3) rotation law.Source: D. Tensor readout · p. 4

III. ZERO-FIELD SPLITTING IN COBALT COMPLEXES

  • The cobalt benchmark predicts a symmetric, traceless rank-2 ZFS tensor and evaluates NequIP, Hamiltonian-informed models, eqSNAP, and GM-NN.The eSpinor variants use point-charge or angular-overlap initialization.Source: D. Tensor readout · p. 4, §III · p. 5
  • Cobalt evaluation uses shared training, validation, in-distribution test, and unseen-ligand test splits; the latter contains only ligands absent from training.Table I reports cobalt prediction MAEs in cm−1, with tensor error averaged over all nine Cartesian entries.Source: §III · p. 5, Table I · p. 5

A. Effective spin Hamiltonian

In four-coordinate Co(II) complexes, eSpinor improves ZFS prediction and ligand generalization, while the supplied Dy passages describe the probe, reference data, and evaluation protocol. The Dy protocol tests predicted crystal-field coefficients, electron-spin correlations, and nuclear relaxation against CASSCF+SOC references.

B. Prediction accuracy and OOD ligand generalization

  • eSpinor (AOM) reduces cobalt tensor MAE versus NequIP by 29% in distribution and 23% for unseen ligands, and has the lowest tensor, D, and |E| errors on both test sets.NequIP+H improves in-distribution tensor error over NequIP but does not consistently improve unseen-ligand errors.Source: B. Prediction accuracy and OOD ligand… · p. 5
  • Within eSpinor, AOM lowers all reported errors versus point-charge initialization, but point-charge eSpinor also beats AOM-initialized NequIP+H on all three metrics across both test sets.The advantage is therefore not attributable to AOM alone.Source: B. Prediction accuracy and OOD ligand… · p. 5
  • For unseen ligands, eSpinor raises D-ranking Spearman correlation to 0.787 from 0.757 for NequIP and 0.675 for NequIP+H.Its 95th-percentile tensor error is 3.41 cm−1, versus 4.73 for NequIP and 5.88 for NequIP+H.Source: B. Prediction accuracy and OOD ligand… · p. 5

A. Hamiltonian and spin propagation

IV. CRYSTAL-FIELD LEARNING AND SPIN DYNAMICS IN A DYSPROSIUM COMPLEX

  • The Dy evaluation uses nested training sets up to 8k geometries and a separate 39k-frame test trajectory to compare crystal-field coefficients, electron-spin correlations, and nuclear relaxation against CASSCF+SOC.The data are evaluated first in water and then after transfer to methanol without retraining.Source: §IV · p. 6

B. Crystal-field prediction

Neural models outperform the tested eqSNAP configuration on crystal-field coefficient MAEs, but those static errors do not determine downstream dynamics. eSpinor performs best on real electron correlations and low-field nuclear relaxation at larger training sizes, while rankings vary with training size and field.

B. Crystal-field prediction

  • Neural models outperform the tested eqSNAP configuration on coefficient MAEs, with similar errors at 8k and no consistent neural-model ranking across ranks or training sizes.Rankwise MAEs average away coefficient-error signs and temporal ordering, which matter during propagation.Source: B. Crystal-field prediction · p. 6

C. Electron-spin correlations

  • The real-correlation advantage persists when every model uses the reference initial thermal state at 3 and 20 T.Individual-window and orientation errors are comparable to NequIP+H’s, but eSpinor has a smaller ensemble-mean signed error.Source: C. Electron-spin correlations · p. 6
  • From 4k to 8k, real-correlation error increases for all three neural models even as coefficient MAEs decrease.The authors say further investigation is needed to identify which changes cause the increase.Source: C. Electron-spin correlations · p. 6
  • 0.553% versus NequIP+H’s 1.140% is eSpinor’s real-correlation error at 8k; it has the lowest mean error from 1k through 8k.Neither model is trained on dynamics.Source: C. Electron-spin correlations · p. 6, Figure 4 · p. 7

D. Nuclear relaxation rates

  • At 8k, eSpinor’s absolute R1N and R2N MAEs are 0.51 and 0.49 s−1, versus 1.40 and 1.38 s−1 for NequIP+H; eSpinor leads at 4k and 8k.At smaller training sizes, eqSNAP leads at 500 samples and NequIP+H at 1k and 2k.Source: Figure 4 · p. 7, D. Nuclear relaxation rates · p. 7
  • At 20 T, Curie relaxation supplies about 93% of reference R2N, and NequIP has the lowest total-rate error.The ranking change accompanies a shift from dipolar to Curie relaxation.Source: D. Nuclear relaxation rates · p. 7

E. Transfer from water to methanol

The section reports eSpinor’s transfer and observable-level results, explains how static tensor accuracy relates to dynamics, and notes the scope and validation limits of the conclusions.

