Isomer Enumeration Using Cyclic Permutation

Isomer enumeration using cyclic permutation – An example

Kakali Datta

Department of Chemistry, M.U.C. Women’s College, West Bengal

Abstract: Polya’s theorem and the groups of symmetries of the geometrical models representing unsubstituted parent compounds has been used to determine the number of isomers of the compounds. The introduction of point group description of a molecule into Polya’s formulation and cycle indices for the structure enable to calculate structural isomers in various kinds of organic compounds.

Key Words: : Isomer, cyclic permutation, symmetry, cycle index, generating function

1. Introduction A general problem of determining the number of isomers of compounds can be solved by Polya’s[ 1] enumeration technique using drawing and models and group theory. Many methods have been developed[ 2, 3-5] to find the number of derivatives arising either by addition to or by substitution in a parent molecule. The methods described in ref. 3-6 are based on the Polya’s theorem. The isomer permutation representation is obtained for fullerene cages in ref.2. Applications of permutation group theory have been made by other mathematicians[7,8 ] with their own methods. The most effective and general method of calculations is one devised and used by Polya[ 1, 9, 10] for enumeration of isomers. We can formulate Polya’s theorem in terms of more familiar approach in group theoretical method of counting numbers of isomers. In this article we demonstrate the method by applying it to cyclopropane.

Counting Method:

A structural model like an equilateral triangle represents the unsubstituted parent compound of cyclopropane( Fig.1).

The rotation which transform an equilateral triangle into itself forms the point group D 3 . The 2 ΄ ˝ corresponding symmetry operations are E, C 3, C 3 , C 2, C 2 ,C 2 which group into three classes. There being three positions in the parent compound at which substitution can occur. With each element of the point group D 3, we can associate a permutation operation forming a

[Article History: Received on 17.03.2017, Accepted on 26.07.2017]

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Isomer enumeration using cyclic permutation – An example

Author: K.Datta permutation group. Such permutation operation corresponding to the symmetry operation E and C 2 are shown below.

Fig. 1

Symmetry operation Permutation

1 2 3  1 2 3 E   =       1 2 3  1 2 3

1 2 3  1 2 3 C2   =     1 3 2  1 3 2

The number of permutations can be written as the product of permutations that involve separate element which are called cycles; in the above example E consists of 3 cycles of length one and C 2 of one cycle of length one and other of length 2. It can be seen that elements belonging to the same class consist of similar sets of cycles.

According to Polya’s theorem the cycle index of a group D 3 is given by[ 1]

1  3 1 1 1  Z(D 3) =  f + 2 f +3 f f  6  1 3 1 2 

The cycle index of the permutation group is the polynomial in the variables f 1, f 2,f 3 .

Polya’s method gives generating function called figure counting series in general denoted by

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2 C(x) = C 0 + C 1x + C 2 x + ...... from which the number of isomers can be counted from the coefficient of x m , the number of elements of weight m. In the Fig.1 each of the positions can be occupied by H or Cl. If we assign the weights 0 and 1 to H and Cl respectively, C 0 = C 1 = 1 and the figure counting series is 1+ x .

j According to Polya’s theorem the polynomial obtained from Z(D 3) by replacing f j by 1+ x , j = 1, 2, 3 is

1 3 3 2 Z(D 3, 1+ x) = [ 1( + x) + 1(2 + x )+ 1(3 + x 1() + x )] 6

On simplifying the expression in the right hand side we obtain the generating function C(x) for the classes of these functions which correspond to the substitution in a skeletal parent compound:

C(x) = 1+ x + x 2+ x 3

The coefficient of x k is the number of substituted cyclopropane with exactly k chlorine atoms, here k = 1, 2, 3.

According to Polya this result can be displayed in chemically familiar notation:

C3H6 + C 3H5Cl + C 3H4Cl 2 + C 3H3Cl 3

From the expression we obtain total number of four substituted cyclopropane by chlorine atoms.

The diagrams which correspond to these compounds are shown in Fig. 2.

Cl Cl Cl

Fig. 2 C 3H6 C3H5Cl C 3H4Cl 2 C3H3Cl 3

Conclusion

The applicability of Polya’s enumeration theorem is more general and wide to count isomers in various kinds of cyclic and acyclic graphs. The solution of the chemical enumeration is expressed in terms of a polynomial or power series whose coefficients display the solution.

[37] Isomer enumeration using cyclic permutation – An example

Author: K.Datta

References: 1. Plya G., Helv. Chim. Acta, 1935, 19 , 22. 2. Fowler P.W., J.Chem.Soc. Faraday Trans. 1995, 91 , 2241. 3. Balaban A.T., Rev.Roum.Chim. 1986, 31 , 679. 4. Hosoya H., Gendai-Kagaku, 1987, 201 , 38. 5. Balasubramanian K., Chem.Phys.Letters, 1993, 202 , 399. 6. Balasubramanian K., J. Phys. Chem., 1993, 97 , 6990. 7. Lunn A.C. and Seulor, J. Phys. Chem., 1929, 33 , 1027. 8. Marchi L.E., Fernelius W.C. and McReynolds J.P., J.Am.Chem.Soc., 1943, 65 , 329. 9. Polya G., Acta Math., 1937, 68 , 145. 10. Polya G., Krist., 1936, 93, 415.

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