The Density Fingerprint of a Periodic Point Set

TL;DR

This paper introduces a density fingerprint for periodic point sets modeling crystals, enabling efficient material search with proven invariance, continuity, and completeness, supported by a fast algorithm and practical applications.

Contribution

It presents a novel density fingerprint for crystals with proven mathematical properties and an efficient algorithm for structure prediction.

Findings

Fingerprint is invariant under isometries

Algorithm is fast and based on Brillouin zones

Application demonstrated in crystal structure prediction

Abstract

Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction.