Symmetry in C$_{60}$

Abstract

The C$_{60}$ molecule is characterized by the peculiar structure, that of soccer-ball, composed of 60 atoms, 90 bonds, and 12 pentagons, and 60 hexagons. The group theoretical symmetry is of the icosahedral type, which contains 120 symmetric operations and 10 irreducible representations, according to which, the eigenstates are classified and the origins of degeneracies in energy spectrum are explained. We begin from the construction of the symmetric operations by means of permutations of atoms, generate the icosahedral group by GAP (The Gap Group 2017), compute the irreducible representations, and compose the projection operators. We execute the electronic structure calculations of the molecule, and analyze the wave-functions from the viewpoint of their irreducible representation. Furthermore, some concepts of symmetries (to which physicists pays little notice) are presented also by the analysis of the computer algebra. Throughout the study, it is necessary that whole subgroups of icosahedral symmetry should be extracted and classified by equivalence, and we will see that the utilization of the computer algebra is effective for this purpose.