1. Normal modes of vibration of a molecule – An Exercise
by Dr.D.UTHRA Head Department of Physics DG Vaishnav College Chennai-106

2. By understanding this presentation you can find
the no. of vibration that a molecule can execute
the type of vibration, whether it is stretching or bending or torsion or out-of plane bending
the no. of symmetric and asymmetric vibration the molecule executes and to which species it belongs to

3. 3N-6 degrees of freedom are required to describe vibration of a non-linear molecule
3N-5 degrees of freedom are required to describe vibration of a linear molecule

4. An exercise! CO2, H2O, NH3, BF3, HCl, CH4, CO, H2S,
Search Ur Allied Chemistry book!!
For the following molecules,
Find the molecule is linear or non-linear
Calculate the number of normal modes of vibration
CO2, H2O, NH3, BF3, HCl, CH4, CO, H2S,
C2H2, PCl3, BCl3, N2F2, B2H6, C6H6, SF6,
C3H4, COS, CH3Cl, H2O2
This is usually a Part A question!

5. Internal Coordinates
Changes in the bond lengths and in the inter-bond angles.
Types of internal coordinates and their notation
bond stretching (Δr )
angle deformation (ΔΦ)
torsion (Δτ )
out of plane deformation (ΔΦ')

6. How to calculate number of vibrations using internal coordinates?
nr = b
nΦ= 4b-3a+a1
nτ = b-a1
nΦ’= no.of linear subsections
in a molecule
where,
b - no.of bonds (ignoring the type of bond)
a - no.of atoms
a1- no.of atoms with multiplicity one
Multiplicity means the number of bonds meeting the atom , ignoring the bond type.
In H2O, multiplicity of H is 1 and that of O is 2. In NH3, multiplicity of N is 3, while H is 1.
In C2H2, it is 1 for H and 2 for C (not 4 b’coz you ignore bond type).
In CH4, it is 1 for H and 4 for C. In CO2, it is 2 for C and 1 for O.

7. Types of vibrations nr - no.of stretching vibrations nΦ - deformations
nτ - torsional vibrations
nΦ’- out of plane bending

8. Exercise Again!
Now using the internal coordinates, calculate the normal modes of vibration again for the molecules
CO2, H2O, NH3, BF3, HCl, CH4, CO, H2S, C2H2, PCl3, BCl3, N2F2, B2H6, C6H6, SF6, C3H4, COS, CH3Cl, H2O2

9. =NR(-1+2cosθ) (for other symmetry operations)
How to calculate normal modes of vibration using character table of the molecule?
Let us try to calculate normal modes of vibration using
the formula
ni' - number of normal modes of vibration
h - order of the point group
ℓi - dimension of the ith species
Ψj(R) – from the character table
χj' (R) =(NR-2)(1+2cosθ) (for proper rotations)
=NR(-1+2cosθ) (for other symmetry operations)
NR- number of atoms that undergo No change due to
operation R
θ - angle of rotation for R
ni'= (1/h)∑R ℓΨj (R)χj' (R)

10. For XY2 bent molecule, calculate normal modes of vibration
by all three methods

11. Using 3N-6 formula N=3 ; For an XY2 bent molecule
3N-6 = 3 modes of vibration

12. Using internal coordinates
In XY2 bent molecule,
b = 2 (no.of bonds)
a = 3 (no.of atoms)
a1= 2 (no.of atoms with multiplicity one)
Hence,
nr = 2
nΦ = 4*2-3*3+2 =1
nτ = 2-2 = 0
3N-6 = nr+nΦ+nτ = 2+1= 3

13. For a XY2 bent molecule, nr = 2 nΦ = 1 implies, this molecule has
two bond stretching vibrations and
one angle deformation (bending)vibration

14. Using Character table h - order of the point group = 4
XY2 bent molecule belongs to C2v point group
From character table,
h - order of the point group = 4
ℓi - dimension of each species =1
C2v E C2 σv
σv ’ A1 1 A2 -1 B1 B2

15. To calculate χj' (R) Θ(in degrees) NR 2cosθ (1+2cosθ) (-1+2cosθ)
C2v E C2 σv
σv ’ A1 1 A2 -1 B1 B2 NR 3
Θ(in degrees)
180
2cosθ 2 -2
(1+2cosθ) -
(-1+2cosθ)
(NR-2)*(1+2cosθ)
NR*(-1+2cosθ)

16. ni'= (1/h)∑R ℓΨ (R)χ' (R)
To find ni‘(A1)
h = 4 ; ℓ =1; Ψ (E) = 1; Ψ (C2) = 1; Ψ (σv) = 1; Ψ (σv ’) = 1;
χ' (E) = 3; χ' (C2) = 1; χ' (σv) = 1; χ' (σv ’) = 3.
To find ni‘(A2)
h = 4 ; ℓ =1; Ψ (E) = 1; Ψ (C2) = 1; Ψ (σv) = -1; Ψ (σv ’) = -1;
ni‘(A1)= (1/4) [ 1*1*3 + 1*1*1 + 1*1*1 + 1*1*3] =(1/4)*8=2
ni‘(A2)= (1/4) [ 1*1*3 + 1*1*1 + 1*-1*1 + 1*-1*3]=(1/4)*0=0
ni‘(B1)= (1/4) [ 1*1*3 + 1*-1*1 + 1*1*1 + 1*-1*3]=(1/4)*0=0
ni‘(B2)= (1/4) [ 1*1*3 + 1*-1*1 + 1*-1*1 + 1*1*3]=(1/4)*4=1
h - order of the C2v point group = 4
ℓi - dimension of each species in C2v point group=1
Ψj(R) – from the character table

17. For an XY2 bent molecule
There are 3 normal modes of vibration as 3N-6 = 3
Using internal coordinates, the nature of these 3 vibrations are found i.e.,
two bond stretching vibrations and
one angle deformation (bending) vibration
Further, using Character table,
ni‘(A1)=2
ni‘(B2)=1 indicate, these 3 vibrations are distributed among A1 and B2 species .
From the knowledge of group theory, as the nature of A species is to have only symmetric vibrations and B is to have only asymmetric vibrations,
There are two symmetric vibrations and one asymmetric vibration
Thus, knowledge of group theory systematically unfolds you the nature of vibration of any molecule!!

18. Normal modes of vibration by using
Tabulate and Tally!
S.No.
Molecule
(formula)
Linear/
Non-Linear N
Normal modes of vibration by using
Tally!
3N-5/3N-6
internal coordinates
Character table

19. Congrats!! You have learnt to calculate normal modes of vibration.
Try for many more molecules.
Make it a fun game, a mental exercise,…

20. C u in the next presentation!!
All the best and
C u in the next presentation!!
-uthra mam