Many of the special properties that fullerenes exhibit are directly related to the very high symmetry of the C60 molecule, where the 60 equivalent carbon atoms are at the vertices of a truncated icosahedron. The regular truncated icosahedron is obtained from the regular icosahedron by passing planes normal to each of the 6 fivefold axes passing through the center of the icosahedron so that the edges of the pentagonal faces thus formed are equal in length to the edges of the hexagonal faces. Figure 1.1(a) shows this soccer-ball configuration for Cm as thought to have been constructed by Leonardo da Vinci in about 1500 [4.1,2], and Fig. 1.1(b) shows the location of the carbon atoms at the vertices of the truncated icosahedron. The first application of the icosahedral group to molecules was by Tisza in 1933 [4.3], In this chapter the group theory for the icosahedron is reviewed, and mathematical tables are given for a simple application of the icosahedral group symmetry to the vibrational and electronic states of the icosahedral fullerenes. The effect of lowering the icosahedral symmetry is discussed in terms of the vibrational and electronic states. Symmetry considerations related to the isotopic abundances of the 12C and 13C nuclei are also discussed. The space group symmetries appropriate to several crystalline phases of C60 are reviewed in §7.1.2. The symmetry properties of symmor-phic and nonsymmorphic carbon nanotubes are discussed in §19.4.2 and §19.4.3.

4.1. Icosahedral Symmetry Operations

The truncated icosahedron (see Fig. 4.1) has 12 pentagonal faces, 20 hexagonal faces, 60 vertices, pad 90 edges. The 120 symmetry operations

for the icosahedral point group are listed in the character table (Table 4.1) where they are grouped into 10 classes. These classes are the identity operator, which is in a class by itself, the 12 primary fivefold rotations (12 C5 and 12 C52) going through the centers of the pentagonal faces, the 20 secondary threefold rotations going through the centers of the 20 hexagonal faces, and the 30 secondary twofold rotations going through the 30 edges joining two adjacent hexagons. Each of these symmetry operations is compounded with the inversion operation. Also listed in the character table are the 10 irreducible representations of the point group Ih.

The C60 molecule has carbon atoms at the 60 equivalent vertices of a truncated icosahedron for which the lengths of the pentagonal edges are slightly longer (a5 = 1.46 A) than the bond lengths shared by two hexagons

Table 4.1

Character table"'6-0 for Ih.

Table 4.1

Character table"'6-0 for Ih.

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