Reducible and irreducible

representations

Udhay kironM.Tech (NST)

2013-15 BATCH

Representation is a set of matrices which represent the operations of a point group.

It can be classified in to two types,1. Reducible representations2. Irreducible representations

Reducible and irreducible representations

Examine what happens after the molecule undergoes each symmetry operation in the point group (E, C2, 2s)

Let us consider the C2h point group as an example. E, C2 ,sh & I are the four symmetry operations present in the group. . The matrix representation for this point group is give below.

In the case of C2h symmetry, the matrices can be reduced to simpler matrices with smaller dimensions (1×1 matrices).

Reducible representation:• A representation of higher dimension which can be reduced in to representation of lower dimension is called reducible representation.•Reducible representations are called block- diagonal matrices.Eg: Each matrix in the C2v matrix representation can be block diagonalizedTo block diagonalize, make each nonzero element into a 1x1 matrix

100

010

001

100

010

001

100

010

001

100

010

001

E C2 sv(xz) sv(yz)

A block diagonal matrix is a special type of matrices, and it has “blocks” of number through its “diagonal” and has zeros elsewhere.

5

Block Diagonal Matrices

sincos0000

2120000

009000

00042cos

0005sin1

000436

The trace of a matrix (χ) is the sum of its diagonal elements.

χ = 31+2sinθ

6

The Trace of a Matrix

sincos0000

2120000

009000

00042cos

0005sin1

000436

i

iiaA)(

• Because the sub-block matrices can’t be further reduced, they are called “irreducible representations”. The original matrices are called “reducible representations”.

Irreducible Representations:If it is not possible to perform a similarity

transformation matrix which will reduce the matrices of representation T, then the representation is said to be irreducible representation.

In general all 1 D representations are examples of irreducible representations.

• The symbol Γ is used for representations where: Γred = Γ1 Γ2 … Γn

Properties of irreducible representations:

1. Dimension of the irreducible representations: If it is uni dimensional (character of E=1), term A or B is

used. For a two dimensional representation, term E is used. If it is 3-D term T is used.

2. Symmetry with respect to principle axis: If the 1-D irrep is symmetrical with respect to the

principle axis () [i.e., the character of the operation is +1], the term A is used. However if the 1-D represent is unsymmetrical with respect to the principal axis (i.e., the character of is -1) the term B is used.

3. Symmetry with respect to subsidiary axis or plane: If the irrep is symmetrical with respect to the subsidiary

axis, or in it absence to plane, subscripts, , ,are used if it is unsymmetrical subscripts ,,, are used.

Symbols used for representing irreducible representation:

4. Prime and double prime marks are used over the symbol of the irrep to indicate its symmetry or anti symmetry with respect to the horizontal plane ().

5. If there is a centre of symmetry in the molecule, subscripts g and u are used to indicate the symmetry or anti symmetry of irreps res. Suppose the point group has no centre of symmetry, g or u subscripts are not used. Term ‘g’ stands for gerade (centrosymmetric) & ‘u’ for ungerade ( non-centrosymmetric).

On the basis of the above, symbols can be assigned to the irreducible representations of point group.