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Irreducible Representations and Character Tables

K.Sridharan

Dean

School of Chemical & Biotechnology

SASTRA University

Thanjavur – 613 401

 

 

 

 

 

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TableofContents1 Irreducible representation and character tables ......................................................................... 4 

1.1  ‐ Irreducible representation and getting the symmetries of translations along the 

different axes ............................................................................................................................... 4 

1.1.1  Importance of irreducible representation ................................................................... 4 

1.1.2 Translation along the x,y and z‐axes and irreducible representation ................................ 4 

1.1.2.1 Translation along the x‐axis ............................................................................................. 4 

1.2  Rotation along the x‐axis ..................................................................................................... 6 

1.2.1 Rotation along the x‐axis .................................................................................................... 6 

1.3 C2v Character Table ................................................................................................................... 7 

2  Identifying the symmetries of translations along the axes in some inorganic molecules ...... 9 

2.1  D4h character table ........................................................................................................ 10 

2.1.1 Meaning of (x,y), (xz,yz) and (Rx,Ry) & C4 operation on the translation along the x‐axis  10 

2.1.2 Operation E on the translation along the x‐axis ............................................................... 11 

2.1.3 C2 operation on the translation along x‐axis .................................................................... 11 

2.1.4 Effect of C2’ operation on x‐ and y‐vectors ....................................................................... 12 

2.1.5 Effect of C2” on x‐ and y‐vectors ....................................................................................... 13 

2.1.6 Effect of operation i on x‐ and y‐vectors .......................................................................... 14 

2.1.7 Effect of S4 operation on the x‐ and y‐vectors ................................................................. 14 

2.1.8 Effect of h operation on x‐ and y‐vectors ....................................................................... 15 

2.1.9 Effect of v on x‐ and y‐vectors ......................................................................................... 15 

2.1.10 Effect of d on x‐ and y‐vectors ...................................................................................... 16 

2.1.11 Identifying the symmetry of translation about x‐axis .................................................... 17 

3.1 Deducing symmetries of rotation about the axes from irreducible representations ............. 18 

3.1.1 Rotation along the z‐axis .................................................................................................. 18 

3.1.1.1 Effect of E on rotation along the z‐axis ..................................................................... 18 

3.1.1.2 Effect of C2 on rotation about z‐axis .......................................................................... 18 

3.1.1.3 Effect of C2’ on rotation about z‐axis ......................................................................... 18 

3.1.1.4 Effect of C2” on rotation about z‐axis ......................................................................... 19 

3.1.1.5 Effect of i on rotation about z‐axis ............................................................................ 19 

3.1.1.6 Effect of S4 on rotation about z‐axis .......................................................................... 19 

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3.1.1.7 Effect of v on rotation about z‐axis ......................................................................... 20 

3.1.1.8 Effect of σd on rotation about z‐axis .......................................................................... 20 

3.1.1.9 Effect of σh on rotation about z‐axis .......................................................................... 20 

4 Applications of Irreducible Representations .............................................................................. 21 

5. References ................................................................................................................................. 24 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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1 Irreducible representation and character tables

1.1 - Irreducible representation and getting the symmetries of translations along the different axes

An irreducible representation contains characters which cannot be reduced

further to a simpler form. In other words, this is the simplest representation.of

characters of different symmetry operations.

1.1.1 Importance of irreducible representation The point group of a molecule consists of a number of symmetry operations.

These symmetry operations constitute a mathematical group. It means that they

exhibit interrelationship as a collection. These mathematical relationships help us

in breaking each group into its irreducible representation.

These irreducible representations help us in analyzing molecular properties such

as optical activity, dipole moments and electronic properties such as IR and

Raman spectroscopy, electronic spectroscopy etc. Dynamic properties such as

translation, rotation etc can also be transformed by symmetry operations of the

point group of the molecule.

1.1.2 Translation along the x,y and z-axes and irreducible representation Let us consider water molecule. It has point group C2v. The symmetry operations

of this point group are E, C2, σv(xz), and σv(yz). Now we can see how the

translation along the three axes is transformed by these symmetry operations.

