Stereoisomer enumeration

Rather similar was the paper [PolG36a] which also derives asymptotic formulae for the number of several kinds of chemical compounds, for example the alcohols and benzene and naphthalene derivatives. Unlike the paper previously mentioned, this one gives proofs of the recursion formulae from which the asymptotic results are derived. A third paper on this topic [PolG36] covers the same sort of ground but ranges more broadly over the chemical compounds. Derivatives of anthracene, pyrene, phenanthrene, and thiophene are considered as well as primary, secondary, and tertiary alcohols, esters, and ketones. In this paper Polya addresses the question of enumerating stereoisomers -- a topic to which we shall return later. 

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. 

Polya s Theorem clearly showed the way to the general enumeration of all acyclic hydrocarbons, irrespective of how many double or triple bonds they might have but it was to be 35 years before this enumeration was carried out. In two papers [ReaR72,76] I obtained the solution to this general problem in both the structural isomer and stereoisomer cases, as generating functions in three variables. Of these variables, x marks the number of carbon atoms, y the number of double bonds, and z the number of triple bonds. The de- rivation of these generating functions was Polya theory all the way — a succession of applications of Polya s Theorem with occasional use of Otter s result. The derivation was really rather tedious, but the generating functions, once obtained, can be used to compute the... 

NouJ79 Nourse, J. G. The configuration symmetry group and its application to stereoisomer generation, specification and enumeration. J. Am. Chem. Soc. lOl (1979) 1210-1216. ObeW67 Oberschelp, W. Kombinatorische Anzahlbestimmungen in Relationen. Math. Ann. 174 (1967) 53-58. 

The dissection of a molecular model into those components that are deemed to be essential for the understanding of the stereochemistry of the whole may be termed factorization (9). The first and most important step toward this goal was taken by van t Hoff and Le Bel when they introduced the concept of the asymmetric carbon atom (10a, 1 la) and discussed the achiral stereoisomerism of the olefins (10b,lib). We need such factorization not only for the enumeration and description of possible stereoisomers, important as these objectives are, but also, as we have seen, for the understanding of stereoselective reactions. More subtle differences also giving rise to differences in reactivity with chiral reagents, but referable to products of a different factorization, will be taken up in Sect. IX. 

Expanding on Nourse s technique, Wieland proposed an enumeration algorithm of stereoisomers in which the valence of the stereocenters can be larger than four. 

II G(/Gx) Type III — G(/Gc) Type IV — G(/G) and Type V — G(/Gy). Note that each of the symbols represents a coset representatitMi corresponding to an equivalence class (Fujita 1991b). These equivalence classes have been used in itemized enumeration of quadruplets of f 5-stereoisomers under the action of... 

---