- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Determination of ligand field parameters of single-molecule magnets from magnetic susceptibility using deep learning based inverse models

Zayan A. Ali, Julius Mutschler, and Oliver Waldmann*

  • *Contact author: oliver.waldmann@physik.uni-freiburg.de
PDF

Phys. Rev. B 112, 064403 – Published 4 August, 2025

DOI: https://doi.org/10.1103/7tfd-22jr

Abstract

Lanthanide-based single-molecule magnets (SMMs) have garnered extensive attention in recent decades for their potential applications in magnetic memory and quantum computing. In ligand field theory, the magnetic properties of a lanthanide ion may have to be characterized by up to 27 ligand field parameters. However, common experimental data such as temperature-dependent susceptibility curves measured on powder samples typically lack distinctive features. This leads to a massive over-parametrization, where multiple parameter sets can equally describe the data. In this work the issue is tackled by applying a deep learning architecture which combines a variational autoencoder (VAE) with an invertible neural network (INN). This VAE-INN architecture is shown to be capable of determining multiple ligand field parameter sets by extracting hidden parameters from the magnetic data and mapping them to the space of ligand field parameters. The approach is tested on simulated and experimental susceptibility data in magnetic models involving second-order ligand field parameters and is gradually extended to higher-order parameters until it is found limited by dimensionality issues. Compared with traditional least-squares fitting methods, the proposed approach exhibits enhanced generalization, robustness, and convergence. The present investigation of the VAE-INN architecture can be hoped to pave the way for further exploration of higher-order ligand field parameters.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (69)