1.1.2.1 Translation along the x-axis Translation is represented by an arrow along the respective axis for the atoms in

the given molecule. Let us consider water molecule as shown in Figure 1.1.2.1.

Z 

Y 

X

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O

HHx

y

zC2

Fig 1.1.2.1 Water molecule – translation along x-axis

Identity operation, E, does not change the directions of arrows. This is called

symmetric and the character is equal to +1

The C2 operation changes the directions of arrows 1800 opposite. This is called

antisymmetric and the character is equal to -1

O

H H

The σv(xz) operation is not changing the direction of the arrows along the x-axis

and hence the character is equal to +1

The σv(yz) operation is not changing the directions of arrows 1800 opposite.

Hence, the character is -1.

Thus the characters of the four symmetry operations can be represented as

follows:

Symmetry operations: E C2 σv(xz) σv(yz).

Characters: +1 -1 +1 -1 From the C2v character table, it is can be seen that this irreducible representation

belongs to B1 symmetry.

Similarly, it can be shown that translation along the y-axis represents B2

symmetry and along z-axis represents A1 symmetry.

The numbers are called characters. Since these numbers cannot be reduced to

lower values, they are called irreducible representations.

This translation operation holds good for p - orbitals also because they can be

compared to arrows: the lobe with positive sign is similar to the head and the

lobe with negative sign can be compared to the tail of an arrow. Hence, the

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symmetry of a px orbital will be the same as that of translation along the x-axis,

that of the py orbital will be the same as that of the translation along the y-axis,

and that of the pz orbital will be the same as that of the pz orbital.

 

1.2 Rotation along the x-axis

A curved arrow ( )  is taken as the basis vector for rotation to understand

the effect of different operations on it.

1.2.1 Rotation along the x-axis Identity operation, E, does not change the directions of arrows. This is called

symmetric and the character is equal to +1

The C2 operation changes the directions of arrows 1800 opposite. This is called

antisymmetric and the character is equal to -1

 

The σv(xz) operation changes the direction of the curved arrow and hence the

character is equal to -1.

The σv(yz) operation does not change the direction of the curved arrow and

hence the character is equal to +1.

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Thus the characters of the four symmetry operations can be represented as

follows:

Symmetry operations: E C2 σv(xz) σv(yz).

Characters: +1 -1 -1 +1 From the C2v character table, it is can be seen that this irreducible representation

belongs to B2 symmetry. That is, the rotation about the x-axis belongs to B2

symmetry. Similarly, it can be shown that the rotation about the y-axis belongs to

B1 symmetry and that about the z-axis belongs to A2 symmetry.

1.3 C2v Character Table  

C2v E C2 σv(xz) σv(yz) A1 1 1 1 1 z x2, y2, z2 A2 1 1 -1 -1 Rz xy B1 1 -1 1 -1 x, Ry xz B2 1 -1 -1 1 y, Rx yz

                                     II I III IV

 

On the left corner of the character table, the point group is shown. Any character

table has four main areas, I, II, III and IV.

Area I consists of the characters of the irreducible representations of the group.

Area II contains the Mulliken symbols. The meanings of 1.these symbols are

given below:

1. Symbols A and B are given to one dimensional representation, E to two

dimensional representation, and T to three dimensional representation.

2. When a one dimensional representation is symmetric with respect to

rotation by 2/n about the principal Cn axis, i.e., χ(Cn) = 1, symbol A is

given and B is given, if it is antisymmetric, χ(Cn) = -1.

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3. Subscripts 1 is attached to A and B, if the operation is symmetric to v or a

C2 perpendicular to the principal axis. and subscript 2 is attached if it is

antisymmetric .

4. Prime is attached to all letters,(A’, B’, etc.) if the operation is symmetric

with respect to h plane. Double prime is attached (A”, B”, etc.) if it is

antisymmetric.

