Continuous Symmetry Measures. 2. Symmetry Groups and the Tetrahedron

We treat symmetry as a continuous property rather than a discrete “yes or no” one. Here we generalize the approach developed for symmetry elements (Part 1: J. Am. Chem. Soc. 1992, 114,7843–7851) to any symmetry group in two and three dimensions. Using the Continuous Symmetry Measure (CSM) method, it is possible to evaluate quantitatively how much of any symmetry exists in a nonsymmetric configuration; what is the nearest symmetry group of any given configuration; and how the symmetrized shapes, with respect to any symmetry group, look. The CSM approach is first presented in a practical easy-to-implement set of rules, which are later proven in a rigorous mathematical layout. Most of our examples concentrate on tetrahedral structures because of their key importance in chemistry. Thus, we show how to evaluate the amount of tetrahedricity (T_d) existing in nonsymmetric tetrahedra; the amount of other symmetries they contain; and the continuous symmetry changes in fluctuating, vibrating, and rotating tetrahedra. The tool we developed bears on any physical or chemical process and property which is either governed by symmetry considerations or which is describable in terms of changes in symmetry.