Chemical enumeration

This illustrates an important distinction in chemical enumeration that between the enumeration of "structural" isomers, in which only the connections between the atoms are considered, and that of stereoisomers, in which the situation of a molecule in space is important, so that as above we can have right- and left-hand forms of a molecule. This distinction will occur, for example, when a carbon atom is bonded to four distinct substituents (it can occur in many other ways). Such a carbon atom is said to be asymmetrical. 

In the realm of chemical enumeration we note Polya s equation (4.4) which gives the generating function for stereoisomers of the alkyl radicals, or equivalently, alcohols — that is, equation (5.2) of this article. His equation (4.3) gives the corresponding result for the structural isomers of these compounds. His equations (4.2) and (4.5) correspond, respectively, to the cases of alcohols without any asymmetric carbon atoms and the number of embeddings in the plane of structural formulae for alcohols in general. The latter problem is not chemically very significant. 

A question which chemical enumerators should not ignore is that of the extent to which their results are realistic in the physical world. Thus in [BlaCSla] it is stated that the number of alkanes (paraffins) with 40 carbon atoms is 62,491,178,805,831. Can we really be sure that all these compounds can exist or could it be that factors not catered for in the enumeration render some of them chemically infeasible In this connection we should note the paper [KleD81], in which it is shown jthat because of such factors the chemical tree enumerations by Polya and others give numbers that are consistently higher than the number of compounds that are in fact chemically possible. This does not detract from the mathematical value of these results it merely shows that care is needed in relating them to problems of real life. 

The power group enumeration theorem has been put to a great number of uses, especially in graph theoretical and chemical enumeration. One simple example will illustrate the value of the theorem. 

It must be admitted that many chemical enumerations that have appeared in the literature are more in the nature of academic exercises than results of practical interest to the chemist. Thus, for example, no chemist really needs to know that the number of alkanes having 60 carbon atoms is 22,158,734,535,770,411,074,184 (see [PerD32]). However, the enumeration of compounds with a given frame can produce results of practical importance. 

The use of Polya s Theorem in a specialized context such as the above, has led to the extension of the theorem along certain useful lines. One such derivation pertains to the situation where the boxes are not all filled from the same store of figures. More specifically, the boxes are partitioned into a number of subsets, and there is a store of figures peculiar to each subset. To make sense of this we must assume that no two boxes in different subsets are in the same orbit of the group in question. A simple extension of Polya s Theorem enables us to tackle problems of this type. Instead of the cycle index being a function of a single family of variables, the 5j, we have other families of variables, one for each subset. An example from chemical enumeration will make this clear. 

The problem that we noted above with clusters appears also in chemical enumeration when we consider compounds formed by attaching radicals which may be chiral or achiral to a frame which is achiral. In this case, too, Polya s Theorem cannot be used, but the problem can be solved by the appropriate use of Burnside s Lemma. It is also amenable to the methods of Redfield, as shown in [DavRSl] and [LloE85]. 

The two specific areas of research in which Polya s Theorem has been most extensively applied are graphical and chemical enumeration, a fact which Polya clearly foresaw in his choice of title. Applications in other fields are far from rare, however, and it is fitting to give a brief account of a few such uses of the theorem. 

ReaR72 Read, R. C. Some recent results in chemical enumeration. 

R-C- Read, Some recent results in chemical enumeration in Graph "Theory and Application Eds- Y- Alavi, D-R- Lick and A-T-White, Springer Verlag, Berlin (1972). 

Chem., 29, 131 (1993). Coronenic Coronoids A Course in Chemical Enumeration. 

I. Novak,/. Chem. Educ., 73, 120 (1996). Chemical Enumeration with Mathematica. 

Early History Isomer Enumeration. - The classic type of chemical enumeration concerns the enumeration of possible molecular structures. Indeed... 

Further Enumerations. - Another, somewhat separate area of chemical enumeration concerns the counting of resonance structures for the purpose of gauging the extent of resonance . Often these structures may have much the flavor of different isomeric structures, though the enumeration is usually taken to... 

Cyvin BN, BrunvoU J, Chen RS, Cyvin SJ (1993) Coronenic Coronoids — A Course in Chemical Enumeration [in] Topological Aspects of Benzenoid Hydrocarbons and Related Structures (Cyvin SJ, Edit). Max—Planck—Institut fur Strahlenchemie, Mulheim a d Ruhr Match 29 131... 

Finally let us mention the approach known as the transfer-matrix method, which is particularly useful for systems that have repeating sub-units, e.g., systems with high symmetry and polymers. Besides applications to benzenoid and non-benzenoid polymers, the transfer-matrix method is suitable for systems with rotational symmetry, such as buckmin-sterfullerene, Ceo- In fact, the first reported value for the number of Kekule valence structures in buck-minsterfullerene Ceo, which is 12 500, was obtained using the transfer-matrix method. The transfer-matrix method has been used for some time in statistical mechanics. " Klein and co-workers ° pioneered the application of the transfer-matrix method for chemical enumerations. 

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