Continuous symmetry measures

We advance the notion that for many realistic issues involving symmetry in chemistry, it is more natural to analyse symmetry properties in terms of a continuous scale rather than in terms of yes or no. Justification of that approach is dealt with at 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 distan...