Structural Analysis Algorithms for Nanomaterials

This thesis presents a reformulation of existing problems in materials science in terms of well-known methods from applied mathematics: graph theory, computational geometry, and mixed integer programming.
The centrosymmetry parameter is reformulated as a graph matching problem, and resolves the inconsistencies in the existing calculation methods as a consequence. By formulating the distance function of lattices as a bipartite graph matching problem, it is shown that the similarity between crystal lattices (root mean square distance, RMSD) can be calculated in polynomial time, which improves upon the existing factorial-time bound. This method is subsequently extended to two-dimensional monolayers.
A method is presented for the identication of ordered crystalline phases in molecular dynamics simulations. A robust classication is obtained by the use of template matching, also formulated as a bipartite matching problem on geometric graphs. This method is adapted for two-dimensional materials, in order that e.g. defect structures in polycrystalline graphene can be studied.
Matrix decompositions are used to develop a geometric lattice matching algorithm, which can exhaustively identify all low-strain interfaces. The stable, low-energy interfaces which are found as a result are intended for use in the design and construction of topological superconductors, which have important applications in quantum computing.
Cluster expansion models are used to nd ground-state structures in gold-silver nanoparticles, which are used in a variety of catalysis processes. In addition to this concrete application, theoretical methods are developed for the optimal construction of cluster expansion models, the exact determination of ground states in a large model, and the exhaustive determination of all possible ground states in a small model.
Lastly, a method for nearly-optimal sampling of orientations is presented. Whilst this has many applications in science and engineering, the use-case described here is the indexing of diffraction patterns for experimental materials characterization. Signicantly improved sampling is achieved by applying methods from computational geometry.