H

eu + tu

a,u + e,u + e2u

ai+ei+e2

3 A, +2 a2

a

Symmetry-lowering

can occur between Ih

-* Th; h -

i; T, D5J\

vibrations of C70, we first find the symmetries for the transformation of the 70 carbon atoms denoted by x*'s'(C70) for the point group D5h where "a.s." refers to atom sites (as in Table 4.5 for icosahedral symmetry). Because of the large number of degrees of freedom in fullerenes, it is advantageous to break up the 70 atoms in C70 into subunits which themselves transform as a subgroup of D5h. This approach allows us to build up large fullerene molecules by summing over these building blocks. The equivalence transformation (^as ) for each of the building blocks can be written down by inspection.

The characters for the equivalence transformation for these subgroup building blocks, which are expressed in terms of sets of atoms normal to the fivefold axis, are listed in Table 4.17. The symmetry operations of the group transform the atoms within each of these subgroups into one another. The s entries in Table 4.17 under the various symmetry operations denote the number of carbon atoms that remain invariant under the various classes of symmetry operations. The irreducible representations of atomic sites are given in Table 4.18. The set C10(capĀ°) denotes the five carbon atoms around the two pentagons (10 atoms in total) through which the fivefold axis passes. Another 10 carbon atoms, that are nearest-neighbors to the 10 atoms on the axial pentagons, transform in the same way as the set C10(capĀ°). The set C10(belt) refers to the 10 equatorial atoms in the five hexagons on the equator that form a subgroup. There are also two sets of 20 carbon atoms on hexagon double bonds, labeled C20(off-belt), that form another subgroup. The characters for the equivalence transformation

Table 4.16

Decomposition of spherical angular momentum states labeled by I (for I < 10) into irreducible representations of lower symmetry groups."

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