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Find the number and symmetries of the Raman and IR active vibrations of the fumarate ion (shown).

1. Determine the point group (The fumarate ion is C_2h)and using the basis of three arrows (along x, y and z directions) on each atom, obtain a representation.

   C2h E C2 i σh Γ

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2. Reduce the representation

   C2h	E	C2	i	σh	Order =
   Γ
   Ag	1	1	1	1
   Bg	1	-1	1	-1
   Au	1	1	-1	-1
   Bu	1	-1	-1	1
   N×χR×χI(Ag)					Σ(N×χR×χI(Ag)) =	Σ(N×χR×χI(Ag))/Order =
   N×χR×χI(Bg)					Σ(N×χR×χI(Bg)) =	Σ(N×χR×χI(Bg))/Order =
   N×χR×χI(Au)					Σ(N×χR×χI(Au)) =	Σ(N×χR×χI(Au))/Order =
   N×χR×χI(Bu)					Σ(N×χR×χI(Bu)) =	Σ(N×χR×χI(Bu))/Order =

   Therefore, the composition of Γ is:
   Γ = Ag + Bg + Au + Bu

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 (R_x, R_y and R_z in the character table) the remaining irreducible representations correspond to vibrational modes.

   C2h	E	C2	i	σh
   Ag	1	1	1	1	Rz	x2+y2,z2
   Bg	1	-1	1	-1	Rx,Ry	xz,yz
   Au	1	1	-1	-1	z
   Bu	1	-1	-1	1	x,y

   Atomic Movements	Modes/Irreducible Representations
   Translational
   Rotational
   Vibrational

   Comments:

   The vibrational modes can be assessed to determine whether they are IR and/or Raman active.

   Spectroscopy	Active Modes		Number of Bands
   IR
   Raman

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