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Chapter 5

Continuous Chirality


 

The concepts behind the Continuous Symmetry Measure can be easily adapted to a Continuous Chirality Measurement. Again the measure is based on the minimal distances that vertices must be moved to attain a specific symmetry (in many cases s reflection). The technique also allows the identification of the most chiral object in a set (e.g. ethane rotamer or simple tetrahedron) as well as monitoring chirality changes along a racimisation pathway and other dynamic systems.(21)

 

5.1 Continuous Chirality Measurement (CCM)

Zabrodsky and Avnir(21) define chirality as the lack of certain symmetries in a molecule’s structure. By definition, the CSM approach measures this lack of symmetry relative to a specified symmetry group or element. Only certain groups will be relevant to chirality considerations. By choosing the correct ones we can calculate a single, quantitative value for the CCM. A single value is preferable since we want a simplified measure of chirality common to all chiral objects. We are only interested in symmetries relevant to these objects’ chirality and there is no need to build complete symmetry profiles. As before, the measure will range from 0 for achiral molecules to 1 for ‘perfectly chiral’ ones (though the latter is of limited relevance).

Letting A’ be the nearest achiral molecule rather than the nearest molecule with a specific symmetry; we can define CCM as the minimum distance we must move our molecule’s vertices in order to reach this achiral state. Therefore we should calculate the CSMs relative to all the chirality-relevant symmetry groups and then choose the least value of these as our CCM. But what symmetry groups are relevant to chirality?

As mentioned at the end of chapter one, a molecule is achiral if it possesses either a reflection plane (s = S1), a centre of inversion (i = S2) or higher order improper rotation axes S2n. By screening over all symmetry groups that contain these elements, we can find our minimum measure and hence our CCM. In practice, we find that most achiral molecules have one or more s or i symmetry elements, and groups containing these are the most important to screen against. The number of S2n containing groups that we also take into account will depend on how accurate we want our final measure to be.

The procedure for calculating the measures is identical to that given before (folding, averaging, unfolding and distance minimising) and this won’t be repeated. Notice that the chirality measure can be considered as a subset of the general symmetry profile and equal to the minimum CSM relative to the s , i or S2n containing groups.

5.2 Conclusion

There is a huge range of possible approaches for measuring symmetry and chirality and many different proposals have been put forward in the literature. This review has covered a few of these measures that the author feels have a great deal of potential but there are others that should be mentioned.

Buda and Mislow’s Hausdorff Chirality Measure (17) is based on the concept of distance between sets. Like CSM, it considers objects as a collection of vertices at known co-ordinates and with known parameters. By describing a molecule as an n-dimensional array of parameters (as often used in computer models), it is possible to algebraically analyse the ‘distance’ between the arrays for two enantiomers. This can then be used as a measure of molecular chirality. A rather similar if mathematically more complex method, is to measure the distance between ‘fuzzy sets’ – for example, the indistinct boundaries of molecular orbitals. Again, this is appropriate for computational modelling and specifically in the area of quantum chemistry.(9,29,30)

Pinsky and Avnir (24) extend the CSM technique to encompass classical polyhedral, of particular relevance to metal complexes and cluster chemistry. Particularly interesting is the analysis of the Jahn-Teller distortion problem mentioned in chapter one. A slightly different CSM to Zabrodsky et al. has been proposed by Stefan Grimme. (25) This uses electronic wavefunctions as the symmetry property and measures their deviation from specific symmetry elements. Interestingly, this CSM uses the reverse method to Zabrodsky et al. Rather than measure the distance between A to A’ where A’ is the closest shape with the required element e , Grimme measures between A and e A. In other words between the shape and the shape transformed by e . While faster, this approach loses the advantage of generality as it is no longer truly a measure relative to a symmetry element and not a shape.

Taking an overview of recent work, Zabrodsky et al.’s CSM and CCM are generally the most common measures used. A wide range of different topics have been investigated, these include; analysis of hyperpolarizability in benzene (26), chiral interconversion pathways of water-trimer enantiomers (27) , shape recognition by Trypsin (31) and Stone-Wales rearrangements in Fullerenes. (32) This range is a reflection of the flexibility and general applicability of the CSM method.

Of the three main approaches reviewed here, property overlap is probably the most useful for general studies. Overlaps are easy and fast to calculate and the technique can be extended to a wide range of properties depending on the subject of investigation. The downside of the technique is that it is very specific to a particular case or measurement technique. Overlap measures can’t be meaningfully compared or contrasted, nor can for instance, chiral measures be compared between chemical series (since the measure to relative to another shape/molecule rather than a common standard). As a result, properly overlap measures are excellent for self contained investigations but are of limited use as a general symmetry measure. Similar comments apply for the rotational chirality measure since it is designed specifically for 2D systems.

Continuous measures are computationally more complex but are more flexible and have the advantage of being entirely general for any object or shape. Hence the CSM could easily be defined as a ‘standard’ measure of symmetry. The downside of this approach is that it is entirely concerned with molecular shape modelled as a array of vertices. This is obviously rather limiting and future work will hopefully see the extension of this measure to encompass a wider range of properties. At the moment there is no obvious ‘best’ symmetry or chirality measure but this is a rapidly evolving field and one that is well worth keeping an eye on.

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