Construction of C3V character table

GROUP THEORY
CONSTRUCTING CHARACTER TABLE IS FOLLOWED BY 4 STEPS through �orthogonality rule
STEP 1 : FIND THE NUMBER OF IRRs
Number of IRs = Number of classes.- In C3v
there is 3 classes so Г1,Г2 Г3

STEP 2: FIND OUT THE DIMENSIONS
Sum of the squares of the dimensions of IRRs = Order of the Group
We have to identify a set of 3 positive integers (I1 I2 I3 dimensions of IRRs) which satisfy this condition







The only value of I which satisfy this condition are 1,1,2 so that I12 = I22
SO 3 IRRs of C3v ,two are 1-D and one is 2-D
STEP 3 : FIND character of two 1-D IRRs

In every point group is 1-D IRR who characters are equal to 1 .this IRRs is called totally symmetric IRR
Thus we have

Which satisfy the rule sum of the square of the characters of all operations in any IRR is equal to the order of the group
FIND characters of another 1-D IRRs�Conditions
All the characters of this IRRs equal to +1 or -1
Also IRR must be Orthogonal to Г1

Г1 has six +1 as characters of the sym operations 1 for E ; 2 (1) for C3 ; 3 (1) for σv
The characters of Г2 is Orthogonal to Г1 so it has three +1 and three -1
For E in 1-D is +1 ; for 2 C3 in 1-D is +1 ; FOR 3 σV is -1

1. Construction of c 3v character TABle - ESWARAN.M 17PCH003

2. C 3v SYMMENTRY OPERATIONS NH 3 C 3v Point group contains symmentry elements of ORDER of the group h =6 classes of the group = 3

3. CONSTRUCTING CHARACTER TABLE IS FOLLOWED BY 4 STEPS through orthogonality rule STEP 1 : FIND THE NUMBER OF IRRs Number of IRs = Number of classes .- In C 3v there is 3 classes so Г 1, Г 2 Г 3 STEP 2: FIND OUT THE DIMENSIONS S um of the squares of the dimensions of IRRs = Order of the Group We have to identify a set of 3 positive integers (I 1 I 2 I 3 dimensions of IRRs) which satisfy this condition

4. The only value of I which satisfy this condition are 1,1,2 so that I 1 2 = I 2 2 SO 3 IRRs of C 3v ,two are 1-D and one is 2-D STEP 3 : FIND character of two 1-D IRRs In every point group is 1-D IRR who characters are equal to 1 .this IRRs is called totally symmetric IRR Thus we have Which satisfy the rule sum of the square of the characters of all operations in any I RR is equal to the order of the group

5. Which satisfy the rule sum of the square of the characters of all operations in any IRR is equal to the order of the group Here 2, 3 are number of elements in the two classes of C3 and σ V operations

6. FIND characters of another 1-D IRRs Conditions All the characters of this IRRs equal to +1 or -1 Also IRR must be Orthogonal to Г 1 Г 1 has six +1 as characters of the sym operations 1 for E ; 2 (1) for C3 ; 3 (1) for σ v The characters of Г 2 is Orthogonal to Г 1 so it has three +1 and three -1 For E in 1-D is +1 ; for 2 C3 in 1-D is +1 ; FOR 3 σ V is -1

7. STEP 4 : FIND character of two 2-D IRR LETS Consider like this Values of x and y can be determined by taking the cross products of Г 1 Г 3 and Г 2 Г 3

8. from this value of x =-1 and y = 0

9. Mulliken symbolism

10. thanks