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Efficient crystal structure prediction based on the symmetry principle

Nature Computational Science volume 5, pages 255–267 (2025)Cite this article

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Abstract

Crystal structure prediction (CSP) is an evolving field aimed at discerning crystal structures with minimal prior information. Despite the success of various CSP algorithms, their practical applicability remains circumscribed, particularly for large and complex systems. Here, to address this challenge, we show an evolutionary structure generator within the MAGUS (Machine Learning and Graph Theory Assisted Universal Structure Searcher) framework, inspired by the symmetry principle. This generator extracts both global and local features of explored crystal structures using group and graph theory. By integrating an on-the-fly space group miner and fragment reorganizer, augmented by symmetry-kept mutation, our approach generates higher-quality initial structures, reducing the computational costs of CSP tasks. Benchmarking tests show up to fourfold performance improvements. The method also proves valid in complex phosphorus allotrope systems. Furthermore, we apply our approach to the diamond–silicon (111)-(7 × 7) surface system, identifying up to 42 metastable structures within an 18 meV Å−2 energy range, demonstrating the efficacy of our approach in navigating challenging search spaces.

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Fig. 1: Illustration of the concept of the symmetry-principle-guided evolutionary structure generator.
Fig. 2: Illustration of how MAGUS found the global minima for MgAl2O4, highlighting the usage of a space group miner and symmetry-kept rattle mutation.
Fig. 3: Illustration of one trajectory of MAGUS’s global search into the γ-B system with an on-the-fly fragments reorganizer and a space group miner.
Fig. 4: Benchmark comparison for different testing systems.
Fig. 5: A trajectory of MAGUS’s global search to identify violet P.
Fig. 6: Illustration of the target structural model, corresponding initial structure and benchmark results of the Si (111)-(7 × 7) surface system.

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Data availability

Source data for Figs. 2–4 and 6, Extended Data Figs. 1 and 3 are available with this paper. All data were generated using the MAGUS code (version 2.0) and are available from gitlab (https://gitlab.com/bigd4/magus) and on Zenodo at https://doi.org/10.5281/zenodo.14730874 (ref. 91).

Code availability

The MAGUS source code can be accessed from gitlab (https://gitlab.com/bigd4/magus) after registration (https://www.wjx.top/vm/m5eWS0X.aspx), or on Zenodo at https://doi.org/10.5281/zenodo.14730874 (ref. 91).

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Acknowledgements

We thank Z. Fan and Y. Wang for fruitful discussion regarding NEP usage. We gratefully acknowledge the financial support from the National Key R&D Program of China (grant number 2022YFA1403201), the National Natural Science Foundation of China (grants T2495231, 12125404 and 123B2049), the Basic Research Program of Jiangsu (grants BK20233001 and BK20241253), the Jiangsu Funding Program for Excellent Postdoctoral Talent (grants 2024ZB002 and 2024ZB075), the Postdoctoral Fellowship Program of CPSF (grant GZC20240695), the AI & AI for Science program of Nanjing University, and the Fundamental Research Funds for the Central Universities. The calculations were carried out using supercomputers at the High-Performance Computing Center of Collaborative Innovation Center of Advanced Microstructures and the high-performance supercomputing center of Nanjing University.

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  1. These authors contributed equally: Yu Han, Chi Ding.

Authors and Affiliations

  1. National Laboratory of Solid State Microstructures, School of Physics and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, China

    Yu Han, Chi Ding, Junjie Wang, Jiuyang Shi, Shaobo Yu, Qiuhan Jia, Shuning Pan & Jian Sun

  2. Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany

    Hao Gao

  3. Fisika Aplikatua Saila, Gipuzkoako Ingeniaritza Eskola, University of the Basque Country (UPV/EHU), Donostia/San Sebastián, Spain

    Hao Gao

  4. Centro de Física de Materiales (CFM-MPC), CSIC-UPV/EHU, Donostia/San Sebastián, Spain

    Hao Gao

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Contributions

Y.H. and C.D. implemented the code, collected and analyzed the data, and led the paper preparation. J. Shi, S.Y., Q.J. and S.P. provided feedback throughout the process, and assisted with the paper writing. J. Sun, H.G. and J.W. conceived the project, supervised the research and contributed to securing funding. All authors participated in the discussion of the results and the writing of the paper.

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Correspondence to Junjie Wang, Hao Gao or Jian Sun.

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Extended data

Extended Data Fig. 1 The symmetry principle in crystal structure prediction for the MgAl2O4 system.

(a) Space group distribution (P1 structures are excluded) and corresponding enthalpy of 3580 local relaxed structures of MgAl2O4, whose initial states are 20 random generated structures for each specific space group. The relaxed structures show a preference for certain space group symmetry. (b) Group-subgroup relationships between the preferred space groups. Each pair of group-subgroup relations with index<4 is connected by an arrow and the space group of GM is colored in red. (c) Spacegoup distribution of resultant structures obtained from applying symmetry-kept rattle mutation to a single metastable Pnma phase followed by local relaxation (duplicates of parent structure are excluded). The subfigure shows the view of the parent and two offsprings. It could conclude that most offsprings lowered their symmetry than parent after local relaxation. (d) A possible explanation can be derived from the double well model PES. When a high symmetry phase locates surrounding by multi lower energy lower symmetry local minima, its symmetry is prone to breaking. The red atoms represent O, the orange atoms represent Mg, and the cyan atoms represent Al.

Source data

Extended Data Fig. 2 Illustration of graph-theory-based structure decomposition method.

(a) Structure of α-B, the boron fragments obtained and their ranking indicators (uniqueness, description length). The two inequivalent atoms in α-B are marked by solid circles of different colors. (b) The ‘neighborhood’ structure within a certain distance and step cutoff of the selected atom marked by magenta. (c) Impact of cutoff parameters for building α-B ‘neighborhood’ structures. As the distance and step cutoff decrease, the time cost decreases as less fragments are identified. The B12 icosahedra can only be found with distance cutoff no less than 4 Å and step cutoff no less than 3, marked by yellow line. (d) Structure of the fibrous red phosphorous and the decomposed fragments. (e) Structure of graphene and the decomposed fragments. (f) The B12 icosahedra community (iv), has the most uniformly distributed betweenness centrality, and appending any neighbor atom to it (i-iii) will disrupt this uniformity. Therefore, the ‘uniqueness’ of betweenness centrality is employed as one of the indicators for fragment ranking. Atoms that have same betweenness centrality are same colored, with the specific values of betweenness centrality indicated in the legend at the bottom of the figure.

Extended Data Fig. 3 Structure of the red phosphorus allotropes and the success rate for identifying them.

(a) Fibrous P structure. Several structures exhibiting characteristics similar to other stable experimental structures are selected and labeled as (ii-vi). The success rate is calculated for identifying structures with machine learning potential energy lower than or equal to that of these reference structures. (b) Violet P structure and the corresponding success rates. The reference structures are shown in Supplementary Fig. 1(d).

Source data

Extended Data Fig. 4 Different representative metastable Si (111)-(7×7) surface reconstruction models found by MAGUS having 96-108 atoms in the surface region.

The surface energy, space group symmetry, and number of atoms in the reconstruction region are indicated. A deeper color and larger atoms represent the upper surface, while lighter color and smaller atoms represent the substrate. The surface energy of reference DAS model is set to 0. More metastable structures are shown in Supplementary Fig. 2.

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Supplementary Sections 1–5, Figs. 1–6 and Tables 1–4.

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Han, Y., Ding, C., Wang, J. et al. Efficient crystal structure prediction based on the symmetry principle. Nat Comput Sci 5, 255–267 (2025). https://doi.org/10.1038/s43588-025-00775-z

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