Isomer counting using point group symmetry

5 Isomer Counting using Point Group Symmetry 

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. 

Ter is, in general, reducible and exhibits traces, x(R) equal to the number of members of the set that are left unshifted by the operations R. r has been tabulated for all point-set orbits of the common groups and is available from the GT calculator orbit-by-orbit or from inspection of the tables in Chapter 3. In what follows, the direct sum irreducible is written 

For isomer counting, the problem to be solved is to determine the number of distinct isomers corresponding to decoration of atoms in a parent molecule. Substitution of an atom by another isotope of the same element or by a different chemical species, addition of a structureless ligand or functional group oriented so as to preserve the local site symmetry, all are to be treated as aspects of the same decoration process. 

The number of distinct isomers, n(X2), is the number of copies of To in T(X2) and it follows from the definition of in reducible form that this is 

---