Counting permutational isomers

The first step is to evaluate the number of permutational isomers, the construction of permutational isomers will be described later. The trivial case is P = R, where the point group does not contain any improper rotation, so that the symmetry operations of the skeleton do not change the substituents, hr this case the skeleton is chiral and we just need to count symmetry classes of distributions S e Y with respect to the action pX. 

11 Remark (The case of a chiral skeleton) If P = R, the skeleton of the molecule is chiral and so the total number of permutational isomers is 

However, when P R, we have to use de Bruijn s approach described above, obtaining 

12 Remark (The case of an achiral skeleton) Now we assume R P, i.e. an achiral skeleton. We indicate the set of substitutable positions with X mid the set of substituents with Y = Y f,iiY. The corresponding actions of the point group are pX and pT. They define induced actions of the diagonal subgroups A(P x P) and A(R x R) on the set Y of distributions, with their sets of orbits 

contains, for each substituent y also the mirror image y, while consists of achiral substituents. First we count the numbers of orbits using the equations derived above  

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We are now in a position to define unlabeled graphs in terms of group actions. For this purpose, we introduce an important action obtained from a given action gX. G. Polya introduced this approach in the seminal paper [234] ([235] contains an English translation). His aim was to count permutational isomers which means the essentially different distributions of admissible substituents over a molecular skeleton, where essentially different means with respect to the symmetry group of the skeleton. We shall describe this in all detail in Chapter 3, where we show that the same approach allows the construction of corresponding molecular graphs, but this needs further notions. 

After counting permutational isomers, we now need to construct them. If all the substituents are achiral or if the skeleton is chiral, then we can use the same methods that we used for the construction of unlabeled m-multigraphs in Example 1.39. These methods were based on the Fundamental Lemma 1.37, which gave a bijection between a set of orbits and a set of double cosets, so that a transversal of the set of orbits can be obtained fi"om a transversal of the set of double cosets. Let us briefly recall the basic facts in the general form, i.e. for general finite actions qX that were used in Remark 1.38. 

Counting the isomers arising by addition to, or substitution in, a basic framework is a mathematical problem with many practical applications in chemistry. In classical organic chemistry, for example, the number of derivatives of a compound was often cited as proof or disproof of structure. Point group theory that uses concepts familiar to most chemists and is easy to apply when the number of addends/substituents is small provides a unified method for deciding, for example, the number of dihydrides C70H2 of fullerene C70, or the number of trihalo-derivatives C2oHi7FClBr of dodecahedrane. All that is needed to determine such matters is the availability of the permutation character. Ter, of the atoms in the parent molecule. 

To count the isomers with 2 substituents of one kind and 1 of another, consider first all possible placings of X2 and then add Y anywhere to each of them, excluding the case where Y falls on an X site. The permutation representation F(X2Y) follows as... 

Now consider the symmetry point group G (or, more precisely, the framework group ) of the above ML coordination compound. This group has IGI operations of which lf l are proper rotations so that IGI/I/ I = 2if the compound is achiral and IGI/I I = 1 if the compound is chiral (i.e., has no improper rotations). The n distinct permutations of the n sites in the coordination compound or cluster are divided into nM R right cosets which represent the permutational isomers since the permutations corresponding to the IWI proper rotations of a given isomer do not change the isomer but merely rotate it in space. This leads naturally to the concept of isomer count, I, namely,... 

The degree of a vertex in the topological representation corresponds to the number of distinet permutational isomers that can be generated in a single isomerization from the isomer represented by the vertex in question this number 6 is called the connectivity. For topologjeally distinct polyhedra depicted in the same topological representation, the isomer counts I and I and connectivities 6 and 6 must satisfy the so-called closure condition, i.e. ... 

In this chapter we enter the geometrical level. A particularly interesting geometrical aspect of molecules is chirality, which requires the description of further methods, aspects and difficulties concerning 3D placements of molecules in space. We describe the enumeration of permutational isomers in detail, extending the description of Polya s methods for counting multigraphs. Constructive aspects are mentioned, existence problems are discussed emd a method for the computation of isomer numbers is demonstrated. 

We want to apply this to the enumeration of permutational isomers, using the same multiplicative weight w that was introduced when we counted symmetry classes of mappings ... 

J.E. Leonard, Studies in Isomerism Permutations, Point Group Symmetries, and Isomer Counting, Ph.D. Thesis, California Inst. Technol., California 1971. 

Now that we have determined the permutation group we can count stereoisomers using the formulas obtained with structural isomers but by replacing the group S3 by C3. For instance, from Eq. [10], the counting series for alkyl groups becomes... 

The application of P6lya s theorem can be illustrated by its use for the counting of substituted benzene isomers. The cycle index (equation S) of the planar hexagonal skeleton can be written as follows using the permutation group ... 

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