12,

"One A2 mode corresponding to translations of the center of mass of the free molecule along the fivefold axis and one A'2 mode corresponding to rotations of the free molecule about the fivefold axis have been subtracted.

''One E{ mode corresponding to translations of the center of mass of the free molecule normal to the fivefold axis and one e'{ mode corresponding to rotations of the free molecule about axes normal to the fivefold axis have been subtracted.

cThe number of times an irreducible representation T, occurs for Cand CT7fl^ is given by symbols nc, nd, and ne which, respectively, denote n+j,n + 2j, and n + 4j, where n is an integer. For example, 4C, 6d, and 15,, respectively, denote 4 + j, 6 + 2j and 15 + 4j.

modes for C70 are classified as 104 cap modes (corresponding to the 60 carbon atoms of the two hemispheres of C60) and 18 belt modes. This division into cap and belt modes becomes more important in the limit of carbon nanotubes which are discussed in §19.7. The symmetries and degeneracies of the distinct mode frequencies for C70 are given in Table 4.22.

Among the modes given in Table 4.22, those that transform according to the Aj, E't or E'[ irreducible representations are Raman-active, with the A\ modes being observed only in the (||, ||) polarization geometry and the E'[ mode observed in the (||,-L) polarization. The E'2 symmetry mode is seen in both polarization geometries. The modes with A2 and E[ symmetries are infrared-active.

4.4.2. Symmetry Considerations for Higher-Mass Fullerenes

Similar arguments can be made to classify the symmetries of the molecular vibrations of the rugby ball-shaped C80 which follows symmetry group D5d. Table 4.19 lists the characters for the equivalence transformations for groups of carbon atoms comprising the Cgo isomer with D$d symmetry. Each of these equivalence transformations forms a reducible representation of DSd and the decomposition of s into irreducible representations of DSd is given in Table 4.20. The vibrations associated with groups D5h and D5d are found using a variant of Eq. (4.6), and the classification of the vibrational modes into irreducible representations of D5h and D5d can be obtained from Tables 4.22 and 4.23. Finally in Table 4.24, we give the number of distinct eigenfrequencies for the Cg0 isomer with D5d symmetry, listed according to their symmetry type, and again distinguishing between the cap and belt modes. It should be noted that the building block approach using point group Dsh can be used to obtain s and ,Ym v for C50, C70, C90, etc., and using group Dsd the corresponding information can simply be found for D5d isomers of C80, C100, etc. The building block approach provides a simple method for constructing the dynamical matrix of large fullerene molecules or for treating their electronic structure when using explicit potentials. Further discussion of the group theoretical aspects of the vibrational modes of the higher-mass fullerenes is given in §11.7. In §19.7, the symmetry of the molecular vibrations of carbon nanotubes is presented.

Carbon has two stable isotopes: 12C, which is 98.892% abundant and has a molecular weight of 12.011 and a zero nuclear spin, and 13C with atomic weight 13.003, a natural abundance of 1.108%, and a nuclear spin of 1/2.

Table 4.23

Symmetries of molecular vibrational modes0,1' '' for groups of carbon atoms with D5d symmetry

Table 4.23

Symmetries of molecular vibrational modes0,1' '' for groups of carbon atoms with D5d symmetry

d5d |
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