Molecular Symmetry and Group Theory

Workshop 6 — Molecular Vibrations

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    • Exercise 1

      Determine the symmetries of all the vibrational modes of the XeF4 molecule (shown).

    To determine which vibrations you would expect to see in the IR spectrum, and which in the Raman spectrum, we use the following procedure:

    1. Determine the point group (XeF4 is D4h) and use the basis of three arrows (along x, y and z directions) on each atom to obtain a representation.

      D4h E 2C4 C2 2C2′ 2C2″ i 2S4 σh 2σv 2σd
      Γ 15


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    2. Reduce the representation

      D4h E 2C4 C2 2C2′ 2C2″ i 2S4 σh 2σv 2σd
      Order =
      		 
      Γ 15    
      A1g 1 1 1 1 1 1 1 1 1 1
      A2g 1 1 1 -1 -1 1 1 1 -1 -1
      B1g 1 -1 1 1 -1 1 -1 1 1 -1
      B2g 1 -1 1 -1 1 1 -1 1 -1 1
      Eg 2 0 -2 0 0 2 0 -2 0 0
      A1u 1 1 1 1 1 -1 -1 -1 -1 -1
      A2u 1 1 1 -1 -1 -1 -1 -1 1 1
      B1u 1 -1 1 1 -1 -1 1 -1 -1 1
      B2u 1 -1 1 -1 1 -1 1 -1 1 -1
      Eu 2 0 -2 0 0 -2 0 2 0 0
      N×χR×χI(A1g)
      Σ(N×χR×χI(A1g)) =
      Σ(N×χR×χI(A1g))/Order =
      N×χR×χI(A2g)
      Σ(N×χR×χI(A2g)) =
      Σ(N×χR×χI(A2g))/Order =
      N×χR×χI(B1g)
      Σ(N×χR×χI(B1g)) =
      Σ(N×χR×χI(B1g))/Order =
      N×χR×χI(B2g)
      Σ(N×χR×χI(B2g)) =
      Σ(N×χR×χI(B2g))/Order =
      N×χR×χI(Eg)
      Σ(N×χR×χI(Eg)) =
      Σ(N×χR×χI(Eg))/Order =
      N×χR×χI(A1u)
      Σ(N×χR×χI(A1u)) =
      Σ(N×χR×χI(A1u))/Order =
      N×χR×χI(A2u)
      Σ(N×χR×χI(A2u)) =
      Σ(N×χR×χI(A2u))/Order =
      N×χR×χI(B1u)
      Σ(N×χR×χI(B1u)) =
      Σ(N×χR×χI(B1u))/Order =
      N×χR×χI(B2u)
      Σ(N×χR×χI(B2u)) =
      Σ(N×χR×χI(B2u))/Order =
      N×χR×χI(Eu)
      Σ(N×χR×χI(Eu)) =
      Σ(N×χR×χI(Eu))/Order =

      Therefore, the composition of Γ is:
      Γ = A1g + A2g + B1g + B2g + Eg + A1u + A2u + B1u + B2u + Eu

    3. Remove the irreducible representations that correspond to translation of the molecule (labelled x, y and z in the character table) and rotation of the molecule (Rx, Ry and Rz in the character table) the remaining irreducible representations correspond to vibrational modes.

      D4h E 2C4 C2 2C2′ 2C2″ i 2S4 σh 2σv 2σd    
      A1g 1 1 1 1 1 1 1 1 1 1   x2+y2,z2
      A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz  
      B1g 1 -1 1 1 -1 1 -1 1 1 -1   x2-y2
      B2g 1 -1 1 -1 1 1 -1 1 -1 1   xy
      Eg 2 0 -2 0 0 2 0 -2 0 0 (Rx,Ry) xz,yz
      A1u 1 1 1 1 1 -1 -1 -1 -1 -1    
      A2u 1 1 1 -1 -1 -1 -1 -1 1 1 z  
      B1u 1 -1 1 1 -1 -1 1 -1 -1 1    
      B2u 1 -1 1 -1 1 -1 1 -1 1 -1    
      Eu 2 0 -2 0 0 -2 0 2 0 0 (x,y)  

      Atomic Movements Modes/Irreducible Representations  
      Translational  
      Rotational
      Vibrational

      Comments:

    4. The vibrational modes can be assessed to determine whether they are IR and/or Raman active.
      Spectroscopy Active Modes Number of Bands  
      IR  
      Raman

    Note that none of the vibrational modes are both IR and Raman active. This is a general rule (known as The Exclusion Rule) for molecules that possess a centre of symmetry (inversion centre i).

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