  1. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  2. X. Gao and L.-M. Duan, Efficient representation of quantum many-body states with deep neural networks, Nat. Commun. 8, 662 (2017).
  3. H.-Y. Huang, R. Kueng, G. Torlai, V. V. Albert, and J. Preskill, Provably efficient machine learning for quantum many-body problems, Science 377, eabk3333 (2022).
  4. A. Samarakoon, D. A. Tennant, F. Ye, Q. Zhang, and S. A. Grigera, Integration of machine learning with neutron scattering for the Hamiltonian tuning of spin ice under pressure, Commun. Mater. 3, 84 (2022).
  5. A. M. Samarakoon, P. Laurell, C. Balz, A. Banerjee, P. Lampen-Kelley, D. Mandrus, S. E. Nagler, S. Okamoto, and D. A. Tennant, Extraction of interaction parameters for RuCl3 from neutron data using machine learning, Phys. Rev. Res. 4, L022061 (2022).
  6. K. T. Schütt, H. Glawe, F. Brockherde, A. Sanna, K. R. Müller, and E. K. U. Gross, How to represent crystal structures for machine learning: Towards fast prediction of electronic properties, Phys. Rev. B 89, 205118 (2014).
  7. L. Fiedler, N. A. Modine, S. Schmerler, D. J. Vogel, G. A. Popoola, A. P. Thompson, S. Rajamanickam, and A. Cangi, Predicting electronic structures at any length scale with machine learning, npj Comput. Mater. 9, 115 (2023).
  8. G. Montavon, M. Rupp, V. Gobre, A. Vazquez-Mayagoitia, K. Hansen, A. Tkatchenko, K.-R. Müller, and O. A. von Lilienfeld, Machine learning of molecular electronic properties in chemical compound space, New J. Phys. 15, 095003 (2013).
  9. Y. Liu, T. Zhao, W. Ju, and S. Shi, Materials discovery and design using machine learning, J. Mater. 3, 159 (2017).
  10. D. Lemm, G. F. von Rudorff, and O. A. von Lilienfeld, Machine learning based energy-free structure predictions of molecules, transition states, and solids, Nat. Commun. 12, 4468 (2021).
  11. A. Merchant, S. Batzner, S. S. Schoenholz, M. Aykol, G. Cheon, and E. D. Cubuk, Scaling deep learning for materials discovery, Nature (London) 624, 80 (2023).
  12. E. O. Pyzer-Knapp, J. W. Pitera, P. W. J. Staar, S. Takeda, T. Laino, D. P. Sanders, J. Sexton, J. R. Smith, and A. Curioni, Accelerating materials discovery using artificial intelligence, high performance computing and robotics, npj Comput. Mater. 8, 84 (2022).
  13. V. Stanev, K. Choudhary, A. G. Kusne, J. Paglione, and I. Takeuchi, Artificial intelligence for search and discovery of quantum materials, Commun. Mater. 2, 105 (2021).
  14. G. Christou, D. Gatteschi, D. N. Hendrickson, and R. Sessoli, Single-molecule magnets, MRS Bull. 25, 66 (2000).
  15. D. Gatteschi, R. Sessoli, and J. Villain, Molecular Nanomagnets (Oxford University Press, Oxford, 2006).
  16. R. Sessoli, H. L. Tsai, A. R. Schake, S. Wang, J. B. Vincent, K. Folting, D. Gatteschi, G. Christou, and D. N. Hendrickson, High-spin molecules: [Mn12O12(O2Cr)16(H2O)4], J. Am. Chem. Soc. 115, 1804 (1993).
  17. P.-H. Lin, T. J. Burchell, L. Ungur, L. F. Chibotaru, W. Wernsdorfer, and M. Murugesu, A polynuclear lanthanide single-molecule magnet with a record anisotropic barrier, Angew. Chem. Int. Ed. 48, 9489 (2009).
  18. A. Dey, P. Kalita, and V. Chandrasekhar, Lanthanide(III)-based single-ion magnets, ACS Omega 3, 9462 (2018).
  19. A. Zabala-Lekuona, J. M. Seco, and E. Colacio, Single-molecule magnets: From Mn12-ac to dysprosium metallocenes, a travel in time, Coord. Chem. Rev. 441, 213984 (2021).
  20. Z. Zhu and J. Tang, Lanthanide single-molecule magnets with high anisotropy barrier: where to from here? Natl. Sci. Rev. 9, nwac194 (2022).
  21. A. Borah and R. Murugavel, Magnetic relaxation in single-ion magnets formed by less-studied lanthanide ions Ce(III), Nd(III), Gd(III), Ho(III), Tm(II/III) and Yb(III), Coord. Chem. Rev. 453, 214288 (2022).
  22. K. Bernot, C. Daiguebonne, G. Calvez, Y. Suffren, and O. Guillou, A journey in lanthanide coordination chemistry: From evaporable dimers to magnetic materials and luminescent devices, Acc. Chem. Res. 54, 427 (2021).
