Symmetry - Methods of Molecular Quantum Mechanics - page 251

+

C

y

α

x

′

y

′

′

′

f ( q ) f ( q )

q q ′

α

ϕ ϕ

0 x 0

Figure 12.1 The function transformed under the positive rotation C a is equal to the
function whose argument is transformed under the negative rotation C a

We obtain the transformations

w

0

¼ C a w ¼ wa;

C a w ¼ wþa

ð 12 :

18 Þ

C a p x ðwÞ¼ p x ð C a wÞ¼ p x ðwþaÞ¼ sin

u cos ðwþaÞ

¼ sin uð cos w cos a sin w sin aÞ¼ p x cos a p y sin a

ð 12

:

19 Þ

C a p y ðwÞ¼ p y ð C a wÞ¼ p y ðwþaÞ¼ sin u sin ðwþaÞ

¼ sin uð sin w cos aþ cos w sin aÞ¼ p x sin aþ p y cos a

ð 12 : 20 Þ

which can be written in matrix form as 5

cos a sin a

C a ð p x p y Þ¼ð C a p x C a p y Þ¼ð p x p y Þ ¼ð p x p y ÞD p ð C a Þ

sin a cos a

ð 12 :

21 Þ

5 The corresponding matrix representative for reflection across the plane specified by

s a is

.

cos 2 a sin 2 a

sin 2 a cos 2 a