1. Construction of Symmetry Coordinates for XY2 bent type molecules
by Dr.D.UTHRA Head Department of Physics DG Vaishnav College Chennai-106

2. This presentation has been designed to serve as a self- study material for Postgraduate Physics students pursuing their programme under Indian Universities, especially University of Madras and its affiliated colleges. If this aids the teachers too who deal this subject, to make their lectures more interesting, the purpose is achieved D.Uthra

3. I acknowledge my sincere gratitude to my teacher
Dr.S.Gunasekaran, for teaching me group theory with
so much dedication and patience & for inspiring me
and many of my friends to pursue research.
- D.Uthra

4. Pre-requisites, before to you get on
Knowledge of symmetry elements and symmetry operations
Knowledge of various point groups
Look-up of character tables
Idea of the three dimensional structure of the molecule chosen

5. Normal modes of vibration of a molecule
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.
Hence,
For an XY2 bent molecule
N=3 ; 3N-6 = 3 modes of vibration
For an XY3 pyramidal molecule
N=4 ; 3N-6 = 6 modes of vibration

6. 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')

7. 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.

8. Let us calculate the number of vibrations for a XY2 bent molecule
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

9. Let us calculate the number of vibrations for a XY3 bent molecule
In XY3 bent molecule,
b = 3 (no.of bonds)
a = 4 (no.of atoms)
a1= 3 (no.of atoms with multiplicity one)
Hence,
nr = 3
nΦ = 4*3-3*4+3 =3
nτ = 3-3 = 0
3N-6 = nr+nΦ+nτ = 3+3=6

10. Symmetry Coordinates
Symmetry coordinates describe the normal modes of vibration of the molecules
They are the linear combinations of the internal coordinates related to various vibrations of the molecule

11. Basic properties of Symmetry Coordinates
must be normalised
must be orthogonal in the given species
has the form
Sj = ∑k Ujkrk
not necessarily unique

12. How are Symmetry Coordinates formed?
Generated by Projection operator method,
S ij = ∑R χi(R)RLk
R- symmetry operation of the point group
χi(R)-character of the species (A1,A2,B1,etc) covering the symmetry operation R (C2, C3,σ,etc)
Lk-generating coordinate( may be single internal coordinate or some linear combination of internal coordinates)
RLk-new coordinate which Lk becomes, after the symmetry operation is performed
You are projecting the operator, that is doing symmetry operations
on various internal coordinates of the chosen molecule, find what happens
to those internal coordinates after projection and sum up the result!

13. Caution : Brush up your knowledge and then proceed
To proceed further…
You need to have the knowledge of
Structure of the chosen molecule
Symmetry operations it undergoes
Point group symmetry it belongs to
Character table of that point group symmetry
Caution : Brush up your knowledge and then proceed

14. Point Group and Symmetry Coordinates
Identify the point group symmetry of the molecule.
By using its Character table, find the distribution of vibrations among its various species , ie find Γvib.
Find symmetry coordinates defining each vibration
! Do you realise the importance of understanding symmetry
operations and point groups and place the molecules correctly in their
point group symmetry. If You are incorrect in finding the point group
symmetry of the molecule, things will not turn out correctly!!

15. XY2 bent molecule
XY2 bent molecule belongs to C2v point group symmetry
Γvib = 2A1 + B2 = 3
Hence, we need 3 symmetry coordinates
This gives you an idea how the
various normal modes of vibration
(calculated with 3N-6, then checked
with internal coordinates) are
distributed among various species.
C2v E C2 σv
σv ’ A1 1 A2 -1 B1 B2

16. How to arrive at the symmetry coordinates?
As seen earlier, these are generated by Projection
operator method,
Step 1 : For every internal coordinate Lk, find RLk.
Step 2 : Use character table and find the product χi(R)RLk for every R
Step 3 : Sum them up
Before that, keep a note of
internal coordinates of the chosen molecule
what changes take place in them after every symmetry operation
character table to which the molecule belongs to
S ij = ∑R χi(R)RLk

17. S ij = ∑R χi(R)RLk In case of XY2 bent molecule Γvib = 2A1 + B2 = 3
Internal coordinates (Lks) are Δd1 , Δd Δd1, Δα
Symmetry operations (Rs) are E, C2 , σv , σv‘ and
their respective χis are 1 or -1 (from character table ),
while i = A1, A2, B1,B2
In A1 species
S1 = ?
S2 = ?
In B2 species
S3 = ?
C2v E C2 σv
σv ’ A1 1 A2 -1 B1 B2
Γvib = 2A1 + B2 = 3