5. If a group gas centre of inversion, then subscript ‘g’ is used if it is

symmetric with respect to inversion and subscript ‘u’ is used if it is

antisymmetric with respect to inversion.

Area III consists of symbols x, y, z, Rx, Ry, and Rz. These represent the

Cartesian coordinates and the rotations about the three axes. If two symbols

are placed within parentheses, [ex: (x,y), (Rx, Ry)], it means that both put

together form the basis and they cannot be separated.

Area IV contains the squares and binary products of the coordinates

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2 Identifying the symmetries of translations along the axes in some inorganic molecules

Example 1: PtCl42-

x

y

z

Pt Cl

Cl

Cl

Cl C'2, v

C'2, v

C"2, d

C"2, d

C4

 

Symmetry elements present:

1. One C4 axis perpendicular to the plane of the paper (i.e. molecular plane)

2. Four C2 axes (two along the Pt-Cl bonds, shown as C2’ and two along the

diagonals shown as C2”

3. One σh plane, that is the plane of the paper (molecular plane)

4. Two σv planes containing the C2’ . axes

5. Two σd planes containing the C2” . axes

Hence, the point group is D4h

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2.1 D4h character table  

D4h E 2C4 C2 2C2’ 2C2

” i 2S4 σh 2σv 2σd A1g 1 1 1 1 1 1 1 1 1 1 x2+y2, z2

A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz B1g 1 -1 1 1 -1 1 -1 1 1 -1 x2-y2 B2g 1 -1 1 -1 1 1 -1 1 -1 1 xy Eg 2 0 -2 0 0 2 0 -2 0 0 (Rx, Ry) (xz, yz) A1u 1 1 1 1 1 -1 -1 -1 -1 -1 A2u 1 1 1 -1 -1 -1 -1 -1 1 1 z B1u 1 -1 1 1 -1 -1 1 -1 -1 1 B2u 1 -1 1 -1 1 -1 1 -1 1 -1 Eu 2 0 -2 0 0 -2 0 2 0 0 (x,y)

2.1.1 Meaning of (x,y), (xz,yz) and (Rx,Ry) & C4 operation on the translation along the x-axis The symbol (x,y) means that translation along the x- and y-axes are inseparable

in a molecule with D4h symmetry and similarly operations on the the px and py

orbitals. The same explanation holds good for the rotation about x- and y-axes,

and the operations on the dxz and dyz orbitals.

Example:

Translation along the x-axis and along the y-axis are represented by arrows in

PtCl42- as shown in Figure 2.1.1.1

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 Fig 2.1.1.1 C4 operation & translation along the x- and y- axes in PtCl4

2-

Thus both the vectors have changed positions in the C4 operation and the

character of this operation is equal to zero, i.e., χ(C4) = 0. Also, the vectors x and

y are inseparable because when an operation is done on x-vector, y-vector is

also affected. Hence, x and y are put in parentheses and written as (x,y).

2.1.2 Operation E on the translation along the x-axis It is a doing nothing operation and the vectors are not disturbed from their

original positions. Hence, χ(E) = 2.

2.1.3 C2 operation on the translation along x-axis The effect of C2 on the translation along the x-axis is shown in Figure 2.1.3.1.

The x- and y-vectors (arrows) are shifted to their negative coordinates. Hence,

χ(C2) = -2.

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Fig 2.1.3.1 Effect of C2 on the translation along the x-axis

2.1.4 Effect of C2’ operation on x- and y-vectors

The C2’ operation converts the x-vector into its negative and the y-vector remains

unchanged. Hence, χ(C2’) = 0 

 

Fig 2.1.4.1 Effect of C2’ on the translation along the x-axis

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2.1.5 Effect of C2” on x- and y-vectors

The C2” operation interchanges the x- and y-vectors. Hence, χ(C2”) = 0

 

Fig 2.1.5.1 Effect of C2” on the translation along the x-axis

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2.1.6 Effect of operation i on x- and y-vectors The inversion operation, i,changes the x- and y-vectors into their negatives. Hence, χ(i) = -2

Fig 2.1.6.1 Effect of i on the translation along the x-axis

2.1.7 Effect of S4 operation on the x- and y-vectors This operation rotates the molecule by 900 and reflects in the molecular plane,

that is, the plane of the paper and χ(S4) = 0.

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Fig 2.1.7.1 Effect of S4 on the translation along the x-axis

2.1.8 Effect of h operation on x- and y-vectors Reflection in the h plane does not affect the x- and y-vectors. Hence, χ(h) = 2

2.1.9 Effect of v on x- and y-vectors The effect will be the same as that of C2

’ because v contains the C2’ axis.

Hence, χ(v) = 0.

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                                                    .

Fig 2.1.9.1 Effect of σv on the translation along the x-axis

2.1.10 Effect of d on x- and y-vectors Dihedral plane σd contains C2

”. Hence, the effect of σd will be the same as that of C2

”. Thus χ(d) = 0

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Fig 2.1.10.1 Effect of σd on the translation along the x-axis

2.1.11 Identifying the symmetry of translation about x-axis Now the characters of different operations can be given as follows:

D4h E 2C4 C2 2C2’ 2C2

” i 2S4 σh 2σv 2σd 2 0 -2 0 0 -2 0 2 0 0

This result is compared with D4h character table to find out the symmetry. It is found that the symmetry is Eu. This appears in the character table as follows:

D4h E 2C4 C2 2C2’ 2C2

” i 2S4 σh 2σv 2σd Eu 2 0 -2 0 0 -2 0 2 0 0 (x,y)

In the same way it can be shown that the translation about z-axis belongs to A2u symmetry.

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3.1 Deducing symmetries of rotation about the axes from irreducible representations A curved arrow is used as the base vector for rotation.

3.1.1 Rotation along the z-axis

3.1.1.1 Effect of E on rotation along the z-axis

 

Fig 3.1.1.1.1 Effect of C4 on rotation about the z-axis

3.1.1.2 Effect of C2 on rotation about z-axis This is nothing but doing C4 twice. The direction or the position of the arrow will

not be changed. Hence, χ(C2) = +1.

3.1.1.3 Effect of C2’ on rotation about z-axis

The effect is shown in Figure 3.1.1.3.1. The direction of the curved arrow is

changed and hence, χ(C2’) = -1

Pt ClCl

Cl

Cl

C4, z

Pt ClCl

Cl

Cl

C2'

-z

C2'

Fig 3.1.1.1.3 Effect of C2’ on rotation about the z-axis

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3.1.1.4 Effect of C2” on rotation about z-axis

The effect is the same as that of C2’. χ(C2

’’) = -1

3.1.1.5 Effect of i on rotation about z-axis The direction of the curved arrow is not changed and shown in Figure 3.1.1.5.1

χ(i’) = +1.

Fig 3.1.1.5.1 Effect of i on rotation about z-axis

3.1.1.6 Effect of S4 on rotation about z-axis

Pt ClCl

Cl

Cl

C4, z

C4

Pt ClCl

Cl

Cl

C4, z

h

Pt ClCl

Cl

Cl

Fig 3.1.1.6.1 Effect of S4 on rotation about z-axis

It is C4 operation followed by reflection in sh plane. The direction of the curved

arrow has not changed. Hence, χ(S4) = +1.

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3.1.1.7 Effect of v on rotation about z-axis The v plane contains the C2 axis. The direction of the arrow changes as shown

in Figure 3.1.1.7.1. Hence, χ(σv) = -1.

Fig 3.1.1.7.1 Effect of σv on rotation about z-axis

3.1.1.8 Effect of σd on rotation about z-axis The σd plane contains C2

” axis. The effect is the same as C2” operation.

χ(σd) = -1.

3.1.1.9 Effect of σh on rotation about z-axis The σh plane is the molecular plane and the direction of the arrow is not changed. Hence, χ(σh) = +1.

Now the characters of the different operation are grouped as under.

