Solution to Inorganic Problem 3.25: Explaining D6h Symmetry of C6O6 2- Anion

[telethrow]

Homework Statement

I am trying to explain why an anion C6O6 2- would have the symmetry D6h. The problem gives you all of the IR and Raman absorptions and asks to explain why D6h is the proper symmetry for C6O6 2- anion.

For you reference the problem is 3.25 in "Inorganic Chemistry" written by Huhney(sorry if i spelled your name wrong)

Homework Equations

The Attempt at a Solution

Do you use character tables to figure this out? Or do I really have to go through the whole process of finding the irreducible set from the reducible set?

 

[TeethWhitener]

Do you use character tables to figure this out? Or do I really have to go through the whole process of finding the irreducible set from the reducible set?

Yes and Yes.
The process is tedious, but not particularly difficult. First, figure out the characters of the representation in the atom basis. Break this down into irreducible representations using the D6h character table. Then, in order to figure out the symmetries of the vibrational modes of the molecule, you need to take the direct product of the atom basis representation with the vector representation in D6h symmetry. (The nice thing about using characters is that you can just multiply them directly). Finally, this will give you a collection of irreducible representations that correspond to the vibrational mode symmetries. The IR active modes are the ones that transform like the coordinates (because of the form of the dipole moment operator), and the Raman active modes transform like the quadratic combinations of the coordinates (because of the form of the polarizibility operator).