Shannon information as a measure of distortion in coordination polyhedra

Variations in the distribution of bond orders within coordination polyhedra in crystals have been used as a scale of departure from perfect polyhedral symmetry and a measure of polyhedral distortion. The distribution of bond orders mathematically resembles the probability distribution; the bond orders can be normalized to their sum within a polyhedron and used to calculate the Shannon information content that depends on the degree of their departure from uniform distribution. The difference in the Shannon information between a perfect and a distorted polyhedron can be defined as a function taking bond orders as arguments to yield a non-negative real value unambiguously ascribed to a polyhedron. The so-defined distortion function is continuous and vanishes for all the bond orders being equal in the case of a perfect polyhedron. For small distortions it can be reduced to a form analogous to the well known chi-squared function.