Continuous symmetry measures

One example of a quantitative measure of molecular chirality is the continuous chirality measure (CCM) [39, 40]. It was developed in the broader context of continuous symmetry measures. A chital object can be defined as an object that lacks improper elements of symmetry (mirror plane, center of inversion, or improper rotation axes). The farther it is from a situation in which it would have an improper element of symmetry, the higher its continuous chirality measure. 

The symmetry of five-coordinate zinc complexes has been analyzed using continuous symmetry measures. This approach and the computational tool allow the degree of polyhedricity to be evaluated quantitatively. The methodology identifies the minimal distance of a given structure to a desired general shape with the same number of vertices.27... 

Salomon, Y. and Avnir, D. (1999) Continuous symmetry measures Finding the closest C2-symmetric object or closest reflection-symmetric object using unit quaternions. J. Comput. Chem. 20, 772-780. 

These are but few examples which illustrate the need for a continuous scale of symmetry. A general approach which answers this need was layed out in refs. [7] and [2], Here we summarize its main features, show how the above mentioned examples are approached, and extend our theory of continuous symmetry measures (CSM) to three new applications ... 

We define the continuous symmetry measure (CSM) as a quantifier of the minimum effort required to turn a given shape into a symmetric shape. This effort is measured by the sum of the square distances each point is moved from its location in the original shape to its location in the symmetric shape. Note that no a priori symmetric reference shape is assumed. 

Following the definition of the continuous symmetry measure (CSM) in Section II, the CSM values are limited to the range 0. .. 1 (where 1 is the normalization scale). The lower bound of the CSM is obvious from the fact that the average of the square of the distances moved by the object points, is necessarily non-negative. The upper bound of the average is limited to 1 since the object is previously... 

XIII. TWO GENERALIZATIONS OF THE ZPA FOLDING-UNFOLDING CONTINUOUS SYMMETRY MEASURES FOR CONTINUA USING THE SNDSM METRIC AND THE HAUSDORFF METRIC... 

TTie dissimilarity of A and Af i p p provides a symmetry deficiency measure analogous to the ZPA continuous symmetry measure of discrete point sets. As a dissimilarity measure, both the SNDSM metric and the Hausdorff metric, or any other dissimilarity measure suitable for continua, are applicable. 

Using the Hausdorff metric for the dissimilarity of sets A and Af f p p, one obtains another generalization of the ZPA continuous symmetry measure of discrete point sets to crisp continuum sets, leading to a new symmetry deficiency measure /Jzpa( > that is a valid measure of... 

XIV. FUZZY SET GENERALIZATIONS OF ZPA FOLDING-UNFOLDING CONTINUOUS SYMMETRY MEASURES BASED ON THE FUZZY FSNDSM METRIC AND FUZZY HAUSDORFF-TYPE METRICS... 

Fuzzy dissimilarity measures, such as the fuzzy FSNDSM metric fs(/l, B), and any one of the fuzzy Hausdorff-type dissimilarity metrics, for example, f(A,B), can be applied to the pair of set A and the folded-unfolded set Aff p p. These fuzzy dissimilarity measures generate fuzzy symmetry deficiency measures analogous to the ZPA continuous symmetry measure of discrete point sets. 

Using any one of the versions of the fuzzy Hausdorff-type metrics for the dissimilarity of sets A and An f/fp, for example, the commitment weighted fuzzy Hausdorff-type dissimilarity metric f(A,B), one obtains another generalization of the ZPA continuous symmetry measure of discrete point sets to crisp or fuzzy sets. The corresponding symmetry deficiency measure f(A,A(f ( p p) provides a measure for the symmetry aspect R for crisp or fuzzy set A, with reference to the given positioning P of R with respect to A and to the choice of the associated partitioning of A. 

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]... 

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. 

CCM = continuous chirality measure CSM = continuous symmetry measure. 

This continuous symmetry measure approach to the problem of non-ideal symmetry has been guided by three principles ... 

The proposed continuous symmetry measure (CSM) method which follows these guidelines is based on the following definition. Given a shape composed of rip points P, (/ = l...rip) and a symmetry group G, the symmetry measure 5(G) is a function of the minimal displacement the points P, of the shape must undergo in order to acquire G symmetry. The CSM method identifies the points P, of the nearest shape having the desired symmetry. Once the nearest P,- values are calculated, a continuous symmetry measure is evaluated as ... 

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

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