CHM 501

23. 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

Answer

Methane belongs to the Td point group.

Td

E

8C3

3C2

6S4

d

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