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Mathematical crystal chemistry II: Random search for ionic crystals and analysis on oxide crystals registered in ICSD

Ryotaro Koshoji*,†

  • *Present address: Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan.
  • †Contact author: cosaji@issp.u-tokyo.ac.jp
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Phys. Rev. Research 7, 033078 – Published 21 July, 2025

DOI: https://doi.org/10.1103/6yl6-fr8b

Abstract

Mathematical crystal chemistry views crystal structures as the optimal solutions of the mathematical optimization problem formalizing inorganic structural chemistry. This paper introduces the minimum and maximum atomic radii depending on the types of geometrical constraints, extending the concept of effective atomic sizes. These radii define permissible interatomic distances instead of interatomic forces, constraining feasible types and connections of coordination polyhedra. The definition derives the continuous optimization problem as dense packings of flexible atomic spheres. Additionally, discrete variables and constraint functions, which give a choice of creatable types of geometrical constraints depending on the spatial order of atoms, are implemented to formalize the feasible atomic environment, such as the composition of coordination polyhedra, resulting in acceleration of structure search and decrease of the number of local minima. The framework identifies unique optimal solutions corresponding to the structures of spinel, pyrochlore (α and β), pyroxene, quadruple perovskite, cuprate superconductor YBa2Cu3O7−x, and iron-based superconductor LaFeAsO. Notably, up to 95% of oxide crystal structure types in the Inorganic Crystal Structure Database align with the optimal solutions preserving experimental structures despite the discretized feasible atomic radii. These findings highlight the role of mathematical optimization problems as a theoretical foundation for mathematical crystal chemistry, enabling efficient structure prediction.

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Mathematical crystal chemistry: A mathematical programming approach to crystal structures of inorganic compounds

Ryotaro Koshoji and Taisuke Ozaki
Phys. Rev. Materials 8, 113801 (2024)

Article Text

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Phys. Rev. Research 7, 033078– Published 21 July, 2025
  • Received 31 March 2025
  • Accepted 7 July 2025
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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.

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