Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Graph-neural-network predictions of solid-state NMR parameters from spherical tensor decomposition

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Graph-neural-network models trained on amorphous silica predict the full tensor quantities that determine solid-state NMR spectra—magnetic shielding and electric field gradients—well enough to reproduce static 1D spectra from…

desk verdict Solid methods paper for in-distribution a-SiO2, with thinner out-of-domain evidence on zeolites and a minor overstatement of the ISD advantage. read the letter →

arxiv 2412.15063 v1 pith:CY672AY3 submitted 2024-12-19 cond-mat.mtrl-sci physics.comp-phstat.ML

Chiheb Ben Mahmoud , Louise A. M. Rosset , Jonathan R. Yates , Volker L. Deringer This is my paper · ORCID
classification cond-mat.mtrl-sciphysics.comp-phstat.ML
keywords graphneuralnetworkssolid-stateNMRmagneticshieldingtensorelectricfieldgradientsphericaldecompositionamorphoussilicazeolitescristobalitephasetransition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that an equivariant graph neural network can learn the full tensor quantities that determine solid-state NMR spectra—the anisotropic magnetic shielding tensor and the electric field gradient—from atomic coordinates alone, not just the isotropic shift. If that works, computational NMR for materials becomes much cheaper: once a model is trained, spectra for large, disordered, or dynamically evolving structures can be generated without repeated quantum-mechanical calculations. The authors test this on a diverse set of 1,000 amorphous SiO2 configurations, show that simulated static 1D NMR spectra match DFT calculations, and then apply the models out-of-domain to hypothetical zeolites and to the α–β inversion in cristobalite, where thermally averaged tensors agree with experimental peak separations and asymmetry parameters. The upshot is a route to ML-driven NMR for both static and dynamic behavior of complex oxides.

What carries the argument

The central machinery is the spherical tensor decomposition of a rank-2 tensor, $T = T^{(0)} \oplus T^{(1)} \oplus T^{(2)}$, in which each piece transforms independently under spatial rotation and all pieces are parity-even. The models are equivariant graph neural networks that map atom-centred local environments to these irreducible components; the tensor-product variant replaces a single decomposition with a learnable linear combination of tensor products of spherical-harmonic features, using Clebsch–Gordan coefficients to extract ranks 0–2. This construction makes the tensor orientation (Euler angles) and anisotropy descriptors (span, skew, asymmetry) available as outputs, not just the isotropic value.

What would settle it

If a zeolite or low-density silica structure with rings larger than those in the training set (say 20- and 22-membered rings) systematically yields ML spectra whose peak positions or line shapes deviate from the DFT reference by much more than the training error, the claimed out-of-domain accuracy would be refuted; a direct test is to compare ML and GIPAW spectra for a library of zeolites with ring sizes 18–24 and check whether error grows with ring-size mismatch.

Watch

Extended reading notes

Core claim

The central claim is that decomposing rank-2 NMR tensors into irreducible spherical components—rank 0 (isotropic), rank 1 (antisymmetric), and rank 2 (anisotropic symmetric)—makes them natural targets for equivariant graph neural networks, and that models trained this way are accurate enough to reproduce static 1D NMR spectra from quantum-mechanical computations. Magnetic shielding uses all three irreducible components; the electric field gradient, being symmetric and traceless, is fully captured by the rank-2 part alone. An alternative tensor-product decomposition, built from learnable linear combinations of Clebsch–Gordan-coupled spherical-harmonic tensors, reaches comparable accuracy with roughly three times fewer parameters. Beyond the training distribution, the models predict spectra for hypothetical zeolites and track the α–β cristobalite transition through simulated magic-angle-spinning spectra, where the ML-derived oxygen electric-field-gradient asymmetry ($\eta_Q = 0.13$) matches an experimental value.

Load-bearing premise

Everything rests on the training set of 1,000 melt-quench-anneal amorphous SiO2 structures covering local environments well enough that hypothetical zeolites and the α–β cristobalite transition, with their different ring-size distributions, stay within the learned representation.

Editorial extensions

If this is right

  • Static 1D NMR spectra for amorphous silica at densities from 2.2 to 2.6 g/cm³ can be generated from ML tensor predictions in good agreement with DFT reference spectra.
  • The same ML models transfer, without retraining, to hypothetical zeolites, with the best spectral agreement occurring when ring sizes resemble the training distribution.
  • For the α–β cristobalite transition, ML-driven molecular dynamics combined with tensor prediction yields thermally averaged 29Si MAS spectra whose peak separation and oxygen EFG asymmetry match experiment, demonstrating a practical route to studying phase-transition dynamics without per-snapshot DFT NMR.
  • Because NMR tensors are learned per atom in an equivariant representation, the same framework applies to other chemical species and to other tensorial properties such as dielectric response or polarization, without retraining from scratch.
  • Predicting anisotropy parameters (span, skew, asymmetry) and tensor orientations from learned tensors is more reliable than specialized scalar models, so spectral shape, not just peak position, becomes accessible.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural stress test would be to train on structures with ring sizes larger than 18 and see whether zeolite and low-density predictions improve, since the paper itself reports that ring-size distribution correlates with error.
  • The oxygen $\sigma^{(1)}$ component is the least accurate target; a weighted loss function or separate loss balance could recover it, potentially improving the oxygen spectra further.
  • The same spherical-tensor scheme could be applied to other tensor-valued NMR quantities such as spin–spin coupling tensors or to non-NMR properties like the Born effective charge tensor, where orientation information matters.
  • Combining the tensor model with a fast machine-learned interatomic potential could enable closed-loop screening: propose a hypothetical structure, generate its spectrum, and refine, all without first-principles NMR.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents two graph-neural-network strategies for predicting solid-state NMR tensors (magnetic shielding and electric field gradient) from spherical tensor decompositions: an irreducible spherical decomposition (ISD) and a tensor-product (TP) combination with learnable weights. The models are trained on a diverse dataset of 1000 amorphous SiO2 structures and benchmarked against GIPAW reference data in terms of irreducible tensor components, anisotropy parameters, Euler angles, and simulated static 1D NMR spectra. The authors then apply the models to larger a-SiO2 configurations, 50 hypothetical zeolites, and the alpha-to-beta cristobalite transition, reporting good agreement with quantum-mechanical and experimental spectra. The central claim is that machine-learned tensor predictions can substitute for DFT-based NMR calculations in silica-based materials, including out-of-domain applications.

