Continuous invariant-based asymmetries of periodic crystals quantify deviations from higher symmetry

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 several Crystal Structure Prediction datasets show that about a half of simulated crystals have high values of CIA, while all experimental structures in these datasets have CIA=0. On another hand, many crystals with high values Z’ in the Cambridge Structural Database (CSD) turned out to be close to more symmetric forms with Z’<=1 due to low values of CIAs.

💡 Research Summary

The paper addresses a long‑standing problem in crystallography: the traditional measure of asymmetry, the relative multiplicity Z′, changes abruptly under infinitesimal atomic displacements, because any small perturbation can arbitrarily scale the primitive cell and thus alter Z′ discontinuously. This makes Z′ unsuitable for quantitative comparison of crystal structures, especially in the context of crystal‑structure prediction (CSP) where simulated structures often drift away from ideal symmetry.

To overcome this, the authors build on a recently developed hierarchy of continuous isometry invariants. The core invariant is the Pointwise Distance Distribution (PDD). For a periodic point set S with motif M, the distances from each motif point to its k nearest neighbours in the infinite set are sorted and assembled into a matrix D(S;k). Identical rows are merged with appropriate weights, yielding a discrete distribution that is invariant under any rigid motion and lattice transformation. Crucially, PDD varies continuously under any small perturbation, a property proved in earlier work.

Because PDD still contains a density‑dependent term that grows like √k, the authors introduce the Pointwise Deviation from Asymptotic (PDA). By subtracting the asymptotic term derived from the point‑packing coefficient PPC(S) (which is inversely proportional to point density), PDA removes the dominant √k scaling and converges to zero in the last column as k → ∞. In practice, a modest k (e.g., 100 nearest neighbours) is sufficient for robust discrimination of crystal structures.

With PDA in hand, the paper defines a distance between two geometric blocks (molecules, ions, or any chemically distinct sub‑clusters) using the Earth Mover’s Distance (EMD). Each block is represented by the set of its rows in PDA; the ground distance for EMD is taken as either the RMS or Chebyshev distance between these row vectors. Labels (atomic types) are enforced by assigning infinite cost to mismatched labels, ensuring only chemically identical points are matched. The resulting EMD(B,C) quantifies the minimal perturbation required to align block B with block C within the ambient periodic set, effectively measuring how far the two blocks are from being symmetry‑equivalent.

The Continuous Invariant‑based Asymmetry (CIA) is then defined for a periodic set S with G geometric blocks in its asymmetric unit. For each block i, the maximal EMD to any other block (d_i = max_j EMD(B_i,B_j)) is computed; CIA is the minimum of these maxima (CIA_min) or, alternatively, the average (CIA_avg). Because the underlying distances are expressed in Å, CIA directly reports the physical deviation of the crystal from a higher‑symmetry configuration.

Experimental validation is performed on several CSP datasets. Approximately 50 % of the simulated crystals exhibit high CIA values (e.g., >0.5 Å), indicating substantial deviation from ideal symmetry after energy relaxation. In contrast, all experimentally determined structures in the same datasets have CIA = 0, confirming that real crystals tend to occupy highly symmetric minima. A separate analysis of the Cambridge Structural Database (CSD) shows that many entries with large Z′ actually possess low CIA, meaning they are geometrically close to Z′ ≤ 1 structures despite being reported as high‑multiplicity. This demonstrates that Z′ can overstate asymmetry, whereas CIA provides a more nuanced, continuous assessment.

Key contributions of the work are: (1) introduction of a physically interpretable, continuous asymmetry metric (CIA) that overcomes the discontinuity of Z′; (2) demonstration that the combination of PDD, PDA, and EMD yields a mathematically rigorous, complete invariant capable of distinguishing virtually all crystal structures; (3) practical evidence that CIA can be computed efficiently (hours on a modest desktop) and applied to large databases for symmetry‑based filtering or validation.

Limitations include the current reliance on a fixed neighbour count (k = 100) and the computational cost of solving optimal transport problems for EMD in very large or highly heterogeneous systems. Future work may explore adaptive k selection, GPU‑accelerated EMD solvers, and extensions of the CIA framework to other periodic properties such as phonon spectra or electronic band structures.

In summary, the paper presents a robust, continuous, and physically meaningful measure of crystal asymmetry, opening new avenues for crystal‑structure prediction, database curation, and the quantitative study of symmetry breaking in solid‑state materials.