E. Transfer from water to methanol

  • eSpinor has the lowest electron-spin and proton-correlation errors on the MeOH trajectory after training on 8k water geometries; both nuclear-rate errors remain under 1%.The test uses the water-trained models without retraining and includes nearby solvent atoms in the inputs.Source: E. Transfer from water to methanol · p. 7, Table III · p. 8

A. OOD ligand generalization

  • For Co(II), eSpinor improves unseen-ligand generalization at fixed metal ion and coordination number, supporting candidate prioritization while broader screening and transfer remain untested.Its advantage over NequIP+H persists with either initializer, and the AOM improvement over point-charge initialization is consistent with AOM’s description of directional metal–ligand interactions.Source: A. OOD ligand generalization · p. 8

B. From coefficients to magnetic observables

  • At 8k Dy training geometries, similar crystal-field MAEs can produce different electron-spin correlations and nuclear relaxation rates because propagation weights errors by their action on electronic states.The common-initial-state diagnostic finds a smaller systematic residual for eSpinor in the sampled ensemble despite comparable individual-sample errors.Source: B. From coefficients to magnetic… · p. 8
  • Relaxation accuracy depends on mechanism: dipolar rates depend on spin correlations and proton motion, whereas Curie rates depend quadratically on anisotropic shielding under tumbling.Consequently, improved dipolar dynamics can coexist with a different high-field ranking, and full susceptibility error alone does not determine Curie-rate accuracy.Source: B. From coefficients to magnetic… · p. 8
  • Water-to-methanol transfer tests the learned mapping on geometries and local environments sampled under another solvent condition, without additional electronic-structure training data.The low inference cost and demonstrated transfer could support longer, denser trajectories, but broader chemical benchmarks and experimental validation are needed for MRI-probe design.Source: B. From coefficients to magnetic… · p. 8

C. SU(2) inductive bias

  • At 4k and 8k, eSpinor achieves lower electron-correlation and low-field relaxation errors than NequIP+H with 15.5% fewer parameters.Spinor propagation also links coefficients in tensor interactions; a comparison against integer-tensor updates with independently learned coefficients would further test this coefficient-sharing contribution.Source: C. SU(2) inductive bias · p. 8

VI. CONCLUSIONS

  • At 4k and 8k water-training geometries, eSpinor gives the lowest mean errors among four models for real electron-spin correlations over tested fields and total nuclear relaxation rates at 1–3 T.At 8k, this advantage persists with coefficient accuracy similar to NequIP+H and extends to methanol without retraining.Source: §VI · p. 8
  • The comparisons measure accuracy relative to electronic-structure references, and establishing practical value for SMM screening or MRI-probe design requires further validation.Current benchmarks cover mononuclear complexes; no suitable labeled multinuclear dataset was identified.Source: §VI · p. 9

Appendix A: Rotation equivariance and eigenbasis independence

The appendix establishes how spinor initialization and subsequent operations preserve rotation equivariance and independence from arbitrary eigenbasis choices, subject to stated degeneracy conditions.

Appendix A: Rotation equivariance and eigenbasis independence

  • The approximate crystal-field Hamiltonian transforms covariantly under rotation because its ligand strengths are invariant and its spherical harmonics and orbital tensor operators transform with matching ranks.The full Hamiltonian retains this covariance when the rotationally invariant L·S term is included.Source: Appendix A · p. 9
  • Rotating the eigenbasis gives an eigenbasis at the rotated geometry with unchanged energy levels, while a separate diagonalization may differ by unitary mixing within energy-degenerate subspaces.For separated Kramers pairs, the allowed basis change is U(2) within each pair; exactly coincident pairs permit a larger unitary freedom.Source: Appendix A · p. 9
  • Gauge-covariant message passing requires equal gate weights within each Kramers pair, and larger degenerate subspaces require weights and diagonal gates equal across all columns.The initialization channel map commutes with right multiplication by the eigenbasis change and transforms consistently under rotation.Source: Appendix A · pp. 9–10
  • Conjugated spinor products cancel the shared unitary eigenbasis change and yield gauge-invariant integer tensors, while mixed spinor–tensor products retain the spinor’s eigenbasis transformation.Induction over interaction layers and the integer-tensor readout gives predictions independent of both the initial eigenbasis and the choice of rotation lift.Source: Appendix A · p. 10

