Symmetry - Methods of Molecular Quantum Mechanics - page 252

Care must be taken in noting that the same operation R may have a
different effect when acting on a different basis. The transformation under
C a

of the d-functions in the xy-plane:

d x 2 -- y 2 / sin 2 d xy / sin 2

u cos 2 w; u sin 2 w ð 12 : 22 Þ

gives in fact

cos 2 a sin2 a

C a ð d x 2 -- y 2 d xy Þ¼ð C a d x 2 y 2 C a d xy Þ¼ð d x 2 y 2 d xy Þ

sin2 a cos 2 a

¼ð d x 2 -- y 2 d xy Þ D d ð C a Þ

ð 12 : 23 Þ

Therefore, it is always necessary to specify the basis which the matrix
representative refers to. These and other matrix representatives of
different symmetry operations may be found elsewhere (Magnasco,
2007).
As a last point on symmetry, it must be recalled that

then RS ¼ T in function space
and Dð R ÞDð S Þ¼Dð T Þ in matrix space

If RS ¼ T in coordinate space ;

ð 12 : 24 Þ

12.2 GROUP THEORETICAL METHODS

Let us now briefly introduce the axioms defining the concept of a
group.
An abstract group G f G 1 ; ...; G h g of order h is given by a closed set

of h elements satisfying the following properties:

; G 2

(i) There is a composition law (usually, but not necessarily, the
multiplication law) such that, for G r and G s belonging to G,
G r G s ¼ G t still belongs to G (we then say that a group is a set
closed with respect to symbolic multiplication).
(ii) The composition law is associative: ð G r G s Þ G t ¼ G r ð G s G t Þ .
(iii) There is an identity (or neutral) element G m , such that G r G m ¼
G m G r ¼ G r .
(iv) Each element has an inverse G 1 , such that G r G 1

¼ G r G r ¼

r r

G m .