Symmetry and Chirality Continuous Measures

The property of an object which cannot be superimposed with its mirror image by translation or rotation. See Symmetry and Chirality Continuous Measures. 

See Symmetric Group and Symmetry and Chirality Continuous Measures Symmetry in Chemistry and Symmetry in Hartree-Fock Theory. 

Computer Graphics and Molecular Modeling Conformational Analysis 1 Electrostatic Catalysis Molecular Docking and Structure-based Design Protein Structure Prediction in ID, 2D, and 3D Structural Chemistry Application of Mathematics Symmetry and Chirality Continuous Measures Symmetry in Chemistry. 

Continuous chirality measure is then defined as follows given a configuration of points P = I, its chirality content is determined by finding the nearest configuration of points Pi - 2 which has an improper element of symmetry, and by calculating the distance between the two sets using Eq. (26). 

Oin search for a discontinuous physical property of a single molecide is discomaged, or at least diverted, by the development of the concept of continuous symmetry measures [127]. Some time ago, Zabrodsky et al. advocated that it was more natural to analyse symmetry properties in terms of a continuous scale rather than in terms of yes and no . In order to justify their view they invoked examples such as symmetry distortions due to vibrations, changes in the aUowedness of electronic transitions due to deviations from an ideal symmetry and so forth. In a similar vein and around the same time Buda et aL [128] defined a degree of chirality as a continuous function that is zero if, and only if, the object is achiral Kanis et al. [129] were motivated to use continuous symmetry measures in their study of the hyperpolarisabihty of non-Unear optic materials in order to provide correlations between structure and physical or chemical properties. Their continuous symmetry metric provided a quantitative characterisation of the extent to which the inversion centre is present for a particular geometry. It then allowed the comparison of computed hyperpolarisabUities with the extent to which the inversion centre remains. 

Starting with the semiempirical approach of Kauzmann et al. (16), Ruch and Schonhofer developed a theory of chirality functions (17,18). These amount to polynomials over a set of variables that correspond to the identity of substituents at various substitution positions on a particular achiral parent molecule. The values of the variables can be adjusted so that the polynomial evaluates to a good fit to the experimentally measured molar rotations of a homologous series of compounds (2). Thus, properties 1 and 2 are satisfied, but the variables are qualitatively distinct for the same substituent at different positions or different substituents at the same positions, violating property 3. Furthermore, there is a different polynomial for each symmetry class of base molecule. Thus, chirality functions are not continuous functions of atom properties and conformation (property 4). 

Two general classes of chirality measures have been recognized in the first, the degree of chirality expresses the extent to which a chiral object differs from an achiral reference object, while in the second it expresses the extent to which two enantio-morphs differ from each other [Buda et al., 1992]. The continuous chirality measure (CCM) recently proposed [Zabrodsky and Avnir, 1995] is an example of chirality measure belonging to the first class and is based on the general definition of continuous symmetry measure defined as ... 

The continuous chirality measure is an example of first class chirality measure based on the general definition of continuous symmetry measure it is defined as [Zabrodsky and Avnir, 1995]... 

Zabrodsky H and Avnir D (1995) Continuous symmetry measures. 4. Chirality./owrwu/ of the American Chemical Society 117 462-473. 

---