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Chimica Inorganica 3

Irreducible Representations and Character Tables

Rather than using geometrical operations, it is often much more convenient to employ a new set of group elements which are matrices and to make the rule of multiplication that of matrix multiplication.

The set of matrices forming the new elements are said to form a representation of the group.

A =

a11 a12 a13 a14 ! a1n

a21 a22 a23 a24 ! a2n

! ! ! ! ! !a

n1 an2 a

n3 an4 ! a

nn

!

"

#####

$

%

&&&&&

Chimica Inorganica 3

Two operators commute when A ! B = B ! A

C4 z( ) !! xz x1, y1, z1( )

"

C4 z( ) x1,#y1, z1( )

"

! d'

! xz !C4 z( ) x1, y1, z1( )

"

! xz y1, x1, z1( )

"

! d

Do C4 z( ) and !xz

commute?

Chimica Inorganica 3

c 11 =a11b 11 +a12b 21 +a13b 31 +…+a1nb n1 = a1kb k1k=1

n

!

c 12 =a11b 12 +a12b 22 +a13b 32 +…+a1nb n2 = a1kb k2k=1

n

!

c 21 =a21b 11 +a22b 21 +a23b 31 +…+a2nb n1 = a2kb k1k=1

n

!! ! !

cnn=a

n1b 1n +an2b 2n +an3b 3n +…+annbnn= a

nkbkn

k=1

n

!

"

#

$$$$$$

%

$$$$$$

a11 a12 a13 ! a1n

a21 a22 a23 ! a2n

! ! ! ! !a

n1 an2 a

n3 ! ann

!

"

#####

$

%

&&&&&

b 11 b 12 b 13 ! b 1nb 21 b 22 b 23 ! b 2n! ! ! ! !bn1 b

n2 bn3 ! b

nn

!

"

#####

$

%

&&&&&

=

c 11 c 12 c 13 ! c 1nc 21 c 22 c 23 ! c 2n! ! ! ! !cn1 c

n2 cn3 ! c

nn

!

"

#####

$

%

&&&&&

A !B = C

cij=a

i1b 1 j +ai2b 2 j +ai3b 3 j +…+ainbnj= a

ikbkj

k=1

n

!

Chimica Inorganica 3

0 1 2 11 1 0 02 0 0 11 0 1 2

!

"

###

$

%

&&&

1 1 0 01 0 1 00 1 0 00 0 0 1

!

"

###

$

%

&&&=

1 2 1 12 1 1 02 2 0 11 2 0 2

!

"

###

$

%

&&&

A B C

1 1 0 01 0 1 00 1 0 00 0 0 1

!

"

###

$

%

&&&

0 1 2 11 1 0 02 0 0 11 0 1 2

!

"

###

$

%

&&&=

1 2 2 12 1 2 21 1 0 01 0 1 2

!

"

###

$

%

&&&

B A D

Chimica Inorganica 3

A ! B !C( ) = A !B( ) !CB !C =D " d

ij= b

ikckj

k=1

n

#

A !D = E " eml= a

msdsl=

s=1

n

# ams

bstctl

t=1

n

# =s=1

n

# ams

s=1

n

#t=1

n

# bstctl

A !B = F " fmt= a

msbst

s=1

n

#

F !C =G " gml= f

mtctl

t=1

n

# = amsbst

s =1

n

# ctl

t =1

n

#G = E

Chimica Inorganica 3

a11 a12 a13 ! a1n

a21 a22 a23 ! a2n

! ! ! ! !a

n1 an2 a

n3 ! ann

!

"

#####

$

%

&&&&&

b 11b 21!bn1

!

"

#####

$

%

&&&&&

=

c 11c 21!cn1

!

"

#####

$

%

&&&&&

c 11 = a1kb k1 ;k=1

n

' ci1 = a

ikbk1

k=1

n

'

Let us now consider an elementary use of the operation of a matrix on a vector

xy

z

!!!

134

"

#

$$$

%

&

'''

1 0 11 1 01 1 0

!

"

###

$

%

&&&

Chimica Inorganica 3

1 0 11 1 01 1 0

!

"

###

$

%

&&&

134

!

"

###

$

%

&&&=

544

!

"

###

$

%

&&&

'''

x 'y 'z '

!

"

###

$

%

&&&

Let us now consider matrices corresponding to symmetry operation of the C3v group:

E,C3,C

32 ,!

v, ˆ !!

v, ˆ !!!

v

The basis set is described by the triangles vertices, points A, B and C. The transformation properties of these points under the symmetry operations of the group are:

Chimica Inorganica 3

EABC

!