D4h E 2C4 C2 2C2’ 2C2

” i 2S4 σh 2σv 2σd 1 1 1 -1 -1 1 1 1 -1 -1

When this is compared with D4h character table, it is found that this has got A2g

symmetry.

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D4h E 2C4 C2 2C2’ 2C2

” i 2S4 σh 2σv 2σd A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz

Similarly, it can be shown that the rotation about x- and y-axis have Eg symmetry.

4 Applications of Irreducible Representations It is a representation which can be further reduced to irreducible form. At first, the

reducible representation for a molecule is derived and then it is reduced. From

this irreducible representation we can find out the representations covering the

translation, rotation and vibration and from this we can find out the IR active and

Raman active vibrations.

Example: Trans-N2F2

Step 1: Structure of the molecule

  Step 2:Symmetry elements present:

1. C2 axis

2. σh plane

3. i

Step 3: Hence, the point group is C2h

Step 4:The C2h character table is given below

C2h E C2 i σh

Ag 1 1 1 1 Rz x2, y2, z2, xy

Bg 1 -1 1 -1 Rx, Ry xz, yz

Au 1 1 -1 -1 z

Bu 1 -1 -1 1 x, y

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Step 5:The number of operations in this group is four:

E, C2, i, σh (as shown by the character table).

Step 6: The characters of the different operations are found out as follows:

Identity operation, E

All the 12 vectors (x,y,z) of the four atoms of the molecule are not disturbed.

Hence, the character, χ(E) = 12

C2 operation

All the four atoms are disturbed from their original places and occupy new

positions.

Hence, the character, χ(C2) = 0

i operation (inversion)

All the four atoms are displaced from their original positions to their new

positions.

Hence, χ(i) = 0

σh operation (reflection in the horizontal plane of symmetry)

All the four atoms retain their original positions. Nothing is changed.

Let us consider the three x, y and z vectors (arrows) of one fluorine atom. When

reflected in the σh plane (i.e. plane of paper), x and z arrows are not affected,

while the y-arrow is inverted.

Thus,

Old x = new x ; character = +1

Old y = - new y; character = -1

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Old z = new z; character = +1

Net character for one atom = +1

Hence, for four atoms, the total character will be equal to 4(+1) = +4.

Hence, χ(σh) = 4.

Step 7: Hence the reducible representation is:

C2h E C2 i σh

Γ 12 0 0 4 Step 8: This is reduced to get the components of this reducible representation

Ag = 1/4 [ (12)(1)(1) + 0 + 0 + (4)(1)(1) ] = 4

Bg = 1/4 [ (12)(1)(1) + 0 + 0 + (4)(-1)(1) ] = 2

Au = 1/4 [ (12)(1)(1) + 0 + 0 + (4)(-1)(1) ] = 2

Bu = 1/4 [ (12)(1)(1) + 0 + 0 + (4)(1)(1) ] = 4

Thus, Γ = 4Ag + 2Bg + 2Au + 4Bu

From the character table, Au represents translation along the z-axis and Bu

represents that along the x- and y-axes. Thus, translation is given by Au + 2Bu.

Similarly, rotations are covered by the representations, Ag + 2Bg.

Translation + rotation are covered by Au + 2Bu + Ag + 2Bg.

This is subtracted from the total representation to find out the normal vibrations:

(4Ag + 2Bg + 2Au + 4Bu) . (Au + 2Bu + Ag + 2Bg) = 3Ag + Au + 2Bu

Of these,

IR active vibrations are Au + 2Bu = 3

Raman active vibrations are 3Ag = 3

Total vibrations = 6

The molecule is non-linear.

Hence, the number of expected vibrations =(3N-6) = (3x4.6) = 6.

Hence, this is correct.

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5. References 1. “Inorganic Chemistry: Principles of Structure and Reactivity”, James

E.Huheey, Ellen A.Keiter, Richard L.Keiter, Okhil K.Medhi, Pearson

Education, Delhi, 2006

2. ‘Chemical Applications of Group Theory”, 2/e, F.Albert Cotton, Wiley

Eastern, New Delhi, 1986