Character Tables: additional problems

 

 

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 --- 2S4 --- C2 --- 2C2' --- 2sd
G-----| 6 ----- 0 ---- 2 ------ 2 ------- 2
G---- | 6 ----- 4 ---- 6 ------ 2 ------- 0

  • 2. For the ammonia molecule: (answer)
    • determine the point group to which it belongs
    • list all the symmetry operations for ammonia (consult character table)

      analysis of the x, y, z coordinates of each atom results in the reducible representation below

C3v | E --- 2C3 --- 3sv
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.
    • 8. ----
    • 2A1 + B1 + B2 + E
    • 3A1 + 2A2 + B1.

     

  • 2.
    • C3v
    • E, C3, sv
    • 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.

 

 

 

 

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Last Updated:October 20, 2006