The Sudden Dis Appearance of Critical Points

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. 

Another alternative is making the topology operate on a sequence of fluctuating electron densities, which in the limit become the real electron density. 

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