Continuous symmetry analyses: C_nv and D_n measures of molecules, complexes, and proteins

The continuous symmetry methodology has been developed to provide a quantitative estimation for the degree of a selected symmetry point-group in a molecular structure. Previous developed measures included the C_S, C_n, S_n (including C_i) symmetries, several polyhedral symmetries, and the related chirality measure. Motivated by the abundance and importance of C_nv symmetry in many key molecules and by the abundance of D_n symmetry in many organometallic complexes and in many protein clusters, measures for these symmetries, based on the general definition of the continuous symmetry measure, were developed. Detailed description of the derivation of the measures, along with plenty of examples for their application, is provided.