Average Minimum Distances of periodic point sets.

Periodic sets of points model all solid crystalline materials (crystals) by representing atoms as labeled points. Crystal structures are determined in a rigid form and are considered up to rigid motions or isometries. Modern tools of Crystal Structure Prediction output thousands of simulated structures, though only few of them can be really synthesized. The first obstacle is the presence of many near duplicate structures that can not be efficiently recognized on the fly by past tools. To continuously quantify a similarity between periodic sets, their isometry invariants should be continuous under perturbations when all discrete invariants such as symmetry groups can break down. This paper studies the isometry classification problem for periodic sets with the new continuity requirement and introduces the Average Minimum Distances, which form an infinite sequence of continuous isometry invariants. Their asymptotic behaviour for a wide class of sets is explicitly described in terms of a point packing coefficient. All results are illustrated by experiments on large datasets of crystals.