A continuous symmetry breaking measure for finite clusters using Jensen-Shannon divergence

A quantitative measure of symmetry breaking is introduced that allows the quantification of which symmetries are most strongly broken due to the introduction of some kind of defect in a perfect structure. The method uses a statistical approach based on the Jensen-Shannon divergence. The measure is calculated by comparing the transformed atomic density function with its original. Software code is presented that carries the calculations out numerically using Monte Carlo methods. The behavior of this symmetry breaking measure is tested for various cases including finite size crystallites (where the surfaces break the crystallographic symmetry), atomic displacements from high symmetry positions, and collective motions of atoms due to rotations of rigid octahedra. The approach provides a powerful tool for assessing local symmetry breaking and offers new insights that can help researchers understand how different structural distortions affect different symmetry operations.

💡 Research Summary

The paper introduces a continuous symmetry‑breaking measure (SBM) that quantifies how strongly each symmetry operation is violated in a finite atomic cluster. The authors model a cluster as a normalized electron‑weighted density function µ(x) built from a superposition of three‑dimensional Gaussian distributions, one for each atom. Each Gaussian is weighted by the atom’s electron count and occupancy, and its variance is taken from an isotropic atomic displacement tensor (ADT) that captures thermal vibrations. By normalizing the total density to unit L1 norm, µ becomes a probability density over ℝ³.

A symmetry operation Tα (e.g., a rotation by angle α, a translation by vector d, a mirror, etc.) is applied by moving the Gaussian means while keeping weights and variances unchanged, yielding a transformed density (Tα)#µ. The degree of symmetry breaking is then defined as the information‑theoretic distance between µ and (Tα)#µ. Two distances are considered: the Kullback‑Leibler (KL) divergence and the Jensen‑Shannon (JS) divergence. The KL‑based SBM, S_KL^Tα