The fullerene Cgo is a highly symmetric molecule belonging to the I_h point grou

Question

The fullerene Cgo is a highly symmetric molecule belonging to the I_h point group. The internuclear modes of motion for this compound span the following irreducible representations 2 A_g + 4T_1g + 4T_2g + 6G_g + 8H_g + 1A_U + 5T_2u + 5 T_2u + 6G_u + 7H_u where T is a triply degenerate irreducible representation, G is a 4-fold degenerate and H is a 5-fold degenerate irreducible representation. Answer the following questions: (a) How many internuclear modes of motion exists for this compound? (b) The translational modes of motion span what irreducible representations? (c) The rotational modes of motion span what irreducible representations? (d) How many IR active modes exist and what irreducible representations doe these modes span? (e) How many vibrational Raman active modes exist and what irreducible representations do these modes span? (f) Are there any modes which cannot be observed using infrared or Raman spectroscopy (i.e., dark modes)? If so, how many and what irreducible representations do these modes span?

Explanation / Answer

a) 2 streaching and bending

b) 5 T1u

c) 4 T1g

d) 5 ir active modes 5 T1u

e) A1 E T2 A1g Eg T2g.

4 T2g

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