Measuring Symmetry in Structural Chemistry

Symmetry is treated as a continuous property and a Continuous Symmetry Measure (CSM) of structures is de ned. The CSM of a structure is de ned to be the minimummean squared distance required to move points of the original structure and change it to a symmetrical structure. This general de nition of a symmetrymeasure enables a comparison of the amount of symmetry of di erent shapes and the amount of di erent symmetries of a single shape. We describe a general method of obtaining the minimal distance to the desired shape and apply it to any symmetry element or symmetry group in two or three dimensions. Various examples of the application of the CSM approach to structural chemistry are presented. These include symmetry analysis of distorted tetrahedra and of rotating ethane, symmetry analysis of contours of equi-property such as molecular orbital representations, reconstruction of incomplete structural data and symmetry analysis of structures with uncertain point-location, such as encountered in x-ray data analysis.