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

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

physics.chem-phOct 21, 2026

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

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

Organizations: Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA · Department of Chemical Engineering, National Tsing Hua University, Hsinchu 300044, Taiwan · Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA

Abstract

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.

Figures & tables

Figure 1: Representation choices for predicting SOC-dependent molecular magnetic properties. eSpinor propagates half-integer spinor features alongside integer-rank tensor features. The ZFS levels are schematic, and the electron-spin correlation $\operatorname{Re}g_{zz}(\tau)$ shows the $3$ T CASSCF+SOC reference [ 8 ] .

Figure 2: eSpinor architecture. (a) Atomic embeddings and Hamiltonian eigenvectors initialize integer tensors

V_{a}

and spinors

h_{a}

for repeated message passing and tensor readout. (b) Convolution uses Clebsch–Gordan tensor products (CGTP). (c) Spinor contractions update integer tensors. Here even/odd denotes integer/half-integer rank and

T=N_{\mathrm{lay}}

. Only

\mathbf{D}

and

B_{m}^{\ell}=B_{q}^{(\ell)}

are benchmark targets. The preliminary tensor convolution used by the learned AOM initializer is omitted.

		In distribution ( $n=862$ )			Unseen ligands ( $n=473$ )
Model	Parameters	$\mathbf{D}$ tensor	$D$	$\|E\|$	$\mathbf{D}$ tensor	$D$	$\|E\|$
eqSNAP	1,870	3.674	16.832	1.729	2.989	12.870	1.670
GM-NN	943,874	2.380	12.090	1.417	2.988	14.710	1.666
NequIP	122,120	1.193	6.645	0.835	1.651	7.611	1.134
NequIP+H	144,923	1.085	5.925	0.825	1.626	8.619	1.211
eSpinor (pc)	149,332	1.004	5.505	0.709	1.370	6.893	1.061

Table 1: Cobalt prediction MAEs (cm -1 ) on the splits in Sec. III . The tensor error averages all 9 Cartesian entries. Parameter counts are ridge coefficients for eqSNAP or trainable neural parameters. The eSpinor variants use point-charge (pc) or angular-overlap (AOM) initialization.

Figure 3: Electron-spin correlations from neural models trained on 8k samples. (a) Raw

\operatorname{Re}g_{zz}

3

T, with CASSCF+SOC as reference. (b)–(d) Signed model-minus-reference residuals at

3

10

, and

20

T on a common scale. Dotted lines mark zero error, and shaded bands show pointwise standard deviations. Curves retain lags

0

–

499

fs without a fitted tail. Results for eqSNAP are included in Table 2 and in Fig. S3 of the Supplemental Material [ 31 ] .

Figure 4: Downstream learning curves for NequIP, NequIP+H, and eSpinor. (a) Real electron-correlation relative

L^{2}

error averaged over all 6 fields. (b), (c) Total nuclear

R_{1N}

and

R_{2N}

errors over H3/H4/H5 and

1

1.5

2

, and

3

T. All errors are relative to CASSCF+SOC on the common 39k test frames. Shaded bands show standard deviations. Figure S3 of the Supplemental Material [ 31 ] includes eqSNAP.

Model	Parameters	Re $g_{zz}$ 1–20 T	Complex $G_{h}$ 1–20 T	$R_{1N}$ 1–3 T	$R_{2N}$ 1–3 T
eqSNAP	1,650	10.935	11.735	22.687	22.386
NequIP	122,080	1.882	2.121	3.448	3.361
NequIP+H	159,116	1.140	1.246	2.294	2.220
eSpinor	134,459	0.553	0.748	0.809	0.774

Table 2: Dy downstream errors (%) relative to CASSCF+SOC for 8k training samples and 39k test frames. Metrics are defined in Secs. IV.3 and IV.4 .

Model	Re $g_{zz}$ 1–20 T	Complex $G_{h}$ 1–20 T	$R_{1N}$ 1–3 T	$R_{2N}$ 1–3 T	$\bm{\chi}T$
eqSNAP	9.289	8.402	11.304	11.219	19.851
NequIP	2.427	2.427	3.209	3.186	5.423
NequIP+H	2.110	1.982	2.296	2.281	5.221
eSpinor	1.240	0.979	0.959	0.954	5.263

Table 3: Transfer errors (%) relative to the MeOH CASSCF+SOC reference using models trained on 8k water geometries. The trajectory and metrics are specified in the Supplemental Material [ 31 ] .

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1: Kramers-pair gaps and prediction continuity for the current Co eSpinor checkpoints with AOM (top) and point-charge (bottom) initialization. (a), (d) Cumulative gap distributions. (b), (e) Cartesian tensor MAE per complex. (c), (f) Median output changes over 24 geometries and 2 directions, taking the larger change from the 2 displacement signs. Shading spans the double-precision minimum and maximum.

Setting	Co	Dy
Convolutions / maximum integer rank / channels per rank	3 / 6 / 4
Radial basis size / cutoff order	8 / 6
NequIP+H initializer	AOM	Point charge
eSpinor initializer	AOM or point charge	Point charge
Graph cutoff (Å) / average-neighbor normalization	6 / 10	4 / 6
Integer radial MLP depth / width	3 / 64	1 / 128

Appendix

Table S1: Settings for NequIP, NequIP+H, and eSpinor, fixed across Dy training budgets.

Figure S2: Cobalt screening for the neural models. eSpinor uses AOM. (a) OOD fraction exceeding tensor-error threshold

e_{i}

, on an axis linear up to 0.1 cm -1 and logarithmic above. (b), (c) Shortlist recovery for 862 in-distribution and 473 OOD complexes.

Figure S3: Water learning curves on 39k test frames. (a)–(c) Crystal-field MAEs. (d), (e) Real electron and complex proton correlations over all 6 fields. (f) Total

R_{2N}

errors over

1

–

3

T. Proton quantities average H3/H4/H5.

R_{1N}

gives the same model ordering at every budget.

	3 T			20 T
Model	$E_{\mathrm{individual}}$	$E_{\mathrm{orientations}}$	$E_{\mathrm{ensemble}}$	$E_{\mathrm{individual}}$	$E_{\mathrm{orientations}}$	$E_{\mathrm{ensemble}}$
NequIP	0.7270	0.4184	0.1962	0.7009	0.3104	0.1859
NequIP+H	0.6929	0.3706	0.1129	0.6718	0.2701	0.1143
eSpinor	0.7195	0.3795	0.0559	0.6882	0.2419	0.0572

Appendix

Table S2: Dimensionless propagation RMS errors at 8k with the reference initial state. Columns follow Eq. ( S21 ).

Figure S4: Field and mechanism comparisons at 8k. (a) Total

R_{2N}

errors over H3/H4/H5 at each field. (b) Dipolar and Curie errors over

1

–

3

T, normalized by their own reference mechanism. (c) Susceptibility errors from Eq. ( S26 ).

Figure S5: Water-to-MeOH transfer across training budgets. (a), (b) Real electron-spin and complex proton-correlation errors over all 6 fields. (c) Total

R_{2N}

errors over H3/H4/H5 and

1

–

3

R_{1N}

gives the same model ordering at every budget. Table 3 includes susceptibility and both rates at 8k.

Equivariant GNNs

Equivariant graph neural networks encode how node and edge features transform under symmetries to make geometry-aware predictions.

See all 42 papers

Graph Neural Network Force Fields for Spin Dynamics in Metallic Magnets
Jul 30, 2026Ali Rayat, Yunhao Fan, Gia-Wei ChernGraph Neural Networks Scientific ML
Department of Physics, University of Virginia, Charlottesville, VA 22904, USA
Predicting magnetism with first-principles AI
Feb 9, 2026Max Geier, Liang FuMaterials Property Prediction Materials Science
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Magnetic HIP-NN for spin dynamics in disordered itinerant magnets
Jun 9, 2026Supriyo Ghosh, Yunhao Fan, Sheng Zhang +2Equivariant Neural Networks Message Passing Neural Networks
Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA · Department of Chemistry, University of Chicago, Chicago, IL 60637, USA · Theoretical Division and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

Jul 30, 2026J/K move · Enter open · S save

Graph Neural Network Force Fields for Spin Dynamics in Metallic Magnets

Ali Rayat, Yunhao Fan, Gia-Wei Chern

Department of Physics, University of Virginia, Charlottesville, VA 22904, USA

Graph Neural Networks Scientific ML Computational Materials Science

Open paper AskPDF arXiv

Metallic magnets exhibit complex spin dynamics governed by electronically generated interactions. Predictive simulations of such dynamics typically require repeated solutions of an underlying electronic problem throughout the time evolution, creating a major computational bottleneck. Here we introduce a graph neural network (GNN) magnetic force-field framework that learns the effective magnetic energy functional governing itinerant spin dynamics directly from electronic calculations. Conceptually analogous to machine-learned interatomic potentials, the proposed framework enables efficient evaluation of spin torques while capturing the nonlinear and spatially extended interactions generated by itinerant electrons. We benchmark the method on representative metallic magnetic systems exhibiting collinear, noncollinear, and noncoplanar magnetic order. The learned force fields accurately reproduce electronically generated spin torques and yield nonequilibrium spin dynamics in excellent agreement with direct electronic simulations. Our results establish graph neural networks as a powerful framework for machine-learned magnetic force fields, providing a pathway toward predictive large-scale simulations of nonequilibrium magnetism across multiple length and time scales.

Abstract

Equivariant GNNs

Explore similar work

Graph Neural Network Force Fields for Spin Dynamics in Metallic Magnets

Predicting magnetism with first-principles AI

Magnetic HIP-NN for spin dynamics in disordered itinerant magnets