1. Lecture 11 REPRESENTATIONS OF SYMMETRY POINT GROUPS 1) Mulliken labels
C2v E C2
sxz
syz G1 1 z G2 -1 Rz G3
x, Ry G4
y, Rx

2. 2) Character tables. Simplest cases
C1 group. Consists of a single operation E; thus its order h=1 and number of classes is 1. There is a single irreducible representation. Its character with any basis function is 1 (E).
Cs group. Consists of two operations, E and sh; thus its order h is 2 and the number of classes is 2. There are two irreducible representations. They can be one-dimensional only, (1)2 + (1)2 = 2.
Ci group. Consists of two operations, E and i. Both its order h and number of classes is 2. Similarly to Cs, the group includes two irreducible one-dimensional representations. C1 E 1 Cs E sh 1
x,y, Rz -1
z,Rx,Ry Ci E i 1 Rx -1
x,y,z

3. 3) The character table for the C3v point group
Symmetry operations constituting C3v point group are E, 2 C3 and 3sv (h = 6).
The number of classes is 3. Thus, there are three irreducible representations. One of them is always with all characters equal to 1 (E). Note that 112 + 2(1)2 + 3(1)2 = 6.
The only combination of dimensions of the three representations which squares sum to 6 is 1, 1 and 2 ( = 6).
The second one-dimension representation G2 is orthogonal to the G1: 1(1)(1)+2(1)(1)+3(1)(-1)=0
The third representation G3 is of order 2. Thus, c3(E) = 2.
Orthogonality of 3rd, 1st and 2nd representations allows us to find c3(C3) = -1 and c3(sv) = 0 and the complete set of characters:
1(1)(2)+2(1)(x)+3(1)(y)=0 (G1 ┴ G3)
1(1)(2)+2(1)(x)+3(-1)(y)=0 (G2 ┴ G3)
give x = -1 and y = 0
C3v E
2C3
3sv G1 1 G2 -1 G3 2

4. 4) Reducible representations
Any reducible representation Gr of a group can always be decomposed into a sum of irreducible ones Gi:
Number of times a particular irreducible representation Gi appears in this decomposition, Ni, can be calculated according to the formula:
here h is the group order, cr(X) is the character of a symmetry operation X in the reducible representation; ci(X) is the character of X in the irreducible representation i and n(X) is the number of symmetry operations in the class.
C2v E C2
sxz
syz A1 1 z A2 -1 Rz B1
x, Ry B2 Gr 9 3
y, Rx