"

###

$

%

&&&=

ABC

!

"

###

$

%

&&&=

1 0 00 1 00 0 1

!

"

###

$

%

&&&

ABC

!

"

###

$

%

&&&

ˆ !!!v

ABC

"

#

$$$

%

&

'''=

CBA

"

#

$$$

%

&

'''=

0 0 10 1 01 0 0

"

#

$$$

%

&

'''

ABC

"

#

$$$

%

&

'''

ˆ !!v

ABC

"

#

$$$

%

&

'''=

BAC

"

#

$$$

%

&

'''=

0 1 01 0 00 0 1

"

#

$$$

%

&

'''

ABC

"

#

$$$

%

&

'''

C32

ABC

!

"

###

$

%

&&&=

CAB

!

"

###

$

%

&&&=

0 0 11 0 00 1 0

!

"

###

$

%

&&&

ABC

!

"

###

$

%

&&&

C3

ABC

!

"

###

$

%

&&&=

BCA

!

"

###

$

%

&&&=

0 1 00 0 11 0 0

!

"

###

$

%

&&&

ABC

!

"

###

$

%

&&&

!v

ABC

!

"

###

$

%

&&&=

ACB

!

"

###

$

%

&&&=

1 0 00 0 10 1 0

!

"

###

$

%

&&&

ABC

!

"

###

$

%

&&&

Chimica Inorganica 3

E xy

!

"#

$

%& =

xy

!

"#

$

%& =

1 00 1

!"#

$%&

xy

!

"#

$

%&

!v' x

y

!

"#

$

%& =

' 12 x '

32 y

12 y '

32 x

!

"##

$

%&&=

' 12 ' 3

2

' 32

12

!

"##

$

%&&

xy

!

"#

$

%&

C3xy

!

"#

$

%& =

' 12 x +

32 y

' 12 y '

32 x

!

"##

$

%&&=

' 12

32

' 32 ' 1

2

!

"##

$

%&&

xy

!

"#

$

%&

!v

xy

!

"#

$

%& =

'xy

!

"#

$

%& =

'1 00 1

!"#

$%&

xy

!

"#

$

%&

If we would have concentrated on coordinates (the z coordinate can be disregarded)

C32 x

y

!

"#

$

%& =

' 12 x '

32 y

' 12 y +

32 x

!

"##

$

%&&=

' 12 ' 3

2

32 ' 1

2

!

"##

$

%&&

xy

!

"#

$

%& !

v" x

y

!

"#

$

%& =

' 12 x +

32 y

12 y +

32 x

!

"##

$

%&&=

' 12

32

32

12

!

"##

$

%&&

xy

!

"#

$

%&

Chimica Inorganica 3

Which is the difference between the two sets of matrices? The former set is not the simplest one. In particular matrices of the first set may be reduced through a similarity transformation.

Similarity transformations yield irreducible representations, Γi, which lead to the useful tool in group theory – the character table. The general strategy for determining Γi is as follows: A, B and C are matrix representations of symmetry operations of an arbitrary basis set (i.e., elements on which symmetry operations are performed). There is some similarity transform operator V such that

ˆV !1 "A " ˆV =A *

ˆV !1 "B " ˆV =B *

ˆV !1 "C " ˆV =C *

V uniquely produces block-diagonalized matrices, which are matrices possessing square arrays along the diagonal and zeros outside the blocks

A * =A1

A2

A3

!

"

###

$

%

&&&B * =

B1

B2

B3

!

"

####

$

%

&&&&

C * =C1

C2C3

!

"

###

$

%

&&&

Chimica Inorganica 3

Matrices A, B, and C are reducible. Sub-matrices Ai, Bi and Ci obey the same multiplication properties as A, B, and C . If application of the similarity transform does not further block-diagonalize A*, B*, and C*, then the blocks are irreducible representations. The character is the sum of the diagonal elements of Γi.

E =1 0 00 1 00 0 1

!

"

###

$

%

&&&

Sv" =

0 0 10 1 01 0 0

!

"

###

$

%

&&&

Sv' =

0 1 01 0 00 0 1

!

"

###

$

%

&&&

C 32

0 0 11 0 00 1 0

!

"

###

$

%

&&&

C 3 =0 1 00 0 11 0 0

!

"

###

$

%

&&&

Sv =1 0 00 0 10 1 0

!

