Fig. 122. Calculated electronic structure of an (a) isolated C^ molecule and (b) face-centered cubic (fee) solid Qo, where the direct band gap at the X point is calculated to be 1.5 eV on the basis of a one-electron model [12.7].
strongly bonding sp2 directed orbitals and 60 tt electrons (i.e., one tt electron per carbon atom) close to the Fermi level. The level filling and overview of the electronic structure can be obtained by filling the electronic states according to their orbital angular momentum quantum numbers, first assuming full rotational symmetry and then imposing level splittings in accordance with the icosahedral symmetry of CW1, treated as a perturbation.
These symmetry issues are discussed in §4.3 and the results for the level filling are given in Table 4.10, where the number of electrons filling each of the angular momentum states is listed together with the cumulative number of filled electronic states and with the splittings of each of the angular momentum states in an icosahedral field. Table 4.10 shows that 50 n electrons fully occupy the angular momentum states through I = 4, so that the 10 remaining tt electrons of C60 are available to start filling the I = 5 state. In full spherical symmetry, the I = 5 state can accommodate 22 electrons, which would correspond to an accumulation of 72 electrons, assuming that all the I = 5 state levels are filled, before any I = 6 levels fill. However, the 1 = 5 state splits in icosahedral symmetry into the Hu+Flu+F2u irreducible representations, as indicated in Table 4.10. The level of lowest energy is the fivefold Hu level, which is completely filled by the 10 available electrons in C60, as indicated in Table 4.10 for 60 electrons. The resulting h]u° ground state configuration, by Hund's rule, is nondegenerate, has angular momentum quantum numbers L — 0, 5 = 0, J = 0, and the many-electron h™ configuration has Ag symmetry within the point group Ih. We use the notation capital Hu to denote the symmetry of the one-electron level and the lowercase h™ to denote the many-electron configuration. Neglecting any thermal excitation, the two threefold Flu and F2u levels (some authors refer to these levels as Tu and T2u, and we use both notations in this volume) in icosahedral symmetry are empty. Whether Qq is considered in terms of a spherical approximation, or if its icosahedral symmetry is taken into account, the many-electron ground state is nondegenerate with a total angular momentum of / = 0. This special circumstance has been suggested as being in part responsible for the high stability of the C60 molecule [12.15].
Figure 12.1 shows that C60 has just enough electrons to fully fill the Hu molecular level, which forms the highest occupied molecular orbital for C60, while the next higher Flu level remains empty and becomes the lowest unoccupied molecular orbital. Molecular local density approximation calculations for C60 yield a value of 1.92 eV for the HOMO-LUMO gap in the free molecule [see Fig. 12.2(a)]. The results of Hiickel calculations, such as shown in Fig. 12.1, have been especially useful for establishing the ordering of the molecular levels of C^ [12.8]. Higher-lying unoccupied states would include the F2u level from the £ — 5 state, as well as Ag + FXg + Gg + Hg levels from the I = 6 state, where the Flg level has been calculated to lie lowest [12.8]. Hiickel calculations (see Fig. 12.1) [12.7,8,17] show that the Flg level of I = 6 lies lower than the F2u level associated with I = 5, implying that the Flg level would fill before the F2u level in a doping experiment.
Furthermore, these ideas can be used to fill states for large fullerenes. Considerations for the filling of states for icosahedral molecules up to C980 on the basis of a spherical approximation are given in Table 4.11, showing that C80 has 8 electrons in the partially filled I = 6 shell and C140 has 12 electrons in the partially filled I = 8 shell [12.15]. These arguments can be extended to consider fullerenes with lower symmetry where the HOMO level splittings in Table 4.10 would have to be expressed in terms of the irreducible representations of the lower symmetry group.
Another example of state filling relates to the C70 fullerene with DSh symmetry. Here we have 70 it electrons to place in the molecular orbitals. As discussed above, filling levels through I = 4 leaves 20 it electrons for the i = 5 shell, which is fully occupied with 22 electrons. In a spherical approximation, two possible electronic configurations which suggest themselves are s2p6dwfuglsh20, which has two holes in the I = 5 (or h) shell. If, on the other hand, we are guided by the level ordering in icosahedral C60, the suggested ground state configuration would be s2p6dwfl*gmhl6p, which has six holes in the t = 5 state and four electrons in the I = 6 state. Group theoretical considerations cannot decide on which of the possible configurations represents the ground state. Considering C70 to consist of the two hemispheres which comprise C60 with a belt of five additional hexagons along the equator of the fullerene, then the level splittings arising from lowering of the symmetry of I D5 are appropriate. In Ih symmetry the first 16 electrons for I = 5 fill the 10-fold Hu level and the 6-fold Flu level, as shown in the Hiickel level-filling picture shown in Fig. 12.1. The next four electrons would partly fill the Flg level. In icosahedral / symmetry, this level, corresponding to the configuration f*g, splits according to F1 in / symmetry into A2 + Ex in D5 symmetry, thus suggesting that the fourfold El level forms the HOMO for C70, while the twofold A2 level forms the unoccupied LUMO. The simple argument given here is, however, not in agreement with molecular orbital calculations [12.7,18,19] for C70, which show that the HOMO level is a nondegenerate level, as discussed further in §12.6.1. The LUMO level for C70 is also a nondegenerate level.
As stated above, a simple approach that is useful for finding the energy levels for any fullerene C„c is based on symmetry considerations, exploiting the observation that the closed cage fullerene molecules are close to being spherical shells. The symmetry lowering imposed by the structure of the specific fullerene under consideration is then treated in perturbation theory and symmetry is used to find the form of the Hamiltonian. The resulting eigenvalues are expressed in terms of a set of expansion parameters which are evaluated either from experiments or from first principles calculations, or from a combination of experiment and theory. The approach is similar to that taken to calculate the crystal field splitting of the rare earth ions in a cubic crystal field [12.20].
The phenomenological one-electron symmetry-based Hamiltonian for a general fullerene is then written as [12.21-23]
where W0(i) is the spherically symmetrical part and is the dominant term, while %?r(i) is the symmetry-lowering perturbation for symmetry group T which describes the symmetry of the fullerene. In the case of icosahedral Ih fullerenes, $fr(') — is the appropriate symmetry-lowering perturba tion. As an example, we will consider the symmetry-lowering perturbation to be an icosahedral perturbation. For icosahedral molecules, %Ih(i) can be written in the form
12.2. Symmetry-Based Models
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