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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The authors declare that the data supporting the findings of this study are available in the paper and Supplementary Information.
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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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.
The authors declare no competing interests.
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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