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Boletín de la Sociedad Chilena de Química

versión impresa ISSN 0366-1644

Bol. Soc. Chil. Quím. v.47 n.2 Concepción jun. 2002 

Bol. Soc. Chil. Quím., 47, 105-112 (2002)



Adelio R. Matamala

Facultad de Ciencias Químicas, Grupo QTC, Universidad de Concepción, Casilla 160-C.
Concepción, CHILE.
(Received: May 7, 2001 - Accepted: January 11, 2002)


In this article Pólya's Combinatorial Method and its application to the enumeration of substitutional isomers are presented in a concise and illustrative way. In particular, two polychlorinated organic compounds have been studied: the polychlorinated naphthalenes (PCN) and the polychlorinated biphenyls (PCB). The latter are an instructive example for showing how the method works with molecules that exhibit internal free rotation.
Keywords: Pólya's Theorem, Enumeration Isomers, Diamutamers.


En este artículo se presenta en forma concisa e ilustrativa el Método Combinatorial de Pólya y su aplicación en la enumeración de isómeros substitucionales. En particular, dos compuestos orgánicos policlorados han sido estudiados: los policloronaftalenos (PCN) y los policlorobifenilos (PCB). Estos últimos constiutyen un buen ejemplo para mostrar cómo trabaja el método en moléculas que presentan rotación libre interna.
Palabras Claves: Teorema de Pólya, Enumeración de Isómeros, Diamutameros.


Chemical compounds that exhibit the same molecular formula (atomic composition) but differ in their structural formulas (chemical constitution) are called isomers. In 1830 this term was coined by Berzelius [1] from the Greek roots 'isoV', equal and 'meroV', part. Frequently, the topic of isomerism is introduced early in most undergraduate organic courses. Generally the discussion about it consist of defining isomerism and giving some examples of isomeric structures. But the question about the number of isomers for a given molecular structure is only solved in these courses by using the well-known "draw and count" method. Obviously it is a useful method in the enumeration of simple structures, but for general cases a more rational approach is required.

The problem of formal isomer enumeration was initiated by the noted mathematician Cayley for the case of alkyl radicals in 1874 [2] and by the chemist Körner on the substitutional isomers of benzene [3]. However, the most notable advance in the history of isomer enumeration came in 1936, when the mathematician George Pólya [4] developed his Enumeration Theorem [5-9]. This theorem has continuously been an important tool for chemical combinatorics [10-17]. The main ingredient for isomer enumeration methods based on Pólya's Theorem is the so-called cycle index. This cycle index encodes the relevant information about symmetry and numbers of binding sites of molecules.

In this article Pólya's Combinatorial Method and its application to the enumeration of substitutional isomers are presented in a concise and illustrative way. In particular, two polychlorinated organic compounds have been studied: the polychlorinated naphthalenes (PCN) and the polychlorinated biphenyls (PCB). The latter is an instructive example for showing how the method works with molecules that exhibit internal free rotation. Generally, enumeration applications of Pólya's Combinatorial Method use the permutation groups assigned to the molecule. However point groups [18-19] are more familiar to chemists than those. In this way, point group formalism has been only considered in all calculations in the present article.

The material which follows has been divided into three sections, one dealing with the Pólya's Combinatorial Method, one with two applications of this method, and finally one section containing a summary and conclusions.

Pólya's Combinatorial Method.
The solution of some combinatorial problems is often most conveniently expressed in terms of a polynomial or power series, the so-called generating function, whose coefficients display the solution. An example will make this clear. Suppose we have a box and a set of beads for building a necklace, each bead being either black (b) or white (w). If we wish to put into the box one bead then a generating functions which display the two possible outcomes in a natural fashion is b+w (see Fig. 1), with the obvious meaning of the variables.

Fig. 1

If there two different boxes then the generating function which shows all four ways of arranging at most one bead per box is (see Fig. 2)

Figure 2

In general for n boxes, the solution is given by the following formula:
Where the coefficient

Gives the number of arrangements in which n -k boxes have a black bead and k have a white.

Now, suppose that we wish to build a necklace using five beads, each of it being either black or white. In this case, Fig 3 shows the diagrams corresponding to all different necklaces with five beads.

Figure 3

Apparently, the problem of counting the numbers of necklaces can be regarded as a problem of five boxes each of it can be filled with a black bead or a white bead. However, the polynomial

does not yet the correct generating function for the present problem because boxes are not differently labeled boxes. They are arranged in a circular array and so the coefficients of (3) are too large and must be reduced by considerations that depend on the symmetries of the pentagon.

