Crystal structure prediction (CSP) is one of the most essential tasks in computational materials design given that the structure of a material highly correlates to its properties and synthesis. CSP algorithms take the composition information as input, iterate through candidate structures, and finally output the crystal structure by optimizing the free energy, which corresponds to the objective function. Mathematically, the target of CSP is to locate the global minimum of this function by efficiently sampling the high-dimensional potential energy surface (PES) as a function of configurations. Most of the available CSP methods were developed to improve the searching efficiency for finding a solution, but without an important question being answered: can we prove the optimality of the solution? In a recent work, Paul Spirakis, Matthew J. Rosseinsky and colleagues proposed a method that introduces optimality guarantees for this problem by mapping the CSP task onto a universal optimization problem: integer programming.
In the study, the authors encoded the atomic information — type and position — into an integer decision variable for each atom. The authors further introduced an exclusive constraint to ensure that two different atoms cannot occupy the same location, and a stoichiometry constraint to enforce the correct composition. For the objective function, the authors used pairwise interactions to describe the total energy of the crystal. These three design choices made it possible to successfully abstract a practical CSP problem to an equivalent binary quadratic program. With a proper discretization of the lattice space, the authors adopted the well-known branch-and-cut methods to explore the PES for identifying guaranteed global optimal configurations. Once a configuration is found, it is further fine-tuned by a subsequent local energy minimization process. The authors applied their approach to predict garnet and spinel structures in order to demonstrate the precise allocation of global minimums, which were in good agreement with experimental data. It is worth mentioning that this method can also potentially take advantage of quantum computing, such as quantum annealers that are designed for solving quadratic binary optimization problems. Although the current demonstration is limited to materials with ionic interactions and cubic structures, it sheds light on a pathway for CSP tasks with optimality guarantees.
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