Appendix B: From spin propagation to nuclear relaxation

The appendix describes a time-domain pipeline that propagates electronic spin correlations and combines them with proton geometry and rotational damping to calculate nuclear relaxation rates.

Appendix B: From spin propagation to nuclear relaxation

  • The effective Hamiltonian combines the crystal-field and Zeeman terms to determine the time propagator, and the electronic correlation is evaluated from the initial thermal state and propagated angular-momentum operators.The Hamiltonian uses cm−1 units, with time expressed in seconds.Source: Appendix B · p. 10
  • The dipolar calculation contracts the electronic correlation with the Dy–proton geometry tensor before trajectory and orientation averaging, retaining joint spin and proton-geometry fluctuations.The resulting geometry-correlation quantity has units m−6(J/T)2 when distances are in metres.Source: Appendix B · p. 10

A. Hamiltonian initialization

  • Dipolar rates use a real fitted correlation tail and solvent-dependent rotational damping to form a spectral density evaluated at the proton Larmor frequency.The appendix specifies the rate at that frequency; it also describes point-charge and AOM ligand-field initializers, with Co AOM channels labeled σ, π, and δ.Source: Appendix B · pp. 10–11, A. Hamiltonian initialization · p. 13

B. Feature updates and tensor outputs

The feature-update scheme preserves spinor propagation while using contractions and geometric features to construct integer-rank outputs.

B. Feature updates and tensor outputs

  • A fixed unitary map converts real features to the complex spherical basis, and the spinor messages are normalized by the square root of the receiving site’s neighbor count.Integer convolutions use a self-connection and the specified table normalization.Source: B. Feature updates and tensor outputs · p. 13
  • The embedding preserves the original spinors, which alone propagate between layers, while gates depend on integer scalars and are independent of the eigenbasis column.Each interaction contracts four learned spinor pairs; the readout uses allowed products with integer features and conjugated spinors.Source: B. Feature updates and tensor outputs · p. 14
  • Readout uses an orthonormal symmetric, traceless Cartesian basis for Co outputs, while Dy neural models use the fixed spherical-basis map and raw Stevens labels.The gate construction uses SiLU MLPs with hidden width 16 and clamped outputs.Source: B. Feature updates and tensor outputs · p. 14

C. Factorized tensor updates

The supplement describes factorized spinor updates, implementation checks, dataset splits, and model settings. It also specifies basis transport and the conventions used to evaluate training variability.

C. Factorized tensor updates

  • For each Kramers pair, the Gram matrix ρ=hh† removes the U(2) basis ambiguity and decomposes into integer tensors.The supplement notes that the analogy to a thermal density matrix is formal because the initializer is approximate and its factor is learned.Source: C. Factorized tensor updates · p. 14

D. Numerical equivariance and implementation tests

  • Float32/complex64 equivariance residuals stay below 1.79 × 10^-6 for tested rotations, gauge changes, and sign flips.Across nine float64/complex128 initializer cases, covariance, paired-energy, and time-reversal residuals are at most 2.82 × 10^-15; spinor and matrix updates agree within 2.38 × 10^-7.Source: D. Numerical equivariance and… · p. 14
  • At sampled finite gaps, double-precision output changes are proportional to small displacements when connectivity is unchanged.The smallest inter-pair gaps are 7.62 cm^-1 for AOM and 8.02 cm^-1 for point-charge initialization; single precision reaches a numerical floor at the smallest steps.Source: D. Numerical equivariance and… · p. 14

E. Parallel transport for multinuclear systems

A. Datasets and averaging

  • The cobalt split reserves 41 ligand identities for OOD testing, yielding 473 complexes with only reserved ligands and 862 in-distribution test complexes.Mixed-ligand complexes are excluded; the filtered dataset contains 8,616 complexes using only retained ligands.Source: A. Datasets and averaging · p. 15
  • Dy training uses nested sets of 500–8,000 geometries, with separate 1,000-frame validation and 39,000-frame test intervals.Errors are averaged across three neural-fit seeds on a fixed split; the reported spread measures training variability on the same trajectory.Source: A. Datasets and averaging · p. 15

