Continuous Symmetry Measures. 4. Chirality

We extend the treatment of symmetry as a continuous molecular structural property (J. Am. Chem. Soc. 1993, 115, 8278) to chirality. Rather than labeling objects as being either chiral or achiral, we provide an exact quantitative measure of this property, which allows one to distinguish (chiral) molecules from each other by their degree of shape chirality. The continuous scale is based on the minimal distances that the vertices of a shape must move in order to attain the nearest achiral symmetry point group (in most cases, C_s, reflection symmetry). A detailed description of the methodology and the practical implementation of the continuous chirality measure (CCM) are given. Its generality and versatility are then demonstrated on a wide variety of chirality related issues and in various chirality measurements. These include the identification of the most chiral objects (the most chiral ethane rotamer, the most chiral tetrahedron, etc.), the chirality evaluation of equicontour representations of molecular orbitals, the calculation of the continuous changes in chirality along racemization pathways (including an all-chiral racemization pathway), the evaluation of chirality of structures with uncertain point locations, the extension of the CCM to diastereomerism (with a comment on prochirality and other stereochemical identifiers), the measurement of the chirality of various phosphates, a fullerene, helicenes, a knot, a Möbius strip, a catenane, and a large random object (a diffusion-limited aggregate), and the calculation of dynamic continuous changes in chirality during fluxional (Walden-type) inversion and in rotating ethane (with a comment on continuous chirality changes along concerted reaction pathways).