Geometrical chirality

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. 

In his Nobel Lecture, Vladimir Prelog defined geometrical chirality by conjoining the two definitions17 ... 

With few exceptions, all nonoriented chemical links reported in the literature are topologically achiral. One of the rare exceptions is the two-component 4-crossing link 4 (Figure 31).114 The molecule in the depicted conformation, and indeed in any imaginable conformation, is geometrically chiral. It is therefore reasonable to conjecture that it is also topologically chiral. Proof that 4 is indeed topologically chiral was provided after the development of suitable polynomials.115... 

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. 

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. 

Fig. 10.1 Geometrically chiral but not stereogenic centers. The ligand p represents the mirror image of p in isolation and the ligand a (or b) is achiral when detached...

Scheme 4-27. Synthesis of the catenane with rigid geometrical chirality.

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