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Voronoi-based similarity distances between arbitrary crystal lattices

Marco Michele Mosca, Vitaliy Kurlin

TL;DR

This work addresses the challenge of comparing crystal lattices across arbitrary systems by introducing two Voronoi-based distances, the extended Hausdorff distance $d_H$ and the scale-invariant distance $d_s$, defined on equivalence classes of lattices under rigid motions. The methods leverage Voronoi cells and optimizations over rotations in $SO(3)$ to achieve invariance to unit-cell representations and scaling, while guaranteeing metric properties and continuity under perturbations. Rigorous definitions and proofs establish the mathematical validity, and a fast linear-time computation approach enables practical use; experiments on a large T2 crystal dataset demonstrate that $d_H$ and $d_s$ uncover geometric differences not captured by energy or density alone, facilitating clustering and visualization. Together, these contributions offer a principled, scalable framework for navigating crystal-structure spaces to accelerate CSP workflows.

Abstract

This paper develops a new continuous approach to a similarity between periodic lattices of ideal crystals. Quantifying a similarity between crystal structures is needed to substantially speed up the Crystal Structure Prediction, because the prediction of many target properties of crystal structures is computationally slow and is essentially repeated for many nearly identical simulated structures. The proposed distances between arbitrary periodic lattices of crystal structures are invariant under all rigid motions, satisfy the metric axioms and continuity under atomic perturbations. The above properties make these distances ideal tools for clustering and visualizing large datasets of crystal structures. All the conclusions are rigorously proved and justified by experiments on real and simulated crystal structures reported in the Nature 2017 paper "Functional materials discovery using energy-structure-function maps".

Voronoi-based similarity distances between arbitrary crystal lattices

TL;DR

This work addresses the challenge of comparing crystal lattices across arbitrary systems by introducing two Voronoi-based distances, the extended Hausdorff distance and the scale-invariant distance , defined on equivalence classes of lattices under rigid motions. The methods leverage Voronoi cells and optimizations over rotations in to achieve invariance to unit-cell representations and scaling, while guaranteeing metric properties and continuity under perturbations. Rigorous definitions and proofs establish the mathematical validity, and a fast linear-time computation approach enables practical use; experiments on a large T2 crystal dataset demonstrate that and uncover geometric differences not captured by energy or density alone, facilitating clustering and visualization. Together, these contributions offer a principled, scalable framework for navigating crystal-structure spaces to accelerate CSP workflows.

Abstract

This paper develops a new continuous approach to a similarity between periodic lattices of ideal crystals. Quantifying a similarity between crystal structures is needed to substantially speed up the Crystal Structure Prediction, because the prediction of many target properties of crystal structures is computationally slow and is essentially repeated for many nearly identical simulated structures. The proposed distances between arbitrary periodic lattices of crystal structures are invariant under all rigid motions, satisfy the metric axioms and continuity under atomic perturbations. The above properties make these distances ideal tools for clustering and visualizing large datasets of crystal structures. All the conclusions are rigorously proved and justified by experiments on real and simulated crystal structures reported in the Nature 2017 paper "Functional materials discovery using energy-structure-function maps".

Paper Structure

This paper contains 19 sections, 5 theorems, 28 equations, 8 figures.

Table of Contents

  1. Introduction: motivations for similarity distances from Crystal Structure Prediction
  2. Rigorous definitions and problem statements for crystal equivalences and distances
  3. The comparison (equivalence) problem for periodic crystals
  4. The equivalence problem and geometric invariants for periodic crystals and lattices
  5. The distance problem for lattices of periodic crystal structures
  6. Strengths and weaknesses of past approaches to a similarity of crystal structures
  7. The COMPACK algorithm for the Cambridge Structural Database (CSD)
  8. The COMPSTRU tool at the Bilbao Crystallographic Server (BCS)
  9. Comparison algorithms based on reduced cells of crystal lattices
  10. New continuous Voronoi-based distances between arbitrary crystal lattices
  11. Offsets and the Hausdorff distance between arbitrary sets and crystal structures
  12. The Voronoi cell of an arbitrary crystal lattice and its geometric stability
  13. The rotational extension of the Hausdorff distance to equivalence classes of lattices
  14. The scaling distance and its rotational extension to equivalence classes of lattices
  15. Computations of new distances for lattices of simulated T2 crystal structures
  16. ...and 4 more sections

Key Result

Lemma 1

(proved in Appendix). For any centrally symmetric polyhedra $P,P' \subset \mathbb R^n$ and a translation $T_v$ by a vector $v \in \mathbb R^n$, the offset parameter is minimal when $T_v$ moves the center $c(P)$ of the polyhedron $P$ to the center $c(P')$ of $P'$.

Figures (8)

  • Figure 1: (1st: any lattice has infinitely many primitive cells, e.g. U,U',U". 2nd: Niggli's reduction of a vector $v_2$ relative to $v_1$ can lead to two cells U' and U", a choice is unstable. 3rd: the yellow offset N(A;1) of A is inside the scaled cell 2A. 4th: yellow offset N(B;1).)
  • Figure 2: (The Voronoi cells of lattices: hexagonal, square, cubic, body-centered cubic (BCC), face-centered cubic (FCC), see definitions in subsection 4.2. the cubic lattice has the standard basis (1,0,0),(0,1,0),(0,0,1); the body-centered cubic (BCC) lattice has the basis (1,0,0),(0,1,0),(0.5,0.5,0.5); the FCC lattice has the basis (1,0,0),(0.5,0.5,0),(0.5,0,0.5). )
  • Figure 3: (Illustrations of extended Hausdorff distances: one Voronoi cell V(L) is optimally rotated and inscribed into a minimal offset Voronoi cell V(L') for pairs L,L' from the left to right: (cubic, BCC), (BCC, cubic), (cubic, FCC), (FCC, cubic), (BCC, FCC), (FCC, BCC). )
  • Figure 4: (Illustrations of scale-invariant distances $d_s(L,L')$ for pairs L,L' from the left to right: (cubic, BCC), (BCC, cubic), (cubic, FCC), (FCC, cubic), (BCC, FCC), (FCC, BCC).)
  • Figure 5: (T2 molecule: triptycenetrisbenzimidazolon; crystals T2$\alpha$, T2$\beta$, T2$\gamma$, T2$\delta$, T2$\epsilon$. biblio:linjiangandy)
  • ...and 3 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 2
  • Lemma 3
  • Theorem 4
  • Theorem 5