Molecular Symmetry and Group Theory Workshop 6 — Molecular Vibrations

• Exercise 2

 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 C2h)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 (Rx, Ry and Rz 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