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Average minimum distances of periodic point sets – foundational invariants for mapping periodic crystals

Average minimum distances of periodic point sets – foundational invariants for mapping periodic crystals

Authors :
Vitaliy Kurlin
Angeles Pulido
Andrew I. Cooper
Marco M. Mosca
Daniel Widdowson
Source :
MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY
Publication Year :
2021
Publisher :
University Library in Kragujevac, 2021.

Abstract

The fundamental model of any solid crystalline material (crystal) at the atomic scale is a periodic point set. The strongest natural equivalence of crystals is rigid motion or isometry that preserves all inter-atomic distances. Past comparisons of periodic structures often used manual thresholds, symmetry groups and reduced cells, which are discontinuous under perturbations or thermal vibrations of atoms. This work defines the infinite sequence of continuous isometry invariants (Average Minimum Distances) to progressively capture distances between neighbors. The asymptotic behaviour of the new invariants is theoretically proved in all dimensions for a wide class of sets including non-periodic. The proposed near linear time algorithm identified all different crystals in the world's largest Cambridge Structural Database within a few hours on a modest desktop. The ultra fast speed and proved continuity provide rigorous foundations to continuously parameterise the space of all periodic crystals as a high-dimensional extension of Mendeleev's table of elements.

Details

ISSN :
03406253
Volume :
87
Database :
OpenAIRE
Journal :
MATCH Communications in Mathematical and in Computer Chemistry
Accession number :
edsair.doi.dedup.....3acd2105deb12a1688ab63d9a5ddcd0d
Full Text :
https://doi.org/10.46793/match.87-3.529w