what are reducible and irreducible representations, properties of irreps, mullikens notations

2.  Representation is a set of matrices which represent the
operations of a point group.
It can be classified in to two types,
1. Reducible representations
2. Irreducible representations
Examine what happens after the molecule undergoes
each symmetry operation in the point group
(E, C2, 2s)

3.  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).

4. • 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
diagonalized
To block diagonalize, make each nonzero element into a 1x1
matrix
 
 
 
 
 
 
 
 
 
 
 
 








 












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



















100
010
001
100
010
001
100
010
001
100
010
001
E C2 sv(xz) sv(yz)

5.  A block diagonal matrix
is a special type of
matrices, and it has
“blocks” of number
through its “diagonal”
and has zeros elsewhere.
5

















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

 


sincos0000
2120000
009000
00042cos
0005sin1
000436

6.  The trace of a matrix (χ)
is the sum of its
diagonal elements.
 χ = 31+2sinθ
6



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

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
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
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 


sincos0000
2120000
009000
00042cos
0005sin1
000436

i
iiaA)(

7. • 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

10.  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.

11.  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.