B. Models and optimization

  • The Dy eqSNAP baseline uses species-pooled bispectra and rankwise ridge fits, while NequIP+H and eSpinor fit their initializers separately.NequIP+H gates weighted initial eigenvectors using integer scalars; eSpinor updates spinors and tensors, with readout multiplicities of (43, 64, 73) at ranks (2, 4, 6).Source: B. Models and optimization · p. 16
  • Models are trained with normalized tensor error and checkpoints are selected by the lowest validation tensor loss.The loss is evaluated using exponential-moving-average weights.Source: B. Models and optimization · p. 16

S3. COBALT TENSOR ERRORS AND ANISOTROPY SCREENING

The supplement defines cobalt tensor and ranking metrics, then describes propagation settings and diagnostics. It reports that real-correlation performance and ensemble residuals reveal distinctions not captured by static tensor errors alone.

S3. COBALT TENSOR ERRORS AND ANISOTROPY SCREENING

  • Cobalt screening evaluates OOD fractions above a tensor-error threshold and shortlist recovery for 862 in-distribution and 473 OOD complexes.The supplement defines tensor error per complex and uses Spearman correlation and shortlist recovery to assess predicted rankings.Source: Figure S2 · p. 17, S3. COBALT TENSOR ERRORS AND… · p. 17

A. Propagation and coordinate conventions

  • Water propagation uses 78 nonoverlapping 500-frame test windows, six field strengths, and 100 equally weighted field orientations at 300 K.Low-field averages cover 1, 1.5, 2, and 3 T; the donor-plane vectors define a molecular frame used for coefficient and Dy–proton coordinates.Source: A. Propagation and coordinate… · p. 17
  • Exporting 39k predictions takes an average of 709.6 s for an 8k eSpinor model on one H200, excluding training and propagation.The two approximate CASSCF+SOC jobs provide a cost scale only, not a controlled speedup comparison.Source: A. Propagation and coordinate… · p. 17

B. Prediction errors and propagation diagnostics

  • Normalized real-correlation advantage persists from 1k to 8k training geometries and across three 8k test-window segments; at 8k, imaginary-correlation errors favor NequIP+H (1.264%) over eSpinor (1.534%).The learning-curve figure covers crystal-field MAEs, real electron and complex proton correlations across six fields, and total R2N errors over 1–3 T.Source: B. Prediction errors and propagation… · p. 18, Figure S3 · p. 18
  • With the reference initial state at 3 and 20 T, eSpinor has a smaller ensemble-mean signed error than NequIP+H despite comparable individual errors.The diagnostic uses all windows and orientations at 8k geometries.Source: B. Prediction errors and propagation… · p. 18

C. Relaxation calculations and sensitivity

The supplement details correlation-tail fitting and sensitivity checks for relaxation calculations, then describes transfer from water-trained models to methanol. It flags both a ground-multiplet approximation and the need for longer trajectories to validate extrapolated tails.

C. Relaxation calculations and sensitivity

  • At 8k, eSpinor leads accuracy for fitted TG and TS, while NequIP leads for β and τ0.Each proton’s real correlation is fitted separately; the explicit complex correlation is retained through 499 fs and followed by a real fitted tail from 500 fs.Source: C. Relaxation calculations and… · p. 19
  • Susceptibility errors against the archive include the ground-multiplet approximation used for model susceptibilities.The model susceptibility treatment is specified within J=15/2.Source: C. Relaxation calculations and… · p. 19
  • Changing η from 10^-3 to 10^-5 and using matched J=15/2 reference susceptibility changes mean rate errors by less than 0.10 percentage points per model, with the 8k low-field ranking unchanged.Longer trajectories are still needed to validate the extrapolated tails.Source: C. Relaxation calculations and… · p. 20

D. Transfer from water to methanol

  • Water-trained models are evaluated on methanol without retraining or selection, using 82,000 matched frames in 164 propagation windows.The water protocol is retained except for a 75 ps rotational correlation time and methanol-specific donor indices; the figure covers correlation errors and total R2N errors across training budgets.Source: D. Transfer from water to methanol · p. 20, Figure S5 · p. 20

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