Significance. If the central claim holds, this work represents a substantial step toward ML-accelerated solid-state NMR prediction, with potential high-throughput impact on structural characterization of framework and amorphous silicates. The paper is methodologically strong: it uses a clear train-validation-test split, normalized error metrics, learning curves, direct spectral comparison to GIPAW, and a systematic comparison of two tensor-decomposition strategies. The transparent discussion of the tensor-product representation and its parameter count is a useful contribution. The main scientific value lies in demonstrating that equivariant GNNs can predict full tensorial NMR parameters, not only isotropic shifts, and that static spectra can be reconstructed from these predictions. However, the out-of-domain claims, especially for zeolites, are supported by only a biased subset of examples, and the quantitative evidence for the full zeolite set is deferred to the Supplementary Material.

major comments (3)
  1. [Section III C (Fig. 6) and Section V] The claim that the ML models generalize to zeolites is not quantitatively supported in the main text. The authors state that they "choose these four representative structures based on the accuracy of the ISD models," which explicitly selects best- and worst-case examples rather than a representative or random sample. No spectral similarity metric or aggregate error distribution for the full 50-zeolite set is reported in the main text; the paper instead notes a "possible correlation between the ring distribution in the zeolites and ML prediction errors" and defers further investigation. Because the training set contains ring sizes mostly between 6 and 16 while zeolites have different ring distributions, the typical out-of-domain performance remains unverified. The authors should either report the full 50-zeolite error distribution (or a quantitative spectral similarity metric) in the main text, or substantially soften the extrapolation claim in the conclusions.
  2. [Section V (Conclusions) vs. Figure 3] The conclusion states that "the irreducible spherical decomposition yielded the lowest prediction errors," but Figure 3 shows no significant difference between the ISD and tensor-product models across the tested magnetic-shielding parameters for either silicon or oxygen. This overstatement should be corrected to say that the two decompositions perform comparably, with only minor differences that are within the spread of the reported errors.
  3. [Abstract and Conclusions] The central claim that "both models produce simulated static 1D NMR spectra in good agreement with quantum-mechanical computations" is directly demonstrated only for the a-SiO2 structures in Figure 5. For zeolites, only a hand-picked subset is shown and no aggregate metric is provided; for cristobalite, the comparison in Figure 7 is to an experimental spectrum rather than to GIPAW calculations. The abstract and conclusions should be rephrased to distinguish the in-distribution demonstration from the exploratory out-of-domain applications.
minor comments (5)
  1. [Section III (error metrics)] The text contains a typographical error: "We use the the root mean square error (RMSE)" should read "We use the root mean square error (RMSE)."
  2. [Section II E] The phrase "based on test errorsor 50 independent configurations" should be corrected to "based on test errors on 50 independent configurations."
  3. [Section III B] There is a duplicate word in "for all tensor product models models when the training set size is 10 structures," which should be "for all tensor product models when the training set size is 10 structures."
  4. [Section II E] The hyperparameter optimization is performed only for the ISD model and a TP model with a l=1 target; it would be helpful to state explicitly whether the same hyperparameters were used for all other TP variants compared in Figures 2 and 3, since differences in hyperparameters could confound the decomposition comparison.
  5. [Section III A] The comparison of models with different numbers of tensor products is made at a fixed maximum rank lmax=4 and a fixed number of features per rank; the authors should note explicitly in the text or a footnote that the 4096-tensor-product model has a different parameter count than the single-tensor-product model, since this affects the interpretation of the performance gains.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ML NMR predictions are benchmarked against independent GIPAW labels on held-out structures, and the self-citations are tooling only.

full rationale

The derivation chain is self-contained with respect to external DFT labels. The ML models are trained on GIPAW-computed magnetic shielding and electric field gradient tensors for 800 a-SiO2 structures, validated on 50, and tested on 150 held-out structures; the reported error metrics are computed on that separate test set, so no fitted parameter is renamed as a prediction. The static 1D spectra are generated from ML-predicted tensors using SOPRANO and compared against GIPAW spectra on the same structures, which is a standard held-out benchmark rather than an identity by construction. Self-citations to the authors' machine-learning potentials and phase classifier (Refs. 64, 72, 76) are used as structure-generation and phase-labeling tools, not as evidence for the NMR predictions themselves, and the NMR claims do not reduce to those citations. The paper itself flags extrapolation limitations in Section III C, noting that the training set contains mostly rings of sizes 6-16 while a low-density test structure contains rings of size 20 and 22 and that zeolites have different ring distributions, and it defers a possible correlation between zeolite ring distribution and prediction errors to future work; these are generalization and reporting concerns, not circularity. No equation reduces to its own input, no uniqueness theorem is imported from the authors' prior work to force the choice of decomposition, and no ansatz is smuggled in to guarantee the central spectral agreement. The external comparisons to experimental spectra of cristobalite and to the published experimental EFG asymmetry provide independent support for the application-level claims.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. Its central claim rests on the standard mathematics of spherical tensor decomposition, the accuracy of PBE-GIPAW references, and the representativeness of the training set for out-of-domain applications. The free parameters are the usual ML hyperparameters and learned weights.