18. Generating the symmetry coordinate with Δd1 as Lk
After operartion
R Lk R Lk
E :
C2 :
σ v :
σv ’ :
C2v E C2 σv
σv ’ A1 1 A2 -1 B1 B2
Δd1
Δd1 d1 d2
Δd1
Δd2
Δd1
Δd2
C2v E C2 σv
σv ’ A1
Δd1
Δd2 A2 B1 B2
Δd1
Δd1

19. S i = ∑R χi(R)RLk C2v E C2 σv σv ’ A1 A2 B1 B2 SB2 = Δd1 - Δd2
Now check your filled up
table having χi(R)RLk entries
Find summation of the χi(R)RLk
entries for every R
C2v E C2 σv
σv ’ A1
Δd1
Δd2 A2
-Δd2
-Δd1 B1
- Δd2 B2
Sd1A1= Δd1+Δd2+Δd2+Δd1 = 2(Δd1+Δd2)
Sd1A2= Δd1+ Δd2- Δd2- Δd1 = 0
Sd1B1= Δd1- Δd2+ Δd2- Δd1 = 0
Sd1B2= Δd1- Δd2- Δd2+ Δd1 = 2(Δd1-Δd2)
From above, you get
SA1 = Δd1+Δd2
SB2 = Δd1 - Δd2

20. Generating the symmetry coordinate with Δd2 as Lk
Similarly generate the symmetry
coordinates with Δd2 as Lk
S i = ∑R χi(R)RLk
Sd2A1=Δd2+Δd1+Δd1+Δd2 =2(Δd2+Δd1)
Sd2A2= Δd2+Δd1-Δd1-Δd2 = 0
Sd2B1= Δd2-Δd1+Δd1-Δd2 = 0
Sd2B2=Δd2-Δd1-Δd1+Δd2 = 2(Δd2–Δd1)
SA1 = Δd1+ Δd2
SB2 = Δd2–Δd1
=-(Δd1-Δd2)
C2v E C2 σv
σv ’ A1
Δd2
Δd1 A2
-Δd1
-Δd2 B1 B2

21. Generating the symmetry coordinate with Δα as Lk
Now generate the symmetry
coordinates with Δ α as Lk
S i = ∑R χi(R)RLk
SαA1 = 4 Δα
SαA2 = 0
SαB1 = 0
SαB2 = 0
SA1 = Δα
C2v E C2 σv
σv ’ A1 Δα A2
-Δα B1 B2

22. SALCS- Symmetry Adapted Linear Combinations
By projection operator method, employing
SALCs for XY2 bent molecule are found to be
SA1 = Δd1+Δd2
SB2 = Δd1 - Δd2
SA1 = Δα
S ij = ∑R χi(R)RLk

23. Condition to normalise and orthogonalise SALCs
For normalisation
For singly degenerate species,
Uak2 = 1/q or Uak = ±(1/q)½
For doubly degenerate species,
Uak2 + Ubk2 = 2/q
For triply degenerate species,
Uak2 + Ubk2 + Uck2 = 3/q
q - total no.of symmetry equivalent internal coordinates involved in that SALC
For orthogonalisation
∑k UakUbk =0

24. For XY2 bent molecule
All three SALCs belong to singly degenerate species.
Applying first condition,
In SA1 = Δd1+Δd2 and SB2 = Δd1 - Δd2,
q=2
SA1 = Δα
q=1

25. Obtain Orthonormal SALCs
After Normalisation
SA1 = 1/√2 (Δd1+ Δd2)
SB2 = 1/√2 (Δd1 - Δd2)
SA1 = Δα
Next, check for orthogonality

26. Orthonormalised Symmetry Coordinates of XY2 bent molecule
3N-6 = 3 distributed as
Γvib. = 2A1 + B2 = 3
A1 Species
S1A1 = 1/√2 (Δd1+ Δd2)
S2A1 = Δα
B2 Species
S3B2 = 1/√2 (Δd1 - Δd2)

27. Recap For an XY2 bent molecule N=3 ; 3N-6 = 3 modes of vibration
In XY2 bent molecule,
b = 2
a = 3
a1= 2
Hence,
nr = 2
nΦ = 4*2-3*3+2 =1
nτ = 2-2 = 0
3N-6 = 2nr+1nΦ = 3
XY2 bent molecule
belongs to C2v point group symetry.
Γvib. = 2A1 + B2 = 3
In case of XY2 bent molecule
Lks are Δd1 , Δd2 , Δα
Rs are E, C2 , σv , σv‘ and their respective
χis are 1 or -1 (from character table ), while
i = A1, A2, B1,B2
Γvib. = 2A1 + B2 = 3 implies
Check for orthogonality
Normalisation
A1 Species
S1 = 1/√2 (Δd1+ Δd2)
S2 = Δα
B2 Species
S3 = 1/√2 (Δd1 - Δd2)

28. All the Best ! uthra mam Hope You enjoyed learning to form
symmetry coordinates !
See U in the next session of group theory!