"Number of modes in icosahedral symmetry. ''Raman-active mode is seen only in ||, || polarization. c Raman-active mode is seen in both ||, || and ||, ± polarizations, ''infrared-active mode.

eI symmetry, thus no gerade or ungerade modes. In this case the F, modes are IR-active and the A and H modes are Raman-active.

found include Ag + Gg + 2Hg + Flu + Flu + Gu + Hu where 2Hg indicates that there are two distinct mode frequencies corresponding to (5x2) = 10 normal modes. We note that for the C60 molecule, every irreducible representation is contained at least once. In carrying out the direct product X&,s' ® Xfu the entries in Table 4.7 for the decomposition of direct products for the point group I are useful. We note that Ih = I <8> i, indicating that the characters for the group Ih are obtained from those of the group I by taking the direct product of group I with the inversion group i.

The Raman-active modes have Ag and Hg symmetry (corresponding to the basis functions for all symmetrical quadratic forms, and the antisymmetric quadratic form transforming as Flg does not contribute to the Raman scattering). The infrared-active modes have Flu symmetry (the linear forms associated with a vector). One can see from the basis functions listed in Table 4.2 that the symmetry of the Raman tensor allows ||, || scattering for Ag modes and both ||, || and ||,-L scattering for Hg modes, where the directions || and _L refer to the polarization directions of the incident and scattered photon electric fields. For example, ||,J_ implies that the polarizations for the incident and scattered beams are orthogonal. Table 4.6 shows that of the 46 distinct vibrational mode frequencies for C60, only 4 are infrared-active with symmetry Flu and only 10 are Raman-active (two with Ag symmetry and eight with Hg symmetry), while the remaining 32 modes are silent in the first-order infrared and Raman spectra. Many of these silent modes can, however, be observed by inelastic neutron scattering, electron energy loss spectroscopy, as vibronic side bands on the photoluminescence spectra, and most sensitively in the higher-order infrared and Raman spectra (see §11.5.3, §11.5.4, and §11.5.6), because of the different selection rules governing these higher-order processes.

Various possible attachments could be made to the Qo molecule without lowering its symmetry [e.g., by attaching 12 equivalent guest species (X) at each end of the 6 fivefold axes, or 20 guest species at each end of the 10 threefold axes, or 30 guest species at each end of the 15 twofold axes]. To preserve the Ih symmetry, all the equivalent sites must be occupied. However, if only some of these sites are occupied, the symmetry is lowered (see §4.4). Table 4.4 gives the characters x™' for the equivalence transformation for special arrangements of guest species that preserve the Ih or / symmetries, and these guest species may be attached through doping or a chemical reaction. The corresponding vibrational modes associated with such guest species are included in Table 4.6, both separately and in combination with the C60 molecule, as for example X12C60, where we might imagine a guest atom to be located at the center of each pentagonal face. Any detailed solution to the normal mode problem will involve solutions of a dynamical matrix in which tangential and radial modes having the same symmetry will mix because of the curvature of the molecule. Symmetry aspects of the vibrational modes of C60 are further treated in §11.3.1.

Vibrational modes that are silent in the first-order spectrum can, however, contribute to the second- and higher-order Raman [4.7] and infrared [4.8] spectra. Anharmonic terms in the potential couple the normal mode solutions of the harmonic potential approximation, giving rise to overtones (rtw,) and combination modes (w, ± <w •), many of which are observable in the second-order spectra. Group theory requires that the direct product of

Table 4.7

Decomposition of direct products" in the icosahedral point group I.

Table 4.7

Decomposition of direct products" in the icosahedral point group I.


r, (a) r2 (F,) r3 (f2) r4 (G)


r, (a)


r2 r,


r4 r,

r5 r2

r2 (ft)


r2 r5

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