CONTINUOUS SYMMETRY MEASURES .2. SYMMETRY GROUPS AND THE TETRAHEDRON

We treat symmetry as a continuous property rather than a discrete ''ye s or no'' one. Here we generalize the approach developed for symmetry elements (Part 1: J. Am. Chem. Soc. 1992, 114, 7843-7851) to any symme try group in two and three dimensions. Using the Continuous Symmetry M easure (CSM) method, it is possible to evaluate quantitatively how muc h of any symmetry exists in a nonsymmetric configuration; what is the nearest symmetry group of any given configuration; and how the symmetr ized shapes, with respect to any symmetry group, look. The CSM approac h is first presented in a practical easy-to-implement set of rules, wh ich are later proven in a rigorous mathematical layout. Most of our ex amples concentrate on tetrahedral structures because of their key impo rtance in chemistry. Thus, we show how to evaluate the amount of tetra hedricity (T(d)) existing in nonsymmetric tetrahedra; the amount of ot her symmetries they contain; and the continuous symmetry changes in fl uctuating, vibrating, and rotating tetrahedra. The tool we developed b ears on any physical or chemical process and property which is either governed by symmetry considerations or which is describable in terms o f changes in symmetry.