CHM 501

24. The total representation for some set of objects in the D6 point group is given below. Find the irreducible representations for this set of objects.

D6

E

2C6

2C3

C2

3C2'

3C2"

 

6

–2

0

–2

0

0

Answer

The entire character table is:

D6

E

2C6

2C3

C2

3C2'

3C2"

a1

1

1

1

1

1

1

a2

1

1

1

1

–1

–1

b1

1

–1

1

–1

1

–1

b2

1

–1

1

–1

–1

1

e1

2

1

–1

–2

0

0

e2

2

–1

–1

2

0

0

 

 

 

 

 

 

 

 

6

–2

0

–2

0

0

n(a1) = [(1)(6)(1) + (2)(–2)(1) + (2)(0)(1) + (1)(–2)(1) + (3)(0)(1) + (3)(0)(1)]/12 = 0

n(a2) = [(1)(6)(1) + (2)(–2)(1) + (2)(0)(1) + (1)(–2)(1) + (3)(0)(–1) + (3)(0)(–1)]/12 = 0

n(b1) = [(1)(6)(1) + (2)(–2)(–1) + (2)(0)(1) + (1)(–2)(–1) + (3)(0)(1) + (3)(0)(–1)]/12 = 1

n(b2) = [(1)(6)(1) + (2)(–2)(–1) + (2)(0)(1) + (1)(–2)(–1) + (3)(0)(–1) + (3)(0)(1)]/12 = 1

n(e1) = [(1)(6)(2) + (2)(–2)(1) + (2)(0)(–1) + (1)(–2)(–2) + (3)(0)(0) + (3)(0)(0)]/12 = 1

n(e2) = [(1)(6)(2) + (2)(–2)(–1) + (2)(0)(–1) + (1)(–2)(2) + (3)(0)(1) + (3)(0)(1)]/12 = 1

The objects transform as b1 + b2 + e1 + e2