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
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 Regzz(τ) shows the 3 T CASSCF+SOC reference [ 8 ] .
Figure 2: eSpinor architecture. (a) Atomic embeddings and Hamiltonian eigenvectors initialize integer tensors Va and spinors ha 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=Nlay . Only D and Bmℓ=Bq(ℓ) 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
D tensor
D
∣E∣
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 Regzz at 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 L2 error averaged over all 6 fields. (b), (c) Total nuclear R1N and R2N 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 gzz 1–20 T
Complex Gh 1–20 T
R1N 1–3 T
R2N 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 gzz 1–20 T
Complex Gh 1–20 T
R1N 1–3 T
R2N 1–3 T
χT
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 ei , 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 R2N errors over 1 – 3 T. Proton quantities average H3/H4/H5. R1N gives the same model ordering at every budget.
3 T
20 T
Model
Eindividual
Eorientations
Eensemble
Eindividual
Eorientations
Eensemble
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 R2N 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 R2N errors over H3/H4/H5 and 1 – 3 T. R1N gives the same model ordering at every budget. Table 3 includes susceptibility and both rates at 8k.
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.
Ali Rayat, Yunhao Fan, Gia-Wei Chern
Department of Physics, University of Virginia, Charlottesville, VA 22904, USA
Computational discovery of magnetic materials remains challenging because magnetism arises from the competition between kinetic energy and Coulomb interaction that is often beyond the reach of standard electronic-structure methods. Here we tackle this challenge by directly solving the many-electron Schrödinger equation with neural-network variational Monte Carlo, which provides a highly expressive variational wavefunction for strongly correlated systems. Applying this technique to transition metal dichalcogenide moiré semicondutors, we predict itinerant ferromagnetism in WSe2/WS2 and an antiferromagnetic insulator in twisted Γ-valley homobilayer, using the same neural network without any physics input beyond the microscopic Hamiltonian. Crucially, both types of magnetic states are obtained from a single calculation within the Sz=0 sector, removing the need to compute and compare multiple Sz sectors. This significantly reduces computational cost and paves the way for faster and more reliable magnetic material design.
Max Geier, Liang Fu
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
We present a magnetic extension of the Hierarchically Interacting Particle Neural Network (HIP-NN) that enables large-scale simulations of electron-mediated spin dynamics in disordered itinerant magnets. The resulting magnetic HIP-NN (mHIP-NN) incorporates rotationally invariant spin correlations directly into hierarchical message-passing layers, enabling the network to learn emergent magnetic energy landscapes and effective local fields from coupled geometric-spin environments while preserving spin-rotation symmetry. As a benchmark application, we consider structurally disordered itinerant s-d exchange models in which the effective magnetic forces arise dynamically from the instantaneous electronic structure and are computationally prohibitive to evaluate using conventional exact-diagonalization-based approaches. We show that mHIP-NN accurately reproduces the local torques governing Landau-Lifshitz-Gilbert dynamics and faithfully captures the nonequilibrium evolution of spatial spin correlations following thermal quenches. Our results establish symmetry-aware hierarchical message-passing networks as an efficient and scalable framework for large-scale simulations of frustrated itinerant spin systems and nonequilibrium magnetic dynamics. More broadly, because the learned energy functional remains fully differentiable with respect to both atomic coordinates and spin variables, the framework also provides a natural foundation for spin-dependent interatomic potentials and coupled atom-spin dynamics.
Supriyo Ghosh, Yunhao Fan, Sheng Zhang +2
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