free parameters (2)
  • NequIP hyperparameters (cutoff radius, number of layers, feature ranks, maximum rank ℓmax=4) = Selected by Bayesian optimization (XPOT) on 100 validation structures; exact values in Supplementary Materials
    Architecture choices affect accuracy and are tuned to the dataset.
  • Linear combination weights w_k1k2 in Eq. (3) for tensor-product representation = Learned during training; number of tensor products varied as 1, 64, 4096
    These weights are free parameters fit to the NMR labels, and the number of tensor products is a model capacity choice.
assumptions (4)
  • standard math Rank-2 Cartesian NMR tensors admit the irreducible spherical decomposition T = T(0) ⊕ T(1) ⊕ T(2) with even parity for all components
    Standard angular momentum algebra and Clebsch-Gordan coefficients, used in Eq. (1) and throughout Section II B.
  • domain assumption PBE-GIPAW DFT provides reference NMR parameters of sufficient accuracy
    All ML labels are PBE-GIPAW values; the paper notes discrepancies with experiment, e.g., the cristobalite peak separation is 2.3 ppm in ML predictions versus 2.6 ppm experimentally (Section IV).
  • domain assumption The 1000 melt-quench-anneal a-SiO2 structures represent the relevant structural space for SiO2 polymorphs and amorphous phases
    Training set covers densities 2.0 to 2.7 g/cm3 and quench rates 1e11 to 1e13 K/s, but the paper acknowledges out-of-domain ring sizes up to 22 in a low-density test structure (Section III C).
  • domain assumption NequIP's equivariant features can be linearly mapped to the tensor components through an e3nn linear layer
    Implementation choice described in Section II C; assumes the learned representation is sufficient for the tensor targets.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Graph-neural-network predictions of solid-state NMR parameters from spherical tensor decomposition." pith.science (2026). https://pith.science/paper/CY672AY3

@misc{pith2026241215063,
  author       = {Pith},
  title        = {Pith review of: Graph-neural-network predictions of solid-state NMR parameters from spherical tensor decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CY672AY3}},
  note         = {Machine review of arXiv:2412.15063}
}
abstract

Nuclear magnetic resonance (NMR) is a powerful spectroscopic technique that is sensitive to the local atomic structure of matter. Computational predictions of NMR parameters can help to interpret experimental data and validate structural models, and machine learning (ML) has emerged as an efficient route to making such predictions. Here, we systematically study graph-neural-network approaches to representing and learning tensor quantities for solid-state NMR -- specifically, the anisotropic magnetic shielding and the electric field gradient. We assess how the numerical accuracy of different ML models translates into prediction quality for experimentally relevant NMR properties: chemical shifts, quadrupolar coupling constants, tensor orientations, and even static 1D spectra. We apply these ML models to a structurally diverse dataset of amorphous SiO$_2$ configurations, spanning a wide range of density and local order, to larger configurations beyond the reach of traditional first-principles methods, and to the dynamics of the $\alpha\unicode{x2013}\beta$ inversion in cristobalite. Our work marks a step toward streamlining ML-driven NMR predictions for both static and dynamic behavior of complex materials, and toward bridging the gap between first-principles modeling and real-world experimental data.

Figures

Figures reproduced from arXiv: 2412.15063 by the authors.

Figure 1
Figure 1. Spherical decomposition strategies for a rank-2 spherical tensor, denoted [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Evolution of ML model errors, reported on the isotropic and anisotropic properties of the MS tensor, as a function [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Comparison between the best models from the ten [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Learning curves for the irreducible tensors of the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: Best and worst ML-predicted simulated 1D static [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: NMR fingerprints of an α-cristobalite to β-cristobalite displacive phase transition. (a) Simulated 29Si MAS spectra of a selection of MD snapshots describing a single α → β transition. The α : β ratio reported on every spectrum describes the ratio of the α-like and β-l…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Distillation of atomistic foundation models across architectures and chemical domains

    physics.comp-ph 2025-06 conditional novelty 6.0 of 10

    Distillation of atomistic foundation models via synthetic data yields 10x-100x faster student potentials with near-teacher accuracy.

Reference graph

Works this paper leans on

89 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    Reif , author S

    author author B. Reif , author S. E. \ Ashbrook , author L. Emsley ,\ and\ author M. Hong ,\ https://doi.org/10.1038/s43586-020-00002-1 journal journal Nature Reviews Methods Primers \ volume 1 ,\ pages 2 ( year 2021 ) NoStop

    doi:10.1038/s43586-020-00002-1 2021

  2. [2]

    author author C. J. \ Pickard \ and\ author F. Mauri ,\ https://doi.org/10.1103/PhysRevLett.88.086403 journal journal Physical Review Letters \ volume 88 ,\ pages 086403 ( year 2002 ) NoStop

    doi:10.1103/physrevlett.88.086403 2002

  3. [3]

    author author S. T. \ Holmes , author J. Schönzart , author A. B. \ Philips , author J. J. \ Kimball , author S. Termos , author A. R. \ Altenhof , author Y. Xu , author C. A. \ O'Keefe , author J. Autschbach ,\ and\ author R. W. \ Schurko ,\ https://doi.org/10.1039/D3SC06026H journal journal Chemical Science \ volume 15 ,\ pages 2181 ( year 2024 ) NoStop

    work page doi:10.1039/d3sc06026h

  4. [4]

    author author D. M. \ Dawson , author J. A. \ Clayton , author T. H. D. \ Marshall , author N. Guillou , author R. I. \ Walton ,\ and\ author S. E. \ Ashbrook ,\ https://doi.org/10.1039/D3SC06924A journal journal Chemical Science \ volume 15 ,\ pages 4374 ( year 2024 ) NoStop

    work page doi:10.1039/d3sc06924a

  5. [5]

    Spearing , author I

    author author D. Spearing , author I. Farnan ,\ and\ author J. Stebbins ,\ https://doi.org/10.1007/BF00204008 journal journal Physics and Chemistry of Minerals \ volume 19 ( year 1992 ),\ 10.1007/BF00204008 NoStop

    work page doi:10.1007/bf00204008 1992

  6. [6]