Certainly, D5h is the point group assigned to a pentagon. But in order to analyze the permutation properties of a pentagon the point group D5 is sufficient. In general, the permutation properties of a n-polygon are determined by the dihedral group Dn of degree n. When the three-dimensional spatial properties of a n-polygon are studied then the group Dnh is required. This is the case for molecules that involve chiral antipodal substituents [15].

The ten symmetries of D5 are:

The action of each symmetry operations on the pentagon 12345 (see Fig. 4)

Figure 4

generate the following permutations:

where the cyclic-permutation notation has been used.(see Ref. [20]).

The cycle index Z(G) for a group G is formally defined by the equation

where |G| is the order of the group G, p is the number of point permuted, the coefficients h are the number of permutations of G consisting of a1 cycles or order one, a2 cycles or order two, and so on. The f's are variables.
Now, returning to our example of the 5-necklace, Z(D5) reads

The Pólya's Theorem establishes that the generating function is obtained by substituting
the variables f's by the figure counting series. In our case, it is given by

Therefore, replacing Eq.(6) in Eq.(5), we obtain

And expanding it, we arrive to the correct generating function:

with the obvious meaning of the symbols.

Applications to Substitutional Isomerism
Substitutional isomers are compounds that exhibit the same central skeleton but differ in the arrangement of ligands. These kind of chemical compounds have been named diamutamers by IUPAC [21]. In this section Pólya's method has been applied in the enumeration of diamutamers. In particular, two examples have been studied in order to show how the method works.

Case 1: Polychlorinated naphthalenes
The first step in the polychlorinated naphthalene enumeration consists of assigning the point group to the naphthalene rings. This molecule belongs to D2h point group (see Fig. 5),but due to the presence of achiral ligands (chloride groups), the group D2 is sufficient in order to describe the permutation properties of the naphthalene rings. The point group D2 contains four symmetry elements: E, C2(x), C2(y) and C2(z).

Figure 5

The next step involves the determination of the cycle index to the group D2 Upon application of each symmetry element in D2; the following permutations are obtained:

Therefore, the cycle index is

The next step consists of defining the figure counting series. In the present case, in accordance to Eq.(5), the f's terms in Eq.(7) are given by

Where the power n of x means that the naphthalene ring contains n chloride ligands. In particular, 1 means zero chloride ligands. Now, replacing the figure counting into Eq.(7),

and expanding it, the generating function of the polychlorinated naphthalenes reads:

Using the "draw and count" method it is possible to check the present result. A list containing all of 75 polychlorinated naphthalene isomers may be consulted in reference [22].

Case 2: Polychlorinated biphenyls
In this case a coplanar configuration for the two phenyl rings (see figure 6), the biphenyl molecule belongs to the same point group that naphthalene molecule.

Figure 6

Repeating the same procedure used for the naphthalene case, the following permutations are obtained:

However, the resulting cycle index

Cannot reproduce the correct generating function. The reason for it is obvious, the free rotation about the C-C bond linking the two phenyl rings must be considered. It contributes with new symmetry elements to the problem. In fact, the free rotation of the (12345) phenyl ring about the C-C bond produces the following permutations

Therefore, using the permutation product, the new permutation elements for the biphenyl molecule read:

These new permutations produce the cycle index

Replacing the figure counting series

In Eq.(11), and expanding the resulting expression, the generating function for the polychlorinated biphenyls is given by the following polynomial:

This result gives the correct number of each PCB isomers. A list containing the 209 polychlorinated biphenyl compounds may be consulted in Ref. [23].


In general, the application of Pólya's Combinatorial Method to the enumeration of diamutamers involves the following steps:

Step 1: The identification of the symmetry point group assigned to the molecules.

Step 2: The derivation of a cycle index that reflects the parent compound.

Step 3: The derivation of a figure counting series that reflects the number of different atoms or groups that can occupy a substitution site.

Site 4: The substitution of the figure counting series (after a minor transformation) into the cycle index.

Step 5: The algebraic expansion of the resultant expression to obtain the generating function.
Pólya's Combinatorial Method is an efficient and powerful tool for the enumeration of the substitutional isomers in chemistry. The apparently tedious expansion of polynomials is easily carried out in software such as MATHEMATICA or MAPLE.


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