"

###

$

%

&&&

Chimica Inorganica 3

Let us consider matrices V and V-1. Do not care how we obtained V!!!

V =

13

26 0

13 ! 1

612

13 ! 1

6 ! 12

"

#

$$$$

%

&

''''

; V !1 =

13

13

13

26 ! 1

6 ! 16

0 12 ! 1

2

"

#

$$$$

%

&

''''

V =

13

26 0

13 ! 1

612

13 ! 1

6 ! 12

13

13

13

26

! 16

! 16

= 16 + 2

6 + 0 ! 0 + 16 + 2

6 =1

C =

13

26 0

13 ! 1

612

13 ! 1

6 ! 12

"

#

$$$$

%

&

''''

; C T =

13

13

13

26 ! 1

6 ! 16

0 12 ! 1

2

"

#

$$$$

%

&

''''

=V !1

Chimica Inorganica 3

Let us consider matrices V and V-1. Do not care how we obtained V!!!

V =

13

26 0

13 ! 1

612

13 ! 1

6 ! 12

"

#

$$$$

%

&

''''

; V !1 =

13

13

13

26 ! 1

6 ! 16

0 12 ! 1

2

"

#

$$$$

%

&

''''

V !1 "C3 "V =

13

13

13

26 ! 1

6 ! 16

0 12 ! 1

2

#

$

%%%%

&

'

((((

"0 1 00 0 11 0 0

#

$

%%%

&

'

((("

13

26 0

13 ! 1

612

13 ! 1

6 ! 12

#

$

%%%%

&

'

((((

=

13

13

13

26 ! 1

6 ! 16

0 12 ! 1

2

#

$

%%%%

&

'

((((

"

13 ! 1

612

13 ! 1

6 ! 12

13

26 0

#

$

%%%%

&

'

((((

=1 0 00 ! 1

232

0 ! 32 ! 1

2

=C3*

A matrix possessing square arrays along the diagonal and zeros outside the blocks

Chimica Inorganica 3

Let us consider matrices V and V-1. Do not care how we obtained V!!!

V !1 "E "V = E* =1 0 00 1 00 0 1

#

$

%%%

&

'

(((

C3* =

1 0 00 ! 1

232

0 ! 32 ! 1

2

#

$

%%%

&

'

(((

V !1C32 "V = C3

2( )* =1 0 00 ! 1

2 ! 32

0 32 ! 1

2

#

$

%%%

&

'

(((

V !1 "!v"V = !

v( )* =1 0 00 1 00 0 !1

#

$

%%%

&

'

(((

V !1 " )!v"V = )!

v( )* =1 0 00 ! 1

232

0 32

12

#

$

%%%

&

'

(((

V !1 " ))!v"V = ))!

v( )* =1 0 00 ! 1

2 ! 32

0 ! 32

12

#

$

%%%

&

'

(((

As above, the block-diagonalized matrices do not further reduce under reapplication of the similarity transform. All are Γirrs.

Chimica Inorganica 3

A 3 × 3 reducible representation, Γred, has been decomposed under a similarity transformation into a 1 (1 × 1) and 1 (2 × 2) block-diagonalized irreducible representations, Γi. The traces (i.e. sum of diagonal matrix elements) of the Γi’s under each operation yield the characters (indicated by χ) of the representation. Taking the traces of each of the blocks:

Note: characters of operators in the same class are identical

This collection of characters for a given irreducible representation, under the operations of a group is called a character table. As this example shows, from a completely arbitrary basis and a similarity transform, a character table is born.

Chimica Inorganica 3

The triangular basis set does not uncover all Γirr of the group defined by {E, C3, C32, σv,

σv’, σv’’}. A triangle represents Cartesian coordinate space (x, y, z) for which the Γis were determined. May choose other basis functions in an attempt to uncover other Γis. For instance, consider a rotation about the z-axis

The transformation properties of this basis function, Rz, under the operations of the group (will choose only 1 operation from each class, since characters of operators in a class are identical):

Chimica Inorganica 3

E: Rz → Rz C3: Rz → Rz σv(xy): Rz → -Rz

These transformation properties generate a Γi that is not contained in a triangular basis. A new (1 x 1) basis is obtained, Γ3, which describes the transform properties for Rz. A summary of the Γi

for the group defined by E, C3, C32, σv, σv’, σv” is:

from triangular basis, i.e. (x, y, z)

from Rz

Is this character table complete? Irreducible representations and their characters obey certain algebraic relationships. From these 5 rