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 ångstroms) 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 of 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.