  23. G. Gabarró-Riera, G. Aromí, and E. C. Sañudo, Magnetic molecules on surfaces: SMMs and beyond, Coord. Chem. Rev. 475, 214858 (2023).
  24. L. Bogani and W. Wernsdorfer, Molecular spintronics using single-molecule magnets, Nat. Mater. 7, 179 (2008).
  25. R. Sessoli, Toward the quantum computer: Magnetic molecules back in the race, ACS Cent. Sci. 1, 473 (2015).
  26. A. Chiesa, P. Santini, E. Garlatti, F. Luis, and S. Carretta, Molecular nanomagnets: A viable path toward quantum information processing? Rep. Prog. Phys. 87, 034501 (2024).
  27. A. Gaita-Ariño, F. Luis, S. Hill, and E. Coronado, Molecular spins for quantum computation, Nat. Chem. 11, 301 (2019).
  28. D. J. Newman, Theory of lanthanide crystal fields, Adv. Phys. 20, 197 (1971).
  29. A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Clarendon Press, Oxford, 1970).
  30. J. J. Moré, The Levenberg-Marquardt algorithm: Implementation and theory, in Numerical Analysis, Lecture Notes in Mathematics, edited by G. A. Watson, Vol. 630 (Springer, Berlin, 1978), p. 105–116.
  31. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press, New York, NY, 1992).
  32. D. N. Woodruff, R. E. P. Winpenny, and R. A. Layfield, Lanthanide single-molecule magnets, Chem. Rev. (Washington, DC, U.S.) 113, 5110 (2013).
  33. C. Benelli and D. Gatteschi, Introduction to Molecular Magnetism (Wiley, Weinheim 2015).
  34. O. Waldmann, Relation between electrostatic charge density and spin Hamiltonian models of ligand field in lanthanide complexes, Inorg. Chem. (Washington, DC, U.S.) 64, 1365 (2025).
  35. B. O. Roos and P.-k. Malmqvist, Relativistic quantum chemistry: The multiconfigurational approach, Phys. Chem. Chem. Phys. 6, 2919 (2004).
  36. L. Ungur and L. F. Chibotaru, Ab initio crystal field for lanthanides, Chem. Eur. J 23, 3708 (2017).
  37. P. P. Hallmen, C. Köppl, G. Rauhut, H. Stoll, and J. van Slageren, Fast and reliable ab initio calculation of crystal field splittings in lanthanide complexes, J. Chem. Phys. 147, 164101 (2017).
  38. A. Lunghi and F. Totti, The role of anisotropic exchange in single molecule magnets: A CASSCF/NEVPT2 study of the Fe4 SMM building block [Fe2(OCH3)2(dbm)4] dimer, Inorganics 4, 28 (2016).
  39. M. Briganti, G. F. Garcia, J. Jung, R. Sessoli, B. Le Guennic, and F. Totti, Covalency and magnetic anisotropy in lanthanide single molecule magnets: The DyDOTA archetype, Chem. Sci. 10, 7233 (2019).
  40. J. Jung, M. A. Islam, V. L. Pecoraro, T. Mallah, C. Berthon, and H. Bolvin, Derivation of lanthanide series crystal field parameters from first principles, Chem. Eur. J 25, 15112 (2019).
  41. X. Wu, J.-F. Li, and B. Yin, The interpretation and prediction of lanthanide single-ion magnets from ab initio electronic structure calculation: The capability and limit, Dalton Trans. 51, 14793 (2022).
  42. A. Soncini and M. Piccardo, Ab initio non-covalent crystal field theory for lanthanide complexes: A multiconfigurational non-orthogonal group function approach, Phys. Chem. Chem. Phys. 24, 18915 (2022).
  43. C. A. Morrison, Classical Point-Charge Model (Springer, Berlin, Heidelberg, 1988), Vol. 47.
  44. C. E. Schäffer and C. K. Jørgensen, The angular overlap model, an attempt to revive the ligand field approaches, Mol. Phys. 9, 401 (1965).
  45. J. J. Baldoví, A. Gaita-Ariño, and E. Coronado, Modeling the magnetic properties of lanthanide complexes: Relationship of the REC parameters with Pauling electronegativity and coordination number, Dalton Trans. 44, 12535 (2015).
  46. J. J. Baldoví, J. J. Borrás-Almenar, J. M. Clemente-Juan, E. Coronado, and A. Gaita-Ariño, Modeling the properties of lanthanoid single-ion magnets using an effective point-charge approach, Dalton Trans. 41, 13705 (2012).
  47. H. Ramanantoanina, W. Urland, F. Cimpoesu, and C. Daul, The angular overlap model extended for two-open-shell f and d electrons, Phys. Chem. Chem. Phys. 16, 12282 (2014).
  48. J. J. Baldoví, J. M. Clemente-Juan, E. Coronado, A. Gaita-Ariño, and A. Palii, An updated version of the computational package SIMPRE that uses the standard conventions for Stevens crystal field parameters, J. Comput. Chem. 35, 1930 (2014).
