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Optimality guarantees for crystal structure prediction

Nature volume 619, pages 68–72 (2023)Cite this article

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Abstract

Crystalline materials enable essential technologies, and their properties are determined by their structures. Crystal structure prediction can thus play a central part in the design of new functional materials1,2. Researchers have developed efficient heuristics to identify structural minima on the potential energy surface3,4,5. Although these methods can often access all configurations in principle, there is no guarantee that the lowest energy structure has been found. Here we show that the structure of a crystalline material can be predicted with energy guarantees by an algorithm that finds all the unknown atomic positions within a unit cell by combining combinatorial and continuous optimization. We encode the combinatorial task of finding the lowest energy periodic allocation of all atoms on a lattice as a mathematical optimization problem of integer programming6,7, enabling guaranteed identification of the global optimum using well-developed algorithms. A single subsequent local minimization of the resulting atom allocations then reaches the correct structures of key inorganic materials directly, proving their energetic optimality under clear assumptions. This formulation of crystal structure prediction establishes a connection to the theory of algorithms and provides the absolute energetic status of observed or predicted materials. It provides the ground truth for heuristic or data-driven structure prediction methods and is uniquely suitable for quantum annealers8,9,10, opening a path to overcome the combinatorial explosion of atomic configurations.

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Fig. 1: CSP using integer programming.
Fig. 2: Using integer programming to predict garnet (Ca3Al2Si3O12) and spinel (MgAl2O4) structures.
Fig. 3: Comparison between heuristic and non-heuristic exploration of a PES.

Data availability

The authors declare that the data supporting the findings of this study are available in the paper and  Supplementary Information.

Code availability

An implementation of the integer programming encoding for the periodic lattice allocation problem and subsequent CSP is available at https://github.com/lrcfmd/ipcsp.

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Acknowledgements

We thank the Leverhulme Trust for funding through the Leverhulme Research Centre for Functional Materials Design. V.V.G. thanks M. W. Gaultois for discussions. We thank R. Savani for feedback on the paper.

Author information

Author notes

  1. Argyrios Deligkas

    Present address: Department of Computer Science, Royal Holloway, University of London, London, UK

Authors and Affiliations

  1. Leverhulme Research Centre for Functional Materials Design, Materials Innovation Factory, University of Liverpool, Liverpool, UK

    Vladimir V. Gusev, Duncan Adamson, Argyrios Deligkas, Dmytro Antypov, Matthew S. Dyer, Paul Spirakis & Matthew J. Rosseinsky

  2. Department of Computer Science, University of Liverpool, Liverpool, UK

    Vladimir V. Gusev, Piotr Krysta, Igor Potapov & Paul Spirakis

  3. Department of Chemistry, University of Liverpool, Liverpool, UK

    Christopher M. Collins, George R. Darling, Matthew S. Dyer & Matthew J. Rosseinsky

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Contributions

All authors took part in discussions to frame the use of modern optimization approaches in CSP. V.V.G. and A.D. conceptualized the idea of periodic lattice atom allocation. V.V.G. developed Ewald summation and QUBO encodings, implemented the approach and evaluated it on classical computers. D. Antypov and C.M.C. performed supplementary analysis of resulting structures. V.V.G. suggested the use of quantum annealers; V.V.G. and D. Adamson performed evaluation. V.V.G., A.D., D. Antypov, M.S.D. and M.J.R. wrote the first draft of the paper. V.V.G., D. Adamson, C.M.C., P.K., I.P., P.S. and M.J.R. wrote the final draft of the paper. All authors were involved in discussions and evaluation of drafts during the writing process. P.S. and M.J.R. directed the research.

Corresponding authors

Correspondence to Paul Spirakis or Matthew J. Rosseinsky.

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Nature thanks C. Richard Catlow and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Table 1 The change in energy of the optimal solution of the periodic lattice atom allocation problem for SrTiO3 with g = 4 and P23 (195) space group symmetry constraint under varying unit cell size
Full size table
Extended Data Table 2 The change in energy of the optimal solution of the periodic lattice atom allocation problem for MgAl2O4 with g = 8 and \({\boldsymbol{Fd}}\bar{{\bf{3}}}{\boldsymbol{m}}\,{\boldsymbol{(}}{\bf{227}}{\boldsymbol{)}}\) space group symmetry constraint under varying unit cell size
Full size table
Extended Data Table 3 The change in energy of the optimal solution of the periodic lattice atom allocation problem for Y2O3 with g = 16 and \({\boldsymbol{Ia}}\bar{{\bf{3}}}\) (206) space group symmetry constraint under varying unit cell size
Full size table
Extended Data Table 4 The change in energy of the optimal solution of the periodic lattice atom allocation problem for Y2Ti2O7 with g = 16 and \({\boldsymbol{Fd}}\bar{{\bf{3}}}{\boldsymbol{m}}\,{\boldsymbol{(}}{\bf{227}}{\boldsymbol{)}}\) symmetry constraint under varying unit cell size
Full size table
Extended Data Table 5 The change in the optimal solution of the periodic lattice atom allocation problem under varying unit cell size for Ca3Al2Si3O12, g = 16, \({\boldsymbol{Ia}}\bar{{\bf{3}}}{\boldsymbol{d}}\) (230)
Full size table
Extended Data Table 6 Buckingham potential parameters for Y2O3, Y2Ti2O7, SrO and SrTiO3 with the cut-off radius of 10Å
Full size table
Extended Data Table 7 Buckingham potential parameters for MgAl2O4
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Extended Data Table 8 Force-field parameters for Ca3Al2Si3O12
Full size table
Extended Data Table 9 Buckingham potential parameters for ZrO2 and ZnS
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Extended Data Table 10 Parameters of the quantum annealing runs
Full size table

Supplementary information

Supplementary Information

This file contains the supplementary discussions and equations.

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Gusev, V.V., Adamson, D., Deligkas, A. et al. Optimality guarantees for crystal structure prediction. Nature 619, 68–72 (2023). https://doi.org/10.1038/s41586-023-06071-y

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