"Note: the symmetry operations about the fivefold axes are in two different classes, labeled 12C5 and 12C52 in the character table. Then iCs = Sin' and /Cy1 = S10 are in the classes labeled 12Sji0 and 125,Q, respectively. Also iC2 = cr„.

bSee Table 4.2 for a complete listing of the basis functions for the Ih point group in terms of spherical harmonics.

"Note: the symmetry operations about the fivefold axes are in two different classes, labeled 12C5 and 12C52 in the character table. Then iCs = Sin' and /Cy1 = S10 are in the classes labeled 12Sji0 and 125,Q, respectively. Also iC2 = cr„.

bSee Table 4.2 for a complete listing of the basis functions for the Ih point group in terms of spherical harmonics.

(fl6 = 1.40 A) (see §3.1). If we take this difference in bond length into account, then the C60 cage forms a truncated icosahedron but not a regular truncated icosahedron where all bond lengths would be equal.

Since the truncated icosahedron is close in shape to a sphere, it is suggestive to relate the basis functions of the icosahedron to those of the sphere, namely the spherical harmonics. The full set of basis functions for group Ih is listed in Table 4.2 in terms of spherical harmonics Yl m with minimal I values. Many physical problems dealing with fullerenes, such as the electronic states or vibrational modes, are treated in terms of spherical harmonics which are basis functions for the full rotational group. The spherical harmonics therefore form reducible representations of the Ih point group for I > 2 and irreducible representations for i — 0,1,2. Odd and even integers I, respectively, correspond to odd and even representations of the group lh. The basis functions for group lh in an /-dimensional manifold are obtained by solving the eigenvalue problem for irreducible tensor operations. The decomposition of the spherical harmonics into irreducible representations of the point group I is given in Table 4.3 for both integral and half-integral values of the angular momentum J. The integral values of J are pertinent to the vibrational spectra (§4.2), while the half-integral J values are also needed to

Basis functions for each of the irreducible representations 01 of point group Ih expressed as spherical harmonics Ylm." For the multidimensional representations the basis functions for each partner are listed [4.4].

(ft Basis function

v^/5y6,6 + VSe/io^,, - v^2/ioy6,^4 —y28/125yg_8 + V39/500y8,_3 + ^143/250^,2 - Jtß/5MY„

-728/125 y8i8 - v/397500y8,3 + 7143/250yg _2 + y63T5ööy8,_7

y87i5y4,_4 + i ,/l7i5y4,_3 + yi47T5y4t2 -yi7i5y4,3 + /Wisiw y87T5y4,4 - /77T51V,

250 '■■'15.15 "r '15,-15,* 250 v115,10 — 715,-10)

-J3/5YV +J2/5YX _2 v/sTiO^-i + /VWYs.i 7375^5,-3 - v^^w yi72(y„ + y5,-5)

Table 4.3

Decomposition of angular momentum basis functions in the full rotation group labeled by J into irreducible representations of the double group of I. Both integral and half-integral angular momentum basis functions are included."

Table 4.3

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