Geometrical chirality measurement

In stark contrast to the numerous functions that are available to measure geometrical chirality, no measure has yet been reported for the quantification of topological chirality. In analogy to geometrical chirality measures, topological chirality measures %(K) must satisfy two minimal conditions They can be equal to zero if and only if the knot or link is achiral, and they have to have the same absolute value for two topological enantiomorphs. 

Of course, it is possible to contemplate experiments that examine photoionization of oriented chiral molecules. An expression has been given for the angle integrated (total) ionization cross-section in such circumstances [48] and CDAD-type measurements have been reported on adsorbed chiral molecules [49, 50], but the interplay of natural and geometric chirality in angle-resolved dichroism measurements remains very much a topic for future investigation. 

Consider, for example, the question What is the shape of the most chiral right triangle in R2 The answer to this question depends on the particular function that is chosen as the chirality measure According to measures based on geometric chirality products,151 on symmetry coordinates,152 on common volumes,153 and on Hausdorff distances,131 154 the smallest internal angles are 18.8°, 30.0°, 37.5°, and 35.2°, respectively.147 The function x(0) = sin 40 mentioned earlier achieves its maximum [ (0) = 1] at 0 = 22.5°. Because there are as many answers to this question as there are functions that can measure the chirality of a triangle, and because there is, in principle, no limit to the number of such functions, the question remains, in a deep sense, unanswerable. 

The same uncertainty attaches to functions designed to measure the chirality of geometrical objects in three dimensions. As a simple example, we saw that the degree of chirality of a helix can be expressed by a function, %(Q) = sin 20, that achieves its maximum value at 0 = 45°. Alternatively, the degree of chirality of a helix can be given by the volume of a cylinder on the surface of which is inscribed... 

Chirality is an important topic in chemistry and biochemistry, due to the natural occurrence of chiral molecules in living organisms. In circular dichroism (CD) one measures the differential absorption of left- and right-handed circularly polarized light, which for chiral species are different. Therefore, CD has turned out to be a powerful tool which provides information on the electronic and geometric structure of chiral molecules. Since most CD spectra are measured in solution we extended our DRF/TDDFT method to also calculate such properties. As a first example we studied... 

We saw that it is natural to attach a quantitative meaning to the chirality of molecular structures and that it is possible to rank the chirality of closely related structures. There is no reason not to apply measures of chirality as welt to geometrical objects, including geometrical models of... 

The Hausdorff chirality measure is a chirality measure of the second class [Buda and Mislow, 1992]. Let Q and Q denote two enantiomorphous, nonempty, and bounded sets of points defined in the geometrical space (x,y,z). Let d(q,q ) denotes the distance between two points G Q and qi G Q. Then, the Hausdorff distance h between sets Q and Q is defined as... 

The chirality of an ordered quadruple of atoms numbered 1,2,3,4 is measured in terms of their (x,y,z) Cartesian coordinates, adopting some geometrical constraints, by the sign of the following determinant [Ursu and Diudea, 2005 Ursu, Diudea et al, 2006] ... 

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