"Note: C5 and C51 are in different classes, labeled 12C5 and 12Cf in the character table. The class R represents a rotation by 2v and classes C; represent rotations by lir/i between 2it and 4tr. In this table r = (1 + V5)/2, -1/t = 1 - r, r2 = 1 + t.

''The basis functions for [',;i = 1,...5 are given in Table 4.2 and for T,;i = 6,... 9 are given in Table 4.9.

"Note: C5 and C51 are in different classes, labeled 12C5 and 12Cf in the character table. The class R represents a rotation by 2v and classes C; represent rotations by lir/i between 2it and 4tr. In this table r = (1 + V5)/2, -1/t = 1 - r, r2 = 1 + t.

''The basis functions for [',;i = 1,...5 are given in Table 4.2 and for T,;i = 6,... 9 are given in Table 4.9.

Table 4.9

Basis functions for the double group irreducible representations Si of the point group Ih expressed as half-integer spherical harmonics .

01 Basis function

antibonding states have antiparallel orbitals. More generally, for fullerenes with nc carbon atoms, the molecular electronic problem involves «c 77 electrons.

The electronic levels for the 7r electrons for a fullerene molecule can be found by starting with a spherical approximation where spherical harmonics can be used to specify the electronic wave functions according to their angular momentum quantum numbers. As stated above, for € > 2 these spherical harmonics form reducible representations for the icosahe-dral group symmetry. By lowering the symmetry from full rotational symmetry to icosahedral symmetry (see Table 4.3), the irreducible representations of the icosahedral group are found. In general, the bonding a levels will lie well below the Fermi level in energy and are not as important for determining the electronic properties as the it electrons near EF.

To obtain the symmetries for the 60 ir electrons for C60 we focus our attention on the 60 bonding v electrons whose energies lie close to the Fermi level. Assigning angular momentum quantum numbers to this electron gas, we see from the Pauli principle that 60 tt electrons will completely fill angular momentum states up through I = 4, leaving 10 electrons in the I = 5 level, which can accommodate a total of 22 electrons. In Table 4.10 we list the number of electrons that can be accommodated in each angular momentum state I, as well as the splitting of the angular momentum states in the icosahedral field.

Table 4.10 thus shows that the I = 4 level is totally filled by nc = 50. The filled states in icosahedral symmetry for i = 4 are labeled by the irre-

r6 r7

Table 4.10

Filled shell 7r-electron configurations for fullerene molecules."

Table 4.10

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