Continuous symmetry measures

Abstract

We advance the notion that for many realistic issues involving symmetry in chemistry, it is more natural to analyze symmetry properties in terms of a continuous scale rather than in terms of yes or no. Justification of that approach is dealt with in some detail using examples such as: symmetry distortions due to vibrations; changes in the allowedness of electronic transitions due to deviations from an ideal symmetry; continuous changes in environmental symmetry with reference to crystal and ligand field effects; non-ideal symmetry in concerted reactions; symmetry issues of polymers and large random objects. A versatile, simple tool is developed as a continuous symmetry measure. Its main property is the ability to quantify the distance of a given (distorted molecular) shape from any chosen element of symmetry. The generality of this symmetry measure allows one to compare the symmetry distance of several objects relative to a single symmetry element and to compare the symmetry distance of a single object relative to various symmetry elements. The continuous symmetry approach is presented in detail for the case of cyclic molecules, first in a practical way and then with a rigorous mathematical analysis. The versatility of the approah is then further demonstrated with alkane conformations, with a vibrating ABA water-like molecule, and with a three-dimensional analysis of the symmetry of a (2 3 21 reaction in which the double bonds are not ideally aligned.