A continuous invariant-based asymmetry of a periodic crystal quantifies its deviation from higher symmetry

TL;DR

This work replaces the discontinuous relative multiplicity $Z'(S)$ as a symmetry measure with the Continuous Invariant-based Asymmetry (CIA), a continuous, geometry-based metric for periodic crystals. CIA leverages the isometry-invariant Pointwise Distance Distribution (PDD) and Earth Mover's Distance (EMD) between geometric blocks to quantify deviations from higher symmetry in physical units and proves invariance and continuity under perturbations. The approach yields fast, scalable screening for crystal structure prediction (CSP) datasets and reveals that many high-$Z'$ structures in the Cambridge Structural Database (CSD) are actually close to more symmetric forms, while simulated crystals often exhibit nonzero CIA, indicating asymmetry-related instability or non-synthesis. Overall, CIA provides a rigorous, continuous, and computationally efficient tool for assessing crystal symmetry and guiding structure prediction and analysis in crystallography.

Abstract

Ideal symmetry is known to break down under almost any noise. One measure of asymmetry in a periodic crystal is the relative multiplicity Z' of geometrically non-equivalent units. However, Z' discontinuously changes under almost any displacement of atoms, which can arbitrarily scale up a primitive cell. This discontinuity was recently resolved by a hierarchy of invariant descriptors that continuously change under all small perturbations. We introduce a Continuous Invariant-based Asymmetry (CIA) to quantify (in physically meaningful Angstroms) the deviation of a periodic crystal from a higher symmetry form. Our experiments on Crystal Structure Prediction datasets show that many simulated crystals may be non-synthesisable not only due to high energy but also due to high CIA values, which are much faster to compute. On another hand, many crystals with high Z' values in the Cambridge Structural Database (CSD) turned out to be close to more symmetric forms with Z'<=1 due to low CIA values.

Figures (34)

• Figure 1: Almost any noise arbitrarily scales up a primitive yellow cell and discontinuously changes the relative multiplicity $Z'$ of molecules, which are represented by black $Y$ graphs whose terminal vertices have initial positions shown by red circles.
• Figure 2: T0, T1, T2, and T2E molecules in the four CSP datasets in this section.
• Figure 3: The histograms of $\mathrm{CIA}$ for simulated crystals represented by 3 base points at 'ends' of molecules in Fig. \ref{['fig:T-molecules']}. Row 1: T0, row 2: T1, row 3: T2, row 4: T2E.
• Figure 4: Energy vs density for simulated T0 crystals, coloured by their $\mathrm{CIA}$.
• Figure 5: Energy vs density for simulated T1 crystals, coloured by their $\mathrm{CIA}$.

Theorems & Definitions (11)

• Definition 1: relative multiplicity $Z'$
• Definition 2: Pointwise Distance Distribution $\mathrm{PDD}$
• Definition 3: invariants $\mathrm{PPC}(S)$ and $\mathrm{PDA}(S;k)$
• Definition 4: Earth Mover's Distance $\mathrm{EMD}$ between geometric blocks
• Definition 5: Continuous Invariant-based Asymmetry $\mathrm{CIA}(S)$
• Lemma 6: invariance of $\mathrm{CIA}$s
• Lemma 7: inequalities for $\mathrm{CIA}$s
• Theorem 8: continuity of $\mathrm{CIA}$ under perturbations
• proof : Proof of Lemma \ref{['lem:CIA_invariance']}
• proof : Proof of Lemma \ref{['lem:CIA_inequalities']}