  49. M. Suta, F. Cimpoesu, and W. Urland, The angular overlap model of ligand field theory for f elements: An intuitive approach building bridges between theory and experiment, Coord. Chem. Rev. 441, 213981 (2021).
  50. J. Mutschler, T. Ruppert, J. Strahringer, S. Schlittenhardt, Z. A. Ali, Y. Peng, C. E. Anson, M. Ruben, A. K. Powell, and O. Waldmann, Transferability of ligand field parameters in a family of 3d−4f M2Ln2 butterfly single-molecule magnets, Inorg. Chem. (Washington, DC, U.S.) 64, 6115 (2025).
  51. A. Lunghi and S. Sanvito, Surfing multiple conformation-property landscapes via machine learning: Designing magnetic anisotropy, J. Phys. Chem. C 10, 5802 (2019).
  52. Y. Duan, L. E. Rosaleny, J. T. Coutinho, S. Giménez-Santamarina, A. Scheie, J. J. Baldoví, S. Cardona-Serra, and A. Gaita-Ariño, Data-driven design of molecular nanomagnets, Nat. Commun. 13, 7626 (2022).
  53. Vu Ha Anh Nguyen and A. Lunghi, Predicting tensorial molecular properties with equivariant machine learning models, Phys. Rev. B 105, 165131 (2022).
  54. V. Zaverkin, J. Netz, F. Zills, A. Köhn, and J. Kästner, Thermally averaged magnetic anisotropy tensors via machine learning based on gaussian moments, J. Chem. Theory Comput. 18, 1 (2022).
  55. I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning, Adaptive Computation and Machine Learning (The MIT Press, Cambridge Massachusetts, 2016).
  56. L. Ardizzone, J. Kruse, S. Wirkert, D. Rahner, E. W. Pellegrini, R. S. Klessen, L. Maier-Hein, C. Rother, and U. Köthe, Analyzing inverse problems with invertible neural networks, in ICLR (Vancouver, Canada, 2018).
  57. Y. Peng, V. Mereacre, C. E. Anson, and A. K. Powell, Butterfly M2IIIEr2 (MIII = Fe and Al) SMMs: Synthesis, characterization, and magnetic properties, ACS Omega 3, 6360 (2018).
  58. G. A. Bain and J. F. Berry, Diamagnetic corrections and Pascal's constants, J. Chem. Educ. 85, 532 (2008).
  59. D. P. Kingma and M. Welling, An introduction to variational autoencoders, Found. Trends Mach. Learn. 12, 307 (2019).
  60. I. Higgins, L. Matthey, A. Pal, C. Burgess, X. Glorot, M. Botvinick, S. Mohamed, and A. Lerchner, beta-VAE: Learning basic visual concepts with a constrained variational framework, in ICLR 2017 (Toulon, France, 2017), https://openreview.net/forum?id=Sy2fzU9gl.
  61. L. Dinh, J. Sohl-Dickstein, and S. Bengio, Density estimation using real NVP, in ICLR 2017 (Toulon, France, 2017).
  62. A. Gretton, K. M. Borgwardt, M. J. Rasch, B. Schölkopf, and A. Smola, A kernel two-sample test, J. Mach. Learn. Res. 13, 723 (2012).
  63. See Supplemental Material at http://link.aps.org/supplemental/10.1103/7tfd-22jr for Figs. S1, S2, S3, S4, S5 and Tables S1, S2.
  64. W. Boulila, M. Driss, E. Alshanqiti, M. Al-Sarem, F. Saeed, and M. Krichen, Weight initialization techniques for deep learning algorithms in remote sensing: Recent trends and future perspectives, in Advances on Smart and Soft Computing, edited by F. Saeed, T. Al-Hadhrami, E. Mohammed, and M. Al-Sarem (Springer Singapore, Singapore, 2022), pp. 477–484.
  65. N. Altman and M. Krzywinski, The curse(s) of dimensionality, Nat. Methods 15, 399 (2018).
  66. A. Beskos, M. Girolami, S. Lan, P. E. Farrell, and A. M. Stuart, Geometric MCMC for infinite-dimensional inverse problems, J. Comput. Phys. 335, 327 (2017).
  67. Edited by C. Sammut and G. I. Webb, Encyclopedia of Machine Learning (Springer, New York and London, 2010).
  68. Y. Chen, L. Lu, G. E. Karniadakis, and L. Dal Negro, Physics-informed neural networks for inverse problems in nano-optics and metamaterials, Opt. Express 28, 11618 (2020).
  69. M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 686 (2019).

Outline

Information

Phys. Rev. B 112, 064403– Published 4 August, 2025
  • Received 28 April 2025
  • Revised 25 June 2025
  • Accepted 14 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation