Enumeration of Sigle Point Derivatives of Fullerene

Abstract This paper reports an analysis of the symmetry of (C60-Ih)[5, 6]Fullerene, so-called 'buchminsterfullerene' or 'fullerene' in short and the enumeration of the compounds derived from a fullerene such as heterofullerene, hydrofullerene, hydoxyfullerane and so on. A fullerene has Ih symmetry so that it has thirty-one axes of rotation and fifteen planes of reflection. It follows that a fullerene has the identity operation, fifty-nine rotation symmetry operations, fifteen reflection symmetry operations and forty-five rotoreflection symmetry operations. For each operation, we represent it as a permutation on positions of sixty carbon atoms, decompose the permutation to a set of cycles and have a cycle index. These operations form a permutation group as well as the identity operation and the rotation symmetry operations form a (sub)group. The former group corresponds to stereoisomers and the latter one corresponds to structural isomers. By Pólya-Redfield theorem, we have a generating function for each group. By means of the generating functions, we finally have the number of the structural and stereoisomers of fullerene derivatives.