Question

QUESTION 2 (a) Consider the attached character table for the point group, D3d. (i) Determine the order of the point group. Show all work. (ii) Determine the order of the point group using any other method. rave? (iii) Which irreducible representations are orthogonal? Show all work. (iv) Give the completely symmetric irreducible representation.

Answer 1

(a) (i) For D_nd or D_nh  point group the order of the group is found out by 4n

order of the group = 4*3 = 12

(ii) By summing up all the symmetry elements grouped into seperate class in the given character table can give the order of the group.

Number of symmetry element = 12

(iii) [A_1g & A_1u]& [E_g & E_u] are orthogonal

D3d	E	2C3	2C2’	i	2S6	3σd
A1g	1	1	1	1	1	1
A1u	1	1	1	-1	-1	-1

A_1g * A_1u = (1*1)+(1*1)+(1*1)+(1*(-1))+(1*(-1))+(1*(-1)) = 0

  

D3d	E	2C3	2C2’	i	2S6	3σd
Eg	2	-1	0	2	-1	0
Eu	2	-1	0	-2	1	0

E_g * E_u = (2*2)+(-1*(-1))+(0*0)+(2*(-2)+(-1*1)+(0*0) = 0

(iv) Completely symmetric irreducible representation is A_1g - since under all symmetry operation symmetry is retained.