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Group theory and molecular symmetry can be used to determine the vibrational modes of molecules with strong symmetry. This type of analysis can also determine which of these vibrational modes will be infrared or Raman active. Some vibrational modes may also be inactive or active for both.
The following is an example analysis of square planar molecule. Some understanding of group theory is assumed. Adapted from literature.

See Figure C.

Determined that the symmetry group for
this molecule to be D
4h with the following
character table.

See Figure D.

MCO6 and Group Theory Figure C.PNG
MCO6 and Group Theory Figure D.PNG

An axis set was assigned to each atom. For a given transformation, if an atom shifts it has no effect. Atoms that remain unshifted attribute +1 for each unchanged axis and -1 for each axis in the opposite direction. In the case that an axis is shifted in a different manner analysis is component based. This obtains a reducible representation Γ.

MCO6 and Group Theory Table .PNG

Reduced the representation in order to obtain a linear combination of symmetric groups.

MCO6 and Group Theory A1g Equation.PNG

h = order of the group = 16 for D4h

Xr= character of the reducible representation

XI = character of the irreducible representation

N= number of symmetry operation in the class

Number of A1g in Γ = 

MCO6 and Group Theory A1g gamma equation

Full analysis results in Γ = A1g + A2g + B1g + B2g + Eg + 2A2u + B2u + 3Eu
These modes where then attributed to translations and rotations. Since the molecule is nonlinear there are three translational and three rotational modes. Translational modes correspond the x, y, and z in the character table while rotations correspond to Rx, Ry,
and R
z. Note that Eg and Eu representations have degeneracies of two.

Translations: A2u + Eu

Rotations: A2g + Eg

The remaining modes are vibrational modes which can be active in infrared and/or Raman. Infrared modes relate to a change in dipole moment and correspond to x, y, and z in the character table. Raman modes relate to a change in polarizability and correspond to binary products such as x  and xy as well as combinations such as x -y . Note that modes can be both infrared and Raman active.

Infrared active: A2u + 2Eu

Raman active: A1g + B1g + B2g

Inactive: B2u


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