as for Table 12.4. Excitonic states can also be formed with a h9uffu configuration where the notation describes the electron-hole pair for the C^ ion. Similar arguments can be applied in this case also to get the symmetries for multiplets corresponding to the h9uffu configuration. We note that the allowed f*u states for the h9uffu configuration are the same as the /,2U holes in the h?uffu configuration (although the physical ordering of the states is expected to be quite different).
The electronic states for other fullerenes can be found using techniques similar to those described above for C«, except that in general the number of interacting electrons increases, the symmetry of the fullerene molecule decreases, and the possibility of large numbers of isomers is introduced.
The filling of the electronic levels for C70 on the basis of a spherical shell electron model is described in §12.1.2. However, molecular orbital calculations for a spherical shell model show that significant level mixing occurs between t = 5 and t = 6 orbital angular momentum states, and more detailed calculations are needed.
Calculations for the electronic structure for a C70 molecule using the local density approximation and density functional theory have been carried out and the calculated density of states [12.18,36] has been compared with measured photoemission spectra [12.37-39]. These calculations indicate that Cro has a slightly higher binding energy per C atom relative to Qo (by ~20 meV) and that five distinct carbon atomic sites and eight distinct bond lengths must be considered for C^ (see §3.2). A bunching of levels is found for the C70 molecular states, just as for Qq, and in both cases the molecular levels show only small splittings of the spherical harmonic states, yielding a higher density of states for certain energy ranges. The energy width (~19 eV) from the deepest valence state to the highest-occupied valence state in C70 is similar to that in Qq, graphite, and diamond. A HOMO-LUMO gap of 1.65 eV is predicted for the free molecule on the basis of a local density approximation (LDA) calculation [12.18], somewhat smaller than the results for similar LDA calculations for the HOMO-LUMO gap for C«, (see Fig. 12.2).
Huckel calculations for C70 show that the states near the Fermi level have the symmetries and degeneracies shown in Fig. 12.3, where the level orderings for the neutral C70 molecule and for the negatively charged Cf0 and C7Ô ions are shown. Here it is seen that in neutral C^ the highest-lying filled state is orbitally nondegenerate (with a\ symmetry), and similarly the lowest-lying unfilled state is also nondegenerate (with a" symmetry). This electronic structure suggests that half-filled levels for electron-doped C,0 correspond to the C70" and ion species, in contrast to the half-filling of the LUMO level in Cgo, which corresponds to Cg0 anions.
The lower symmetry of C70 (D5h) implies a more limited utility of group theoretical arguments. The energies of the calculated molecular orbitals for the neutral C70 molecule are shown in Fig. 12.3(a). However, the convergence of the perturbation scheme in §12.2 suggests that we consider the 70 electron configuration to have the following occupied states near the Fermi level hl°ffuf*g, implying that the molecular states associated with two Flg holes will have symmetries given by their direct product, yielding Flg ® F\g = Ag + F]g + Hg states. When spin is included, the Pauli-allowed states have lAg +3Flg +1Hg symmetries. The splitting of these states in Dsh
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