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"As an example in using the table, T(J =5) = T2{Fl) + r3(F2) + TS(H).

describe the electronic states when electron spin is included in the wave function (§4.3).

Fivefold symmetry is not often found in solid-state physics, because it is impossible to construct a Bravais lattice based on fivefold symmetry. Thus fullerenes in the solid state crystallize into solids of lower point group symmetries, such as the fee lattice (e.g., C60 at room temperature) or the hep lattice (e.g., some phases of C70). Nevertheless, the local point group symmetry of the individual fullerene molecules is very important because they crystallize into highly molecular solids in which the electronic and vibrational states are closely related to those of the free molecule. Therefore we summarize in this chapter the group theoretical considerations that are involved in finding the symmetries and degeneracies of the vibrational and electronic states of the C60 molecule, with some discussion also given to higher-mass fullerenes. Group theoretical applications of the space groups associated with the crystalline solids are made elsewhere in the book (see §7.1, §19.4).

To describe the symmetry properties of the vibrational modes and of the electronic levels, it is necessary to find the equivalence transformation for the carbon atoms in the molecule. The characters for the equivalence transformation for the 60 equivalent carbon atom sites (a.s.) for the C60 molecule in icosahedral Ih symmetry are given in Table 4.4. Also listed in Table 4.4 are the number of elements for each class (top) and the characters for the equivalence transformation for the 12 fivefold axes, the 20 threefold axes, and the 30 twofold axes which form classes of the icosahedral Ih group. The decomposition of the reducible representations of Table 4.4 into their irreducible constituents is given in Table 4.5, which directly gives the number of it orbitals for each irreducible representation. For example, if guest species X are attached to each fivefold axis at an equal distance from the center of the icosahedron to yield a molecule X12C60, then the full icosahedral Ih symmetry is preserved. Table 4.4 also lists for a few higher icosahedral fullerenes: C80 for which (n, m) = (2,0); C140 for which (n, m) = (2,1); and C240 for which (n, m) = (2,2) (see §3.3). We note that fullerenes with either m — 0, or n = 0, or those with m = n have inversion symmetry and therefore are described by group lh. Other icosahedral fullerenes with m ^ n ^ 0, lack inversion symmetry (e.g., C]40) and are described by the point group I. The decomposition of the equivalence transformation s which is a reducible representation of the group Ih (or I) into its irreducible constituents is given in Table 4.5 for every entry in Table 4.4, and the even and odd constituents are listed on separate lines.

Table 4.4

Characters x",s' f°r the equivalence transformation of various atomic sites in icosa-hedral Ih symmetry. The corresponding irreducible representations of group Ih are listed in Table 4.5.

Table 4.4

Characters x",s' f°r the equivalence transformation of various atomic sites in icosa-hedral Ih symmetry. The corresponding irreducible representations of group Ih are listed in Table 4.5.

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