E + T

in which parity is conserved in a cubic symmetry-lowering centrosymmetric crystal field perturbation. Group theory requires that only the G and H modes are split in the cubic crystal field. Since in Th symmetry, Ag, Eg, and Tg are all Raman-active, cubic crystal field effects can give rise to new Raman-active modes. Since the infrared-active modes have Tu symmetry for the group Th and FXu symmetry in the lh group, no crystal field-induced mode splitting is expected, although the F2u, Gu, and Hu modes can be made infrared-active by crystal field effects in accordance with Eq. (11.2). The symmetry-lowering effect discussed above would be expected to give rise to mode splittings of a few wave numbers, depending on the magnitude of the intermolecular interactions.

In contrast, there is a second crystal field effect, associated with the increase in size of the crystalline unit cell by a factor of 4 in volume, giving rise to zone-folding effects which would be expected to result in former zone edge and midzone modes being folded into the zone center, and giving rise to Raman and infrared modes with much larger "splittings," because these mode splittings depend on the width of the branch of the phonon dispersion relations. The symmetries for the mode splittings under zone folding are determined by considering the direct product of the equivalence transformation with each of the modes in Th symmetry, and the results are given in Table 7.10. Whereas experimental evidence for zone-folding effects is well documented for the intermolecular vibrational modes (see §11.4), there is no clear evidence for such effects in the intramolecular vibrational spectrum for solid C60.

Having summarized the expected crystal field-induced effects, we now offer two reasons why crystal field symmetry-lowering effects are not important to a first approximation for the analysis of intramolecular vibrational spectra for C60. The first argument relates to merohedral disorder and the second to Brillouin zone folding. Regarding merohedral disorder, we note that if a molecule is orientationally disordered, then both the misaligned molecule and all its nearest-neighbors lose the Th crystal field symmetry. Thus a 10% merohedral disorder at low temperature (see §7.1.4) should be sufficient to essentially smear out all the predicted Th crystal field splittings of the G and H symmetry modes.

The second argument relates to zone folding the phonon dispersion relations in reciprocal space giving rise to new Raman-active modes (see also §7.1.2). Even though the dispersion of the phonon branches is small for fullerene solids (~50-100 cm"1), the zone folding would be expected to result in a multiplicity of lines with relatively large splittings in the Raman-active modes (~tens of cm"1). Since large splittings of this magnitude are not seen experimentally, we conclude that the coupling of the dynamical matrix to the cubic crystal potential is weak.

11.3.3. Isotope Effects in the Vibrational and Rotational Spectra of C60 Molecules

As an illustration of the various isotope effects on the rotational and vibrational energy levels, we discuss here the most spectacular of the isotope effects, specifically the modification of the rotational levels of 12C60 [11.47], We then list a number of other isotope effects that are expected for C60, some of which have not yet been reported experimentally. To treat these isotope effects, we consider the total wave function of the 60 carbon atoms

(including their nuclei) M', which can be expressed by the product wave function ns'

where ^vib, ^rot and refer, respectively, to the electronic, vibrational, rotational, and nuclear spin factors for the molecule. The ground state electronic structure of a C60 molecule requires to have Ag symmetry.

In the case of 12C60, the total wave function, should be totally symmetric for any permutations of the 12C nuclei within the molecule, because each 12C nucleus is a boson with nuclear spin J = 0. In contrast, the 13C60 nuclei have totally antisymmetric states for an odd number of nuclear exchanges, since each 13C nucleus is a fermion with nuclear spin J = 1/2. For a C60 molecule which contains both 12C and 13C isotopes, the proper statistics should be applied to the permutations among the 12C atoms and among the 13C atoms. There are no restrictions regarding the statistics for permutations between 12C and 13C atoms, since each is considered to be a different particle, and permutation exchanges between unlike particles are not symmetry operations of a group. For illustrative purposes, we consider initially only 12C60 and 13C60 molecules, i.e., CM molecules consisting only of 12C isotopes or of 13C isotopes.

