Question

How do you use character table to generate a reducible representation of the 3n translational rotation and vibrational degrees of freedom

Answer 1

Solution :

To understand how to generate a reducible representation of the 3n translation, rotation and vibrational degrees of freedom from a character table, let us consider an example of a molecule, BF₃.

STEP 1: Identify the point group of the molecule.
BF₃ belongs to D_3h point group.

STEP 2 : Determination of degrees of freedom
BF₃ is a non-linear molecule , with 4 atoms.
Using the equation 3N, there are (3 x 4=12) 12 degrees of freedom. Similarly,using the equation 3n-6, we find that there are (12-6=6) 6 vibrational degrees of freedom.

STEP 3 : Determination of irreducible representations of
The three axes are put on each atom. is then evaluated by observing the effect on the axes by all the symmetry operations.

	E	2C3	3C2	σ h	2S3	3σ v
	12	0	-2	4	-2	2

The contributions from each symmetry species are as follows:

Therefore, Γ _tot = A₁ ’ + A₂ ’ + 3E’ + 2A₂” + E”. From the above contriutions of each symmetry, we can conclude that, Γ _tot has twelve degrees of freedom as already shown in step 2.

STEP 3: DETERMINATION OF :
We know that, Γ _tot = Γ _trans + Γ _rot + Γ _vib
From the character table,   Γ _trans = E’ + A₂ ”
Also, Γ _rot = A₂’ + E”
Therefore, using Γ_vib = Γ _tot – Γ _trans - Γ_rot, we arrive at Γ _vib = A₁’ + 2E’ + A₂”
Γ _vib has six degrees of freedom.

STEP 4 : SPLITTING INTO STRETCHES AND BONDS
BF₃ has three bonds, so it has 3 stretches and 3 bends.

STEP 5 : DETERMINATION OF IRREDUCIBLE REPRESENTATION OF
One axis is put on each bond.

	E	2C3	3C2	σ h	2S3	3σ v
	3	0	1	3	0	1

The contributions of each symmetry element are as follows:

Γ _stretch = A₁’ + E’

STEP 6 : DETERMINATION OF
Γ _bend = Γ _vib - Γ _stretch
Γ_bend = E’ + A₂ ”