Fig. 11.4. Comparison of the continuum shell model predictions for axisymmetric frequencies of vibrations (open circles) and the experimental, infrared and Raman scattering results [11.30,31] (filled circles). The dashed lines trace the two branches of the basic dispersion curve of the spherical shell model [11.28], The higher mode numbers physically correspond to multiphonon processes.

2 3 Mode Number

Fig. 11.4. Comparison of the continuum shell model predictions for axisymmetric frequencies of vibrations (open circles) and the experimental, infrared and Raman scattering results [11.30,31] (filled circles). The dashed lines trace the two branches of the basic dispersion curve of the spherical shell model [11.28], The higher mode numbers physically correspond to multiphonon processes.

Fig. 11.5. Displacements for the eight basic low-lying axisymmetric modes for a thin-shell model of C«, are shown in schematic form, six on the upper (fl0-a5) branch, and two on the lower branch (b2,b3), with bt denoting the acoustic mode. The "breathing" mode a0 is considered to be the fundamental, while the b3 mode departs the most from the initial spherical symmetry [11.28], formable shell are displayed. Symmetry labels and experimental frequencies are provided to aid in the identification of these shell modes with actual C60 modes.

Several normal mode calculations for the Q,, molecule involving discrete equilibrium positions for the C atoms have been carried out and can be compared with experiment. These calculations fall into two categories: classical force constant treatments involving bond-bending and bond-stretching displacements [11.22,24,32-35] and ab initio or semiempirical calculations [11.5,36-40]. The minimal set of force constants for C60 includes two bond-stretching constants (for bonds along the pentagonal and hexagonal edges) and the two corresponding bond-bending constants. Bonding to more distant neighbors has been considered in the actual calculations. For example, a five-force-constant model for the CM vibrational problem has been carried out [11.33,34] with values for the force constants determined from other aromatic hydrocarbon molecules, while another empirical model [11.22], also using a bond-bending and bond-stretching force constant model, includes pairwise interactions through third-nearest-neighbor interactions. By adjusting the force constants to match the 10 observed first-order Raman-active mode frequencies, very good agreement was obtained between the calculated and experimental mode frequencies determined by Raman and infrared spectroscopy, inelastic neutron scattering, and electron energy loss spectroscopy. Also available are ab initio calculations for the molecular modes of C60 [11.6], which are also in reasonably good agreement with experiment.

Although these vibrational calculations are all made for an isolated molecule, the calculated results are usually compared to experimental data on C60 molecular vibrations in the solid state. The theoretical justification for this is the weak van der Waals bonds between C60 molecules in the face-centered cubic (fee) solid. The direct experimental evidence in support of this procedure comes from comparison between the observed vibrational spectra of C60 in the gas-phase [11.41], the solid phase [11.19,42], and in solution [11.42], the mode frequencies showing only small differences from one phase to another. Gas-phase values (527, 570, 1170, 1407 cm"1) for the four Fiu infrared-active mode frequencies were found to be slightly lower (1-22 cm-1), when compared to the mode frequencies in Table 11.1 for the solid state.

To evaluate the status of various theoretical calculations, we compare in Table 11.2 various theoretical values for the mode frequencies for the 14 first-order Raman (Ag and Hg) and infrared-active (Flu) modes which are known to ±1 cm"1. Overall agreement between experiment (see Table 11.1) and some of the calculations in Table 11.2 is quite good [see calculations by Negri et al. [11.43] (ab initio), Jishi et al. [11.22] (adjustable force constants),

Comparison of experimental and calculated Raman, and infrared mode frequencies for various theoretical models. Further comparison of calculated models to experimental determinations can be made by reference to Tables 11.1 and 11.5.

Table 11.2

Mode |
Expt. |
Wu° |
W6 |
Sc |
A" |
N' |
J' |
Qs |
F" |

"iK) |
496 ±2 |

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