Any symmetry operation of the icosahedral group Ih can be expressed by a permutation of 60 elements, and the group Ih is a subgroup of the symmetric group (or permutation group) 5(60). It is then necessary to consider symmetries of the wave function under the permutations which belong to group Ih.

The rotational energy levels of a C60 molecule in vacuum are given by

£y(rot) = ^jJ(J + 1) = 3.3 x 10~2J(J + 1) [K], (11.4)

where I is the moment of inertia of a C60 molecule (7 = 1.0 xlO-43 kg-m2), J denotes the angular momentum quantum number of the molecule, and the rotational energies are given in degrees Kelvin. The A7 = ±1 transitions between the rotational states of the molecule £/(rot) are much lower in frequency than the lowest intramolecular vibration.

Evidence for P, Q, and R branches associated with the rotational-vibrational states of the C60 molecules comes from high temperature (~ 1060°C) gas-phase infrared spectroscopy [11.41]. The rotational-vibrational spectra associated with the free C60 molecule are exceptional and represent a classic example of the effect of isotopes on the molecular vibrational-rotational levels of highly symmetric molecules [11.48].

Calculations show that isotope effects should be very important in the vibrational spectroscopy of the gas-phase of C60 at very low temperatures (< 1 K), where quantum effects are dominant [11.47]. Because of the large moment of inertia of C60, classical behavior (which suppresses the isotope effect) becomes dominant at relatively low temperatures (> 1 K). However, differences in the symmetry of the 12C and 13C nuclei in a C60 molecule are expected to be relevant to the rotational motion of the molecule at low temperature, imposing severe restrictions on the allowed rotational states of 12C 60, which contains only 12C isotopes [11.47,49,50]. The symmetry differences between 12C and 13C also serve to break down the icosahedral symmetry of C60 molecules which contain both 12C and 13C isotopes, and lead to the observation of weak symmetry-breaking features in the Raman and infrared spectra (see §11.5.3 and §11.5.4). As mentioned in §4.5, the natural abundance of the 13C isotope is 1.1%, the remaining 98.9% being 12C.

In the case of 12C60, there is no nuclear spin for 12C and thus the nuclear spin wave function transforms as the irreducible representation Ag in icosahedral Ih symmetry. Using Eq. (11.3) and the fact that ^vib, all transform as Ag at low temperature (< 1 K), the rotational states ^rot for 12Qo are also required to have Ag symmetry. Thus the rotational motion for 12C 60 is restricted to J values which contain the irreducible representation Ag of the point group Ih. From Table 4.3, where the decomposition of the angular momentum states J of the full rotation group into the irreducible representations of Ih is given, we obtain the J values which contain

Ag symmetry, namely J = 0,6,10,12,16,____Thus the lowest Pauli-allowed rotationally excited state for 12C60 must have J — 6. Using Eq. (11.4), we estimate the energy of the J = 6 level to be ~1.6 K. It is interesting that all rotational states from / = 1 to / = 5 are not allowed in 12C60 by the Pauli principle. Since the nuclear spin for 13C is 1/2 and the total nuclear spin angular momentum for the 13C60 molecule can vary between 0 and 30, in general, contains all possible irreducible representations of Ih, and consequently there are no restrictions on or on J for the free 13C60 molecule. This symmetry requirement for 12C60 leads to major differences between the low-lying rotational levels for 12C60 relative to 13C60 (and even relative to 12C5913C) for low J values.

Referring to Fig. 11.6 we see a schematic diagram for possible vibrationrotation spectra associated with an infrared-active Flu mode for C60. The top trace shows P, Q, and R branches for the 13C60 molecule. The rotational structure shows mostly uniform splittings in the P (A/ = +1) and R (A/ = -1) infrared branches with spacings of 25(1 - £) = 0.0064, 0.0083, 0.0076, and 0.0062 cm"1, respectively, for each of the four fundamental infrared-active modes Flu(l),.. .,F,„(4) (see Table 11.1). In contrast, the 12C60 molecule shows a sparse spectrum, reflecting the stringent



11. Vibrational Modes [P-Branch )

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