4 The characters for the total representation of the C-H bonds in CH4 are given below. Find the irreducible representations for this total representation.

E C3 C2 S4 σd
4 1 0 0 2

Methane belongs to the Td point group.
Td
E
8C3
3C2
6S4
6σd
a1
1
1
1
1
1
a2
1
1
1
-1
-1
e
2
-1
2
0
0
t1
3
0
-1
1
-1
t2
3
0
-1
-1
1

n(a1) = [(1)(1)(4) + (1)(8)(1) + (1)(3)(0) + (1)(6)(0) + (1)(6)(2)]/24 = 1
n(a2) = [(1)(1)(4) + (1)(8)(1) + (1)(3)(0) + (11)(6)(0) + (11)(6)(2)]/24 = 0
n(e) = [(2)(1)(4) + (-1)(8)(1) + (2)(3)(0) + (0)(6)(0) + (0)(6)(2)]/24 = 0
n(t1) = [(3)(1)(4) + (0)(8)(1) + (-1)(3)(0) + (1)(6)(0) + (-1)(6)(2)]/24 = 0
n(t2) = [(3)(1)(4) + (0)(8)(1) + (-1)(3)(0) + (-1)(6)(0) + (1)(6)(2)]/24 = 1

so the C-H bonds transform as a1 + t2