Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jonathan McManus, Vitaliy Kurlin

TL;DR

This work tackles the problem of exactly computing the bridge length $\beta(S)$ for periodic point sets, a crucial ingredient for a complete, continuous isometry invariant of crystals. It introduces a graph-theoretic framework based on periodic Euclidean graphs and labelled quotient graphs, and then delivers a practical algorithm that determines $\beta(S)$ in time $O(m^2 a(U)^n N)$, with $m$ the motif size, $a(U)$ the unit-cell aspect ratio, and $N$ the cost of the Smith Normal Form. The paper provides correctness proofs and tight complexity analyses, and demonstrates the method on real and simulated crystals, showing that $\beta(S)$ is substantially smaller than classical upper bounds and thereby enabling faster, more accurate geometric data analyses in crystallography. Altogether, this work advances exact, scalable bridge-length computation as a key step toward continuous isometry classification and more efficient inverse material design.

Abstract

The fundamental model of any periodic crystal is a periodic set of points at all atomic centres. Since crystal structures are determined in a rigid form, their strongest equivalence is rigid motion (composition of translations and rotations) or isometry (also including reflections). The recent classification of periodic point sets under rigid motion used a complete invariant isoset whose size essentially depends on the bridge length, defined as the minimum `jump' that suffices to connect any points in the given set. We propose a practical algorithm to compute the bridge length of any periodic point set given by a motif of points in a periodically translated unit cell. The algorithm has been tested on a large crystal dataset and is required for an efficient continuous classification of all periodic crystals. The exact computation of the bridge length is a key step to realising the inverse design of materials from new invariant values.

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

TL;DR

This work tackles the problem of exactly computing the bridge length for periodic point sets, a crucial ingredient for a complete, continuous isometry invariant of crystals. It introduces a graph-theoretic framework based on periodic Euclidean graphs and labelled quotient graphs, and then delivers a practical algorithm that determines in time , with the motif size, the unit-cell aspect ratio, and the cost of the Smith Normal Form. The paper provides correctness proofs and tight complexity analyses, and demonstrates the method on real and simulated crystals, showing that is substantially smaller than classical upper bounds and thereby enabling faster, more accurate geometric data analyses in crystallography. Altogether, this work advances exact, scalable bridge-length computation as a key step toward continuous isometry classification and more efficient inverse material design.

Abstract

The fundamental model of any periodic crystal is a periodic set of points at all atomic centres. Since crystal structures are determined in a rigid form, their strongest equivalence is rigid motion (composition of translations and rotations) or isometry (also including reflections). The recent classification of periodic point sets under rigid motion used a complete invariant isoset whose size essentially depends on the bridge length, defined as the minimum `jump' that suffices to connect any points in the given set. We propose a practical algorithm to compute the bridge length of any periodic point set given by a motif of points in a periodically translated unit cell. The algorithm has been tested on a large crystal dataset and is required for an efficient continuous classification of all periodic crystals. The exact computation of the bridge length is a key step to realising the inverse design of materials from new invariant values.

Paper Structure

This paper contains 6 sections, 10 theorems, 1 equation, 5 figures, 1 table.

Table of Contents

  1. Introduction: practical motivations and the problem statement
  2. Auxiliary concepts of graph theory for the bridge length algorithm
  3. Algorithm for the bridge length of a periodic point set
  4. Correctness and time complexity of the bridge length algorithm
  5. Experiments on real and simulated crystals, and a discussion
  6. A faster 'online' algorithm for the Smith Normal Form

Key Result

Theorem 6

For any periodic point set $S \subset \mathbb{R}^n$ with a motif of $m$ points in a unit cell $U$, the bridge length $\beta(S)$ can be computed in time $O(m^2 a(U)^nN)$, where $N$ is the time complexity of the Smith Normal Form, $a(U)$ is the aspect ratio from Definition dfn:parameters.

Figures (5)

  • Figure 1: Left: the orthonormal basis $\vb*{v_1},\vb*{v_2}$ generates the green lattice $\Lambda$ and the unit cell $U$ containing the blue motif $M$ of three points. The periodic point set $S=\Lambda+M$ is obtained by periodically repeating $M$ along all vectors of $\Lambda$. Right: different motifs $M,M'$ in the same cell generate periodic sets that differ by only translation.
  • Figure 2: All Minimum Spanning Trees on extended motifs of a periodic point set S have the longest edge (in blue) of length 3, which could be made arbitrarily long, relative to a preserved minimum inter-point distance of 1 and bridge length $\beta(S)=2$ due to shorter edges from the top right point in every cell across a cell boundary.
  • Figure 3: Left: the periodic point set $S$ with the basis vectors $\vb*{v_1}=(5,0)$, $\vb*{v_2}=(0,5)$ and motif points $p=(2,1)$, $q=(3,4)$. Middle: the periodic Euclidean graph $G\subset\mathbb R^2$ with three types of straight-line edges: green, blue, orange of lengths $\sqrt{5},\sqrt{10},\sqrt{20}$, respectively. Right: the labelled quotient graph $Q$ has directed edges $e_g,e_b,e_o$ with translational vectors indicating integer shifts of cells, see Definitions \ref{['dfn:periodic_graph']}, \ref{['dfn:quotient_graph']}, \ref{['dfn:LQG']}.
  • Figure 4: Left: the 3 vectors $\vb*{v_1}=(0,1)$, $\vb*{v_2}=(2,0)$, $\vb*{v_3}=(3,0)$ form a basis of $\mathbb Z^2$. Other images: none of the 3 pairs $(\vb*{v_2}, \vb*{v_3})$, $(\vb*{v_1}, \vb*{v_2})$, $(\vb*{v_1}, \vb*{v_3})$ form a basis (insufficient for full connectedness) of $\mathbb Z^2$. Some straight edges are shown curved for better visibility.
  • Figure 5: T2 molecule and 5 crystals synthesized from T2. The first four T2-$\alpha$, T2-$\beta$, T2-$\gamma$, T2-$\delta$ were reported in pulido2017functional, the last T2-$\epsilon$ in zhu2022analogy.

Theorems & Definitions (33)

  • Definition 1: lattice, unit cell, motif, periodic point set
  • Definition 2: bridge length $\beta(S)$
  • Definition 4: Minimum Spanning Tree
  • Definition 5: parameters $r(U)$, $R(S)$, $a(U)$
  • Theorem 6
  • Definition 7: $G\subset\mathbb R^n$
  • Definition 8: quotient graph
  • Definition 9: labelled quotient graph
  • Lemma 10: lifting
  • proof
  • ...and 23 more