Computing the bridge length: the key ingredient in a continuous isometry classification of periodic point sets

Area of Science:

• Crystallography
• Materials Science
• Computational Chemistry

Background:

• Periodic crystals are modeled as point sets, with rigid motion (translations, rotations, reflections) defining their equivalence.
• Current classification methods rely on invariants like isosets, whose computation can be complex.
• The bridge length, a measure of the minimum connection distance between any two points in a set, is crucial for isoset-based classification.

Purpose of the Study:

• To propose and validate a practical algorithm for computing the bridge length of periodic point sets.
• To facilitate efficient and continuous classification of crystal structures.
• To enable inverse design of materials by leveraging novel invariant values.

Main Methods:

• Development of a practical algorithm to calculate the bridge length from a unit cell motif.
• Testing the algorithm on a large dataset of crystal structures.
• Utilizing the bridge length as a complete invariant for classifying periodic point sets under rigid motion.

Main Results:

• A practical algorithm for computing the bridge length of periodic point sets has been successfully developed.
• The algorithm's efficacy was demonstrated through extensive testing on a large crystal dataset.
• The bridge length computation is shown to be a critical step for efficient crystal classification.

Conclusions:

• The developed algorithm provides an efficient means to compute the bridge length, a vital invariant for crystal structure analysis.
• This work is essential for advancing the continuous classification of periodic crystals.
• Accurate bridge length computation paves the way for inverse material design strategies.