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DOI: Fujita and first author. The full non-rigid group of hexamethylethane is the semi-direct product of the direct products of six copies of the cyclic group Z3 by the dihedral group of order 12 see, Asian J.
In this paper, we continue our study on finite groups see Int. Physics, Group Theory, and Nonlinear Optics , 17 and all the integer-valued characters of the above molecule are successfully derived. Introduction An object is called symmetrical if some movement or operation leaves the object in a position indistinguishable from its original position. Conflicts of Interest The authors declare no conflicts of interest.
PÓLYA'S COMBINATORIAL METHOD AND THE ISOMER ENUMERATION PROBLEM
When the three-dimensional spatial properties of a n-polygon are studied then the group D nh 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 see Fig. 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.
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In our case, it is given by Therefore, replacing Eq. Applications to Substitutional Isomerism Substitutional isomers are compounds that exhibit the same central skeleton but differ in the arrangement of ligands.
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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 D 2h point group see Fig. Figure 5 The next step involves the determination of the cycle index to the group D 2 Upon application of each symmetry element in D 2 ; 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. In particular, 1 means zero chloride ligands. Now, replacing the figure counting into Eq.
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.
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It contributes with new symmetry elements to the problem. In fact, the free rotation of the 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. A list containing the polychlorinated biphenyl compounds may be consulted in Ref.
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.