Continuous distance-based isometry invariants of periodic crystal structures

The number of known crystal structures, as well as simulated structures created for purposes such as crystal structure prediction, has risen exponentially in recent decades. Crystals are often classified by symmetry groups, however given they are finite and discrete, many otherwise distinct crystals are classified together, and nearly identical structures can be classified as different. The aim of this research is to develop a 'continuous' classification of periodic structures, where all crystals live in a rigorously parameterisable 'Crystal Isometry Space'.

A key axiom is that since crystals are rigid objects, 'rigid motion' (translation + rotation) does not change a crystal, and therefore our crystal descriptors should not change under rigid motion either. This work considers the wider class of transformations 'isometry' (rigid motion + reflection). A crystal descriptor which does not change under isometry is called an 'isometry invariant'. As opposed to discrete symmetry groups, our 'continuous' isometry invariants guarantee that small changes in atomic positions result in small changes in the invariant.

The contribution of this thesis is a collection of continuous isometry invariants of crystals each with various properties of interest to crystallography or geometry. We will show how fast comparisons of these invariants were able to produce an extensive list of cross-entries between the Cambridge Structural Database and Crystallography Open Database, how we discovered erroneous entries in these databases, and how we demonstrated the 'Crystal Isometry Principle': all known non-disordered organic crystals are determined uniquely by the geometry of its atomic centres alone.