    Sattig , author S

    author author M. Sattig , author S. Reutter , author F. Fujara , author M. Werner , author G. Buntkowsky ,\ and\ author M. Vogel ,\ https://doi.org/10.1039/C4CP02057J journal journal Phys. Chem. Chem. Phys. \ volume 16 ,\ pages 19229 ( year 2014 ) NoStop

    work page doi:10.1039/c4cp02057j

  7. [7]

    author author C. Y. \ Huang , author H. Y. \ Lee , author Y. C. \ Chang , author C. K. \ Hong , author Y. R. \ Ou , author C. N. \ Kuo ,\ and\ author C. S. \ Lue ,\ https://doi.org/10.1103/PhysRevB.106.195101 journal journal Physical Review B \ volume 106 ,\ pages 195101 ( year 2022 ) NoStop

    doi:10.1103/physrevb.106.195101

  8. [8]

    Polenova , author R

    author author T. Polenova , author R. Gupta ,\ and\ author A. Goldbourt ,\ https://doi.org/10.1021/ac504288u journal journal Analytical Chemistry \ volume 87 ,\ pages 5458 ( year 2015 ) NoStop

    work page doi:10.1021/ac504288u 2015

Show all 89 references
  1. [9]

    author author S. E. \ Ashbrook \ and\ author D. McKay ,\ https://doi.org/10.1039/C6CC02542K journal journal Chemical Communications \ volume 52 ,\ pages 7186 ( year 2016 ) NoStop

    2016 doi

  2. [10]

    Valenzuela Reina , author F

    author author J. Valenzuela Reina , author F. Civaia , author A. F. \ Harper , author C. Scheurer ,\ and\ author S. S. \ Köcher ,\ https://doi.org/10.1039/D4FD00075G journal journal Faraday Discussions \ ,\ ( year 2024 ) ,\ note publisher: The Royal Society of Chemistry NoStop

    2024 doi

  3. [11]

    author author A. F. \ Harper , author S. P. \ Emge , author P. C. M. M. \ Magusin , author C. P. \ Grey ,\ and\ author A. J. \ Morris ,\ https://doi.org/10.1039/D2SC04035B journal journal Chemical Science \ volume 14 ,\ pages 1155 ( year 2023 ) NoStop

    doi

  4. [12]

    author author C. J. \ Pickard \ and\ author F. Mauri ,\ https://doi.org/10.1103/PhysRevB.63.245101 journal journal Physical Review B \ volume 63 ,\ pages 245101 ( year 2001 ) NoStop

    2001 doi

  5. [13]

    author author J. R. \ Yates , author C. J. \ Pickard ,\ and\ author F. Mauri ,\ https://doi.org/10.1103/PhysRevB.76.024401 journal journal Physical Review B \ volume 76 ,\ pages 024401 ( year 2007 ) NoStop

    2007 doi

  6. [14]

    Behler \ and\ author M

    author author J. Behler \ and\ author M. Parrinello ,\ https://doi.org/10/c7kbsq journal journal Physical Review Letters \ volume 98 ,\ pages 146401 ( year 2007 ) ,\ note publisher: American Physical Society NoStop

    2007

  7. [15]

    author author A. P. \ Bartók , author M. C. \ Payne , author R. Kondor ,\ and\ author G. Csányi ,\ https://doi.org/10/dkchb7 journal journal Physical Review Letters \ volume 104 ,\ pages 136403 ( year 2010 ) ,\ note publisher: American Physical Society NoStop

    2010

  8. [16]

    Zhang , author J

    author author L. Zhang , author J. Han , author H. Wang , author R. Car ,\ and\ author W. E ,\ https://doi.org/10.1103/PhysRevLett.120.143001 journal journal Physical Review Letters \ volume 120 ,\ pages 143001 ( year 2018 ) NoStop

    2018 doi

  9. [17]

    Drautz ,\ https://doi.org/10.1103/PhysRevB.99.014104 journal journal Physical Review B \ volume 99 ,\ pages 014104 ( year 2019 ) ,\ note publisher: American Physical Society NoStop

    author author R. Drautz ,\ https://doi.org/10.1103/PhysRevB.99.014104 journal journal Physical Review B \ volume 99 ,\ pages 014104 ( year 2019 ) ,\ note publisher: American Physical Society NoStop

    2019 doi

  10. [18]

    author author K. T. \ Schütt , author P. Kessel , author M. Gastegger , author K. A. \ Nicoli , author A. Tkatchenko ,\ and\ author K.-R. \ Müller ,\ https://doi.org/10.1021/acs.jctc.8b00908 journal journal Journal of Chemical Theory and Computation \ volume 15 ,\ pages 448 ( ...

    doi

  11. [19]

    Li , author J

    author author Z. Li , author J. R. \ Kermode ,\ and\ author A. De Vita ,\ https://doi.org/10.1103/PhysRevLett.114.096405 journal journal Physical Review Letters \ volume 114 ,\ pages 096405 ( year 2015 ) NoStop

    2015 doi

  12. [20]

    author author O. T. \ Unke , author S. Chmiela , author H. E. \ Sauceda , author M. Gastegger , author I. Poltavsky , author K. T. \ Schütt , author A. Tkatchenko ,\ and\ author K.-R. \ Müller ,\ https://doi.org/10.1021/acs.chemrev.0c01111 journal journal Chemical Reviews \ vo...

    doi

  13. [21]

    author author V. L. \ Deringer , author M. A. \ Caro ,\ and\ author G. Csányi ,\ https://doi.org/10.1002/adma.201902765 journal journal Advanced Materials \ volume 31 ,\ pages 1902765 ( year 2019 ) NoStop

    2019 doi

  14. [22]

    Cheng , author G

    author author B. Cheng , author G. Mazzola , author C. J. \ Pickard ,\ and\ author M. Ceriotti ,\ https://doi.org/10/gg99h2 journal journal Nature \ volume 585 ,\ pages 217 ( year 2020 ) ,\ note number: 7824 Publisher: Nature Publishing Group NoStop

    2020

  15. [23]

    author author V. L. \ Deringer , author N. Bernstein , author G. Csányi , author C. Ben Mahmoud , author M. Ceriotti , author M. Wilson , author D. A. \ Drabold ,\ and\ author S. R. \ Elliott ,\ https://doi.org/10/ghsb84 journal journal Nature \ volume 589 ,\ pages 59 ( year 2...

