a1 ft

Si or tlfl)

or Utthf)

i«2Aslh?)

"The angular momentum for a spherical shell of v electrons is denoted by £, while nc denotes the number of it electrons for fullerenes with closed-shell (]Ag) ground state configurations in icosahedral symmetry. The last column gives the symmetries of all the levels of the I value corresponding to the highest occupied molecular orbital (HOMO); the superscript on the symmetry label indicates the total spin and orbital degeneracy of the level and all of the listed levels are assumed to be occupied.

ducible representations g® and hlg° to accommodate a total of 18 electrons. On filling the I — 4 level, possible ground states occur when either the gg level is filled with 8 electrons at nc = 40 or the hg level is filled with 10 electrons at nc ~ 42, or when the complete shell I = 4 is filled (i.e., ggfig°) at nc = 50. Following the same line of reasoning, the 22-fold degenerate i — 5 level in full rotational symmetry will be filled by C72, which splits into the irreducible representations Hu +Flu + F2u of the icosahedral group with filled shell occupations for these levels of 10, 6, and 6 electrons, respectively. Ten electrons in the I = 5 angular momentum states of C60 are sufficient to completely occupy the hu level, leaving the flu and f2u levels completely empty, so that the highest occupied molecular orbital (HOMO) corresponds to the hu level and the lowest unoccupied molecular orbital (LUMO) corresponds to the fUt level, in agreement with Hiickel calculations for the one-electron molecular orbitals [4.13]. It should be noted that for Hiickel calculations the next lowest unoccupied molecular orbital is not an f2u level but rather an fig level, associated with the angular momentum state t = 6. The reason why an I = 6 derived level becomes lower than an i = 5 derived level is due to the form of the atomic potential. In fact, the C60 molecule has sufficiently large icosahedral splittings so that for the low-lying excited states, some of the I = 6 states become occupied before the I — 5 shell is completely filled. Such level crossings occur even closer to the HOMO level as nc increases. The resulting electronic levels for the C60 molecule and its molecular ions are further discussed in §12.4.

Not only C60, but also other higher-mass fullerenes, have icosahedral symmetry. As discussed in §3.2 and §3.3, all icosahedral (n, m) fullerenes can be specified by C„c where nc = 20(n2 + nm + m2). Using the same arguments as for C60, the angular momentum states and electronic configurations for the nc it electrons in these larger fullerenes can be found (see Table 4.11 for nc < 780). In this table, the symmetry of each icosahedral fullerene is given. Fullerenes with (n, m) values such that n — 0, m = 0 or m = n have Ih symmetry (including the inversion operation), while other entries have I symmetry, lacking inversion. Also listed in this table is £max, the maximum angular momentum state that is occupied, from which ntut, the maximum number of electrons needed to fill a spherical shell can be calculated according to

The number of valence electrons in the unfilled shell nv is included in Table 4.11, which also lists the full electronic configuration of the it electrons, the /Hund value for the ground state configuration according to Hund's rule and the icosahedral symmetries for the valence electron states. Then by decomposing these angular momentum states into irreducible representations

Table 4.11

Symmetries and configurations of the tt electrons for icosahedral fullerenes

Table 4.11

Symmetries and configurations of the tt electrons for icosahedral fullerenes

'-ma |
"lot |
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