Ryotaro Koshoji

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While previous studies developed many numecial techniques for crystal structure prediction using metaheuristics such as evolutionary algorithms, we are developing a minimum model for crystal structure prediction without using ab-initio simulation or AI techniques [1-4]. The minimum model formulates the empirical rules of inorganic structural chemistry using generalized disjunctive programming (GDP) [3], so that the minimization problem of Gibbs free energy can be replaced by an auxiliary problem which can be solved easily. GDP is formulated using continuous and Boolean variables to involve the algebraic equations, disjunctions, and logic propositions. This is one of the specific forms of mixed-integer nonlinear programming, which is formulated using continuous and integer variables, in the sense that all the binary variables are only used for selecting the enforced constraints on continuous variables. The typical form of GDP can naturally formulate the rules to design prototypes of crystal structures [3], because inorganic structural chemistry elucidates the spatial orders of atoms based on both the continuous and discrete aspects: The former is packings of atomic spheres, and the latter is connections of coordination polyhedra. This classical model provides a good qualitative explanation of the principles underlying crystal structures, which corresponds to the mathematical structure of crystal structure prediction given by Gibbs free energy. While many numerical techniques such as graph neural networks use the discrete aspects for estimation of not only physical properties but also crystal structure prediction, we have showed that it is important to combine the continuous and discrete aspects to develop an efficient method for designing crystal structures [1-3]. This theory minimizes the number of input parameters for designing crystal structures: We have only to designate the atomic radii and feasible coordination numbers of each chemical element with its oxidation state. Since this theory
is an extension of inorganic structural chemistry, the feasible interatomic distance of every pair of atoms is given by the sum of their atomic radii, where the type of interatomic force is controlled by Boolean variables. The prototypes of crystal structures, which are the optimal solutions of the GDP, satisfy all the constraints on interatomic distances and coordination numbers of each atom. While the original problem for crystal structure prediction includes the eigenvalue problem for many-body wave functions of electrons, the auxiliary problem finds a reasonable initial guess of crystal structures based on only the geometrical constraints. The solution method for the GDP is developed based on the general solution techniques for GDP or integer programming. The GDP is divided into two subproblems: One is the optimization problem of continuous variables for fixed Boolean variables, and the other is the optimization problem of Boolean variables for fixed continuous variables. The former approximates the constraints on interatomic distances by linear penalty potentials to optimize the continuous
variables that are the position of atoms and lattice parameters. The latter is almost equivalent to an edge-coloring problem with two colors, so we assume that this subproblem belongs to class P. The numerical results show that the computational cost of this algorithm is very small; for example, not only simple ternary oxides such as the normal spinel but also simple quaternary oxides such as BaDyFe4O7 can be discovered in about one second by only using ten cores of Intel® coreTM Ultra 9 185H (2.30 GHz), 32GB [3,4]. We also found that the number of optimal solutions are very small; almost all the unstable structures are efficiently removed as infeasible solutions [1-4]. While we must solve a few theoretical problems such as the identification of all the necessary atomic environments for reproducing all the crystal structures, we believe that this minimum model enables creating the table of all the
possible crystal structures like the periodic table. The table will also enable making a list of all the possible chemical compounds with using ab-initio simulations, leading to the computational discovery of materials.

[1] R. Koshoji and T. Ozaki, Phys. Rev. Mater. 8, 113801 (2024).
[2] R. Koshoji, Physical Review Research 7, 033078 (2025).
[3] R. Koshoji, Phys. Rev. Res, 8, 033246 (2026).
[4] https://marici.mathematical-crystal-chemistry.solutions

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University of Tokyo
PH2
Contact: Simon Trebst