    2021

  16. [24]

    Deng , author C

    author author Y. Deng , author C. Wang , author X. Xu ,\ and\ author H. Li ,\ https://doi.org/10.1016/j.taml.2023.100481 journal journal Theoretical and Applied Mechanics Letters \ volume 13 ,\ pages 100481 ( year 2023 a ) NoStop

    2023

  17. [25]

    author author G. C. \ Sosso , author G. Miceli , author S. Caravati , author F. Giberti , author J. Behler ,\ and\ author M. Bernasconi ,\ https://doi.org/10.1021/jz402268v journal journal The Journal of Physical Chemistry Letters \ volume 4 ,\ pages 4241 ( year 2013 ) NoStop

    doi

  18. [26]

    author author P. M. \ Piaggi , author J. Weis , author A. Z. \ Panagiotopoulos , author P. G. \ Debenedetti ,\ and\ author R. Car ,\ https://doi.org/10.1073/pnas.2207294119 journal journal Proceedings of the National Academy of Sciences \ volume 119 ,\ pages e2207294119 ( year...

    doi

  19. [27]

    Rupp , author A

    author author M. Rupp , author A. Tkatchenko , author K.-R. \ Müller ,\ and\ author O. A. \ von Lilienfeld ,\ https://doi.org/10.1103/PhysRevLett.108.058301 journal journal Physical Review Letters \ volume 108 ,\ pages 058301 ( year 2012 ) NoStop

    doi

  20. [28]

    Talapatra , author B

    author author A. Talapatra , author B. P. \ Uberuaga , author C. R. \ Stanek ,\ and\ author G. Pilania ,\ https://doi.org/10.1038/s43246-023-00373-4 journal journal Communications Materials \ volume 4 ,\ pages 46 ( year 2023 ) NoStop

    2023 doi

  21. [29]

    Isayev , author C

    author author O. Isayev , author C. Oses , author C. Toher , author E. Gossett , author S. Curtarolo ,\ and\ author A. Tropsha ,\ https://doi.org/10.1038/ncomms15679 journal journal Nature Communications \ volume 8 ,\ pages 15679 ( year 2017 ) NoStop

    doi

  22. [30]

    author author D. M. \ Wilkins , author A. Grisafi , author Y. Yang , author K. U. \ Lao , author R. A. \ DiStasio ,\ and\ author M. Ceriotti ,\ https://doi.org/10.1073/pnas.1816132116 journal journal Proceedings of the National Academy of Sciences \ volume 116 ,\ pages 3401 ( ...

    doi

  23. [31]

    Grumet , author C

    author author M. Grumet , author C. Von Scarpatetti , author T. Bučko ,\ and\ author D. A. \ Egger ,\ https://doi.org/10.1021/acs.jpcc.4c00886 journal journal The Journal of Physical Chemistry C \ volume 128 ,\ pages 6464 ( year 2024 ) NoStop

    doi

  24. [32]

    Gigli , author M

    author author L. Gigli , author M. Veit , author M. Kotiuga , author G. Pizzi , author N. Marzari ,\ and\ author M. Ceriotti ,\ https://doi.org/10.1038/s41524-022-00845-0 journal journal npj Computational Materials \ volume 8 ,\ pages 209 ( year 2022 ) NoStop

    doi

  25. [33]

    Grisafi , author A

    author author A. Grisafi , author A. Fabrizio , author B. Meyer , author D. M. \ Wilkins , author C. Corminboeuf ,\ and\ author M. Ceriotti ,\ https://doi.org/10.1021/acscentsci.8b00551 journal journal ACS Central Science \ volume 5 ,\ pages 57 ( year 2019 ) NoStop

    doi

  26. [34]

    Chandrasekaran , author D

    author author A. Chandrasekaran , author D. Kamal , author R. Batra , author C. Kim , author L. Chen ,\ and\ author R. Ramprasad ,\ https://doi.org/10.1038/s41524-019-0162-7 journal journal npj Computational Materials \ volume 5 ,\ pages 22 ( year 2019 ) NoStop

    doi

  27. [35]

    Ben Mahmoud , author A

    author author C. Ben Mahmoud , author A. Anelli , author G. Csányi ,\ and\ author M. Ceriotti ,\ https://doi.org/10.1103/PhysRevB.102.235130 journal journal Physical Review B \ volume 102 ,\ pages 235130 ( year 2020 ) ,\ note publisher: American Physical Society NoStop

    2020 doi

  28. [36]

    Dey \ and\ author D

    author author M. Dey \ and\ author D. Ghosh ,\ https://doi.org/10.1021/acs.jpca.3c05322 journal journal The Journal of Physical Chemistry A \ volume 127 ,\ pages 9159 ( year 2023 ) NoStop

    2023 doi

  29. [37]

    Fiedler , author N

    author author L. Fiedler , author N. A. \ Modine , author S. Schmerler , author D. J. \ Vogel , author G. A. \ Popoola , author A. P. \ Thompson , author S. Rajamanickam ,\ and\ author A. Cangi ,\ https://doi.org/10.1038/s41524-023-01070-z journal journal npj Computational Mat...

    doi

  30. [38]

    Scherbela , author L

    author author M. Scherbela , author L. Gerard ,\ and\ author P. Grohs ,\ https://doi.org/10.1038/s41467-023-44216-9 journal journal Nature Communications \ volume 15 ,\ pages 120 ( year 2024 ) NoStop

    2024 doi

  31. [39]

    Sun , author Y

    author author Q. Sun , author Y. Xiang , author Y. Liu , author L. Xu , author T. Leng , author Y. Ye , author A. Fortunelli , author W. A. \ Goddard ,\ and\ author T. Cheng ,\ https://doi.org/10.1021/acs.jpclett.2c02222 journal journal The Journal of Physical Chemistry Letter...

    doi

  32. [40]

    Golze , author M

    author author D. Golze , author M. Hirvensalo , author P. Hernández-León , author A. Aarva , author J. Etula , author T. Susi , author P. Rinke , author T. Laurila ,\ and\ author M. A. \ Caro ,\ https://doi.org/10.1021/acs.chemmater.1c04279 journal journal Chemistry of Materia...

    doi

  33. [41]

    Batatia , author P

    author author I. Batatia , author P. Benner , author Y. Chiang , author A. M. \ Elena , author D. P. \ Kovács , author J. Riebesell , author X. R. \ Advincula , author M. Asta , author W. J. \ Baldwin , author N. Bernstein , author A. Bhowmik , author S. M. \ Blau , author V. ...

    doi arXiv

  34. [42]

    Deng , author P

    author author B. Deng , author P. Zhong , author K. Jun , author J. Riebesell , author K. Han , author C. J. \ Bartel ,\ and\ author G. Ceder ,\ https://doi.org/10.1038/s42256-023-00716-3 journal journal Nature Machine Intelligence \ volume 5 ,\ pages 1031 ( year 2023 b ) NoStop

    doi

  35. [43]

    Merchant , author S

    author author A. Merchant , author S. Batzner , author S. S. \ Schoenholz , author M. Aykol , author G. Cheon ,\ and\ author E. D. \ Cubuk ,\ https://doi.org/10.1038/s41586-023-06735-9 journal journal Nature \ volume 624 ,\ pages 80 ( year 2023 ) NoStop

    doi

  36. [44]

    Yang , author C

    author author H. Yang , author C. Hu , author Y. Zhou , author X. Liu , author Y. Shi , author J. Li , author G. Li , author Z. Chen , author S. Chen , author C. Zeni , author M. Horton , author R. Pinsler , author A. Fowler , author D. Zügner , author T. Xie , author J. Smith...

    doi arXiv

  37. [45]

    Kwon , author D

    author author Y. Kwon , author D. Lee , author Y.-S. \ Choi , author M. Kang ,\ and\ author S. Kang ,\ https://doi.org/10.1021/acs.jcim.0c00195 journal journal Journal of Chemical Information and Modeling \ volume 60 ,\ pages 2024 ( year 2020 ) NoStop

    doi

  38. [46]

    Han , author H

    author author J. Han , author H. Kang , author S. Kang , author Y. Kwon , author D. Lee ,\ and\ author Y.-S. \ Choi ,\ https://doi.org/10.1039/D2CP04542G journal journal Physical Chemistry Chemical Physics \ volume 24 ,\ pages 26870 ( year 2022 ) NoStop

    doi

  39. [47]

    Bånkestad , author K

    author author M. Bånkestad , author K. M. \ Dorst , author G. Widmalm ,\ and\ author J. Rönnols ,\ https://doi.org/10.1039/D4RA03428G journal journal RSC Advances \ volume 14 ,\ pages 26585 ( year 2024 ) NoStop

    2024 doi

  40. [48]

    Rupp , author R

    author author M. Rupp , author R. Ramakrishnan ,\ and\ author O. A. \ Von Lilienfeld ,\ https://doi.org/10.1021/acs.jpclett.5b01456 journal journal The Journal of Physical Chemistry Letters \ volume 6 ,\ pages 3309 ( year 2015 ) NoStop

    doi

  41. [49]

    author author F. M. \ Paruzzo , author A. Hofstetter , author F. Musil , author S. De , author M. Ceriotti ,\ and\ author L. Emsley ,\ https://doi.org/10.1038/s41467-018-06972-x journal journal Nature Communications \ volume 9 ,\ pages 4501 ( year 2018 ) NoStop

    doi

  42. [50]

    Chaker , author M

    author author Z. Chaker , author M. Salanne , author J.-M. \ Delaye ,\ and\ author T. Charpentier ,\ https://doi.org/10.1039/C9CP02803J journal journal Physical Chemistry Chemical Physics \ volume 21 ,\ pages 21709 ( year 2019 ) NoStop

    doi

  43. [51]

    author author M. C. \ Venetos , author M. Wen ,\ and\ author K. A. \ Persson ,\ https://doi.org/10.1021/acs.jpca.2c07530 journal journal The Journal of Physical Chemistry A \ volume 127 ,\ pages 2388 ( year 2023 ) NoStop

    2023 doi

  44. [52]

    Charpentier ,\ https://doi.org/10.1039/D4FD00129J journal journal Faraday Discussions \ ( year 2024 ),\ 10.1039/D4FD00129J ,\ note publisher: The Royal Society of Chemistry NoStop

    author author T. Charpentier ,\ https://doi.org/10.1039/D4FD00129J journal journal Faraday Discussions \ ( year 2024 ),\ 10.1039/D4FD00129J ,\ note publisher: The Royal Society of Chemistry NoStop

    2024 doi

  45. [53]

    author author A. F. \ Harper , author S. S. \ Köcher , author K. Reuter ,\ and\ author C. Scheurer ,\ https://doi.org/10.26434/chemrxiv-2024-j0kp2 title Performance metrics for tensorial learning: prediction of Li4Ti5O12 nuclear magnetic resonance observables at experimental a...

