Chacter tables
 1. Using the D2d charcter table do the following:
(answer)
 Determine the order of the D2d group
 Reduce the following representations to their component irreducible representations
D2d  E  2S_{4}  C_{2}  2C_{2}'  2s_{d}
G 6  0  2  2  2
G  6  4  6  2  0
 2. For the ammonia molecule:
(answer)
C3v  E  2C_{3}  3s_{v}
G 12  0 2
 reduce the representation to its irreducible components
 assign the irreducible representations to translational modes (which ones represent x, y or z functions)
 assign remaining irreducible representations to rotational modes (which ones represent Rz, Rx, or Ry functions)
 finally assign remaining irreducible representations to vibrational modes (this will be stretches and bends)
 which of the vibrational modes are IR active? (which ones represent x, y or z functions)
 go to Vibrational modes of NH3 to see the actual modes
Answers
 1.
 2A1 + B1 + B2 + E
 3A1 + 2A2 + B1.
 2.
 C3v
 E, C3, s_{v}
 3A1 + A2 + 4E
 Translation: A1 + E.
 Rotation: A2 + E
 Remaining are vibrations: 2A1 + 2E.
 6 types of vibrations. Looking at the character table shows that all are active. E represents a degenerate pair of vibrations that would appear as one band in the ir spectrum. Thus, there are 6 vibrations and 4 bands in the ir spectrum.

