Symmetry - Methods of Molecular Quantum Mechanics - page 255

group, namely: A ¼ X 1 BX. The set of all conjugate elements defines a
class. Conjugate operations are always of the same type (rotations with
rotations, reflections with reflections, etc.). The number of classes equals
the number of irreps.

12.2.3 Representations and Characters

Let G f G 1

; ...; G h g be a group of h elements and fDð G 1 Þ; Dð G 2 Þ;
...; Dð G h Þg a group of matrices isomorphic to G. We then say that the
groupofmatrices gives a representation (German, darstellung, hence D )of
the abstract group. If we have a representation of a group in the form of a
groupofmatrices, thenwe also have an infinite number of representations.
In fact, we can always subject all matrices of a given representation to a
similarity transformation, thereby obtaining a new representation, and so
on, the multiplication rule being preserved during the similarity transfor-
mation. If, by applying a similarity transformationwitha unitarymatrix U
toarepresentationofagroupGinthe formofagroupofmatrices,weobtain
a new representation whose matrices have a block-diagonal form, we say
that the representation has been reduced.
The set of functions that are needed to find a (generally,

; G 2

reducible)

representation forms a basis for the representation. The functions

forming a basis for the irreducible representations (irreps) of a symmetry
group are said to be symmetry-adapted functions, and transform in the
simplest and characteristic way under the symmetry operations of the
group. It is of basic importance in quantum chemistry to find such
symmetry-adapted functions, starting from a given basis set through use
of suitable projection operators, as we shall see.
As already said before, the character is the trace of the matrix repre-
sentative of the symmetry operator

G

R and is denoted by

x

(R). The

characters have the following properties:

1. They are invariant under any transformation of the basis.
2. They are the same for all symmetry operations belonging to the same
class.
3. The condition for two representations to be equivalent is that they
have the same characters.

12.2.4 Three Theorems on Irreducible Representations

1. The necessary and sufficient condition that a representation be

irreducible is that the sum over all operations R of the group of the

G