    2024 doi

  46. [54]

    Batzner , author A

    author author S. Batzner , author A. Musaelian , author L. Sun , author M. Geiger , author J. P. \ Mailoa , author M. Kornbluth , author N. Molinari , author T. E. \ Smidt ,\ and\ author B. Kozinsky ,\ https://doi.org/10.1038/s41467-022-29939-5 journal journal Nature Communica...

    doi

  47. [55]

    Haeberlen ,\ @noop title High resolution NMR in solids: selective averaging ,\ series Advances in magnetic resonance : Supplement \ No

    author author U. Haeberlen ,\ @noop title High resolution NMR in solids: selective averaging ,\ series Advances in magnetic resonance : Supplement \ No. number 1 \ ( publisher Academic Press ,\ address New York ,\ year 1976 ) NoStop

    1976

  48. [56]

    author author R. K. \ Harris , author E. D. \ Becker , author S. M. \ Cabral De Menezes , author P. Granger , author R. E. \ Hoffman ,\ and\ author K. W. \ Zilm ,\ https://doi.org/10.1016/j.ssnmr.2008.02.004 journal journal Solid State Nuclear Magnetic Resonance \ volume 33 ,\...

    2008 doi

  49. [57]

    Mason ,\ https://doi.org/10.1016/0926-2040(93)90010-K journal journal Solid State Nuclear Magnetic Resonance \ volume 2 ,\ pages 285 ( year 1993 ) NoStop

    author author J. Mason ,\ https://doi.org/10.1016/0926-2040(93)90010-K journal journal Solid State Nuclear Magnetic Resonance \ volume 2 ,\ pages 285 ( year 1993 ) NoStop

    1993 doi

  50. [58]

    pages ii NoStop

    in\ https://doi.org/10.1016/B978-0-12-025561-0.50001-0 booktitle High Resolution Nmr in Solids Selective Averaging \ ( publisher Elsevier ,\ year 1976 )\ p. pages ii NoStop

    1976 doi

  51. [59]

    Thomas , author T

    author author N. Thomas , author T. Smidt , author S. Kearnes , author L. Yang , author L. Li , author K. Kohlhoff ,\ and\ author P. Riley ,\ https://doi.org/10.48550/ARXIV.1802.08219 title Tensor field networks: Rotation - and translation-equivariant neural networks for 3D po...

    doi arXiv

  52. [60]

    Bonhomme , author C

    author author C. Bonhomme , author C. Gervais , author F. Babonneau , author C. Coelho , author F. Pourpoint , author T. Azaïs , author S. E. \ Ashbrook , author J. M. \ Griffin , author J. R. \ Yates , author F. Mauri ,\ and\ author C. J. \ Pickard ,\ https://doi.org/10.1021/...

    doi

  53. [61]

    author author F. A. \ Anet , author D. J. \ O'Leary , author C. G. \ Wade ,\ and\ author R. D. \ Johnson ,\ https://doi.org/10.1016/0009-2614(90)85237-7 journal journal Chemical Physics Letters \ volume 171 ,\ pages 401 ( year 1990 ) NoStop

    doi

  54. [62]

    Geiger \ and\ author T

    author author M. Geiger \ and\ author T. Smidt ,\ https://doi.org/10.48550/ARXIV.2207.09453 title e3nn: Euclidean Neural Networks , \ ( year 2022 ),\ note version Number: 1 NoStop

    2022 doi arXiv

  55. [63]

    author author A. P. \ Thompson , author H. M. \ Aktulga , author R. Berger , author D. S. \ Bolintineanu , author W. M. \ Brown , author P. S. \ Crozier , author P. J. \ in 't Veld , author A. Kohlmeyer , author S. G. \ Moore , author T. D. \ Nguyen , author R. Shan , author M...

  56. [64]

    author author L. C. \ Erhard , author J. Rohrer , author K. Albe ,\ and\ author V. L. \ Deringer ,\ https://doi.org/10.1038/s41524-022-00768-w journal journal npj Computational Materials \ volume 8 ,\ pages 90 ( year 2022 ) NoStop

    2022 doi

  57. [65]

    Carré , author J

    author author A. Carré , author J. Horbach , author S. Ispas ,\ and\ author W. Kob ,\ https://doi.org/10.1209/0295-5075/82/17001 journal journal EPL (Europhysics Letters) \ volume 82 ,\ pages 17001 ( year 2008 ) NoStop

    2008 doi

  58. [66]

    Profeta , author F

    author author M. Profeta , author F. Mauri ,\ and\ author C. J. \ Pickard ,\ https://doi.org/10.1021/ja027124r journal journal Journal of the American Chemical Society \ volume 125 ,\ pages 541 ( year 2003 ) NoStop

    2003 doi

  59. [67]

    author author S. J. \ Clark , author M. D. \ Segall , author C. J. \ Pickard , author P. J. \ Hasnip , author M. I. J. \ Probert , author K. Refson ,\ and\ author M. C. \ Payne ,\ https://doi.org/10.1524/zkri.220.5.567.65075 journal journal Zeitschrift für Kristallographie - C...

    doi

  60. [68]

    author author J. P. \ Perdew , author K. Burke ,\ and\ author M. Ernzerhof ,\ https://doi.org/10/bppfwt journal journal Physical Review Letters \ volume 77 ,\ pages 3865 ( year 1996 ) ,\ note publisher: American Physical Society NoStop

    1996

  61. [69]

    author author D. F. \ Thomas Du Toit \ and\ author V. L. \ Deringer ,\ https://doi.org/10.1063/5.0155618 journal journal The Journal of Chemical Physics \ volume 159 ,\ pages 024803 ( year 2023 ) NoStop

    2023 doi

  62. [70]

    Svenningsson \ and\ author L

    author author L. Svenningsson \ and\ author L. J. \ Mueller ,\ https://doi.org/10.1016/j.ssnmr.2022.101849 journal journal Solid State Nuclear Magnetic Resonance \ volume 123 ,\ pages 101849 ( year 2023 ) NoStop

    2022

  63. [71]

    Shankar ,\ @noop title Principles of quantum mechanics \ ( publisher Springer ,\ year 2013 ) NoStop

    author author R. Shankar ,\ @noop title Principles of quantum mechanics \ ( publisher Springer ,\ year 2013 ) NoStop

    2013

  64. [72]

    author author L. C. \ Erhard , author J. Rohrer , author K. Albe ,\ and\ author V. L. \ Deringer ,\ https://doi.org/10.1038/s41467-024-45840-9 journal journal Nature Communications \ volume 15 ,\ pages 1927 ( year 2024 a ) NoStop

    1927 doi

  65. [73]

    https://github.com/CCP-NC/soprano title Soprano , \ NoStop

  66. [74]

    author author A. Stukowski ,\ https://doi.org/10.1088/0965-0393/18/1/015012 journal journal Modelling and Simulation in Materials Science and Engineering \ volume 18 ,\ pages 015012 ( year 2010 ) NoStop

    2010 doi

  67. [75]

    Baerlocher , author D

    author author C. Baerlocher , author D. Brouwer , author B. Marler ,\ and\ author L. McCusker ,\ https://www.iza-structure.org/databases/ title Database of Zeolite Structures , \ NoStop

  68. [76]

    author author L. C. \ Erhard , author D. Utt , author A. J. \ Klomp ,\ and\ author K. Albe ,\ https://doi.org/10.1088/1361-651X/ad64f3 journal journal Modelling and Simulation in Materials Science and Engineering \ volume 32 ,\ pages 065029 ( year 2024 b ) NoStop

    doi

  69. [77]

    Walker , author S

    author author R. Walker , author S. Zerfoss , author S. Holley ,\ and\ author L. Gross ,\ https://doi.org/10.6028/jres.061.026 journal journal Journal of Research of the National Bureau of Standards \ volume 61 ,\ pages 251 ( year 1958 ) NoStop

    doi

  70. [78]

    https://doi.org/10.1524/zkri.1973.138.1-4.274 journal journal Zeitschrift für Kristallographie - Crystalline Materials \ volume 138 ,\ pages 274 ( year 1973 ) NoStop

    1973 doi

  71. [79]

    author author A. F. \ Wright \ and\ author A. J. \ Leadbetter ,\ https://doi.org/10.1080/00318087508228690 journal journal Philosophical Magazine \ volume 31 ,\ pages 1391 ( year 1975 ) NoStop

    1975 doi

  72. [80]

    Hatch \ and\ author S

    author author D. Hatch \ and\ author S. Ghose ,\ https://doi.org/10.1007/BF00202234 journal journal Physics and Chemistry of Minerals \ volume 17 ,\ pages 554 ( year 1991 ) NoStop

    1991 doi

  73. [81]

    Yuan \ and\ author L

    author author F. Yuan \ and\ author L. Huang ,\ https://doi.org/10.1103/PhysRevB.85.134114 journal journal Physical Review B \ volume 85 ,\ pages 134114 ( year 2012 ) NoStop

    2012 doi

  74. [82]

    Sun , author A

    author author J. Sun , author A. Ruzsinszky ,\ and\ author J. Perdew ,\ https://doi.org/10.1103/PhysRevLett.115.036402 journal journal Physical Review Letters \ volume 115 ,\ pages 036402 ( year 2015 ) NoStop

    2015 doi

  75. [83]

    author author S. E. \ Ashbrook , author D. M. \ Dawson ,\ and\ author V. R. \ Seymour ,\ https://doi.org/10.1039/C4CP00578C journal journal Phys. Chem. Chem. Phys. \ volume 16 ,\ pages 8223 ( year 2014 ) NoStop

    2014 doi

  76. [84]

    author author A. G.M. Rankin , author F. Pourpoint , author N. T. \ Duong , author L. Delevoye , author J.-P. \ Amoureux ,\ and\ author O. Lafon ,\ in\ https://doi.org/10.1016/B978-0-12-823144-9.00192-8 booktitle Comprehensive Inorganic Chemistry III \ ( publisher Elsevier ,\ ...

    doi

  77. [85]

    Trillot , author J

    author author S. Trillot , author J. Lam , author S. Ispas , author A. K. A. \ Kandy , author M. E. \ Tuckerman , author N. Tarrat ,\ and\ author M. Benoit ,\ https://doi.org/10.1016/j.commatsci.2024.112848 journal journal Computational Materials Science \ volume 236 ,\ pages ...

    2024

  78. [86]

    Roy , author J

    author author S. Roy , author J. P. \ Dürholt , author T. S. \ Asche , author F. Zipoli ,\ and\ author R. Gómez-Bombarelli ,\ https://doi.org/10.1038/s41467-024-50407-9 journal journal Nature Communications \ volume 15 ,\ pages 6030 ( year 2024 ) NoStop

    doi

  79. [87]

    Ben Mahmoud , author J

    author author C. Ben Mahmoud , author J. L. A. \ Gardner ,\ and\ author V. L. \ Deringer ,\ https://doi.org/10.1038/s43588-024-00636-1 journal journal Nature Computational Science \ volume 4 ,\ pages 384 ( year 2024 ) NoStop

    2024 doi

  80. [88]

    author author O. T. \ Unke \ and\ author M. Meuwly ,\ https://doi.org/10.1021/acs.jctc.9b00181 journal journal Journal of Chemical Theory and Computation \ volume 15 ,\ pages 3678 ( year 2019 ) NoStop

    2019 doi

  81. [89]

    Ivkovi\' c , author J

    author author . Ivkovi\' c , author J. Jover ,\ and\ author J. Harvey ,\ https://doi.org/10.1039/D4DD00168K journal journal Digital Discovery \ volume 3 ,\ pages 2242 ( year 2024 ) ,\ note publisher: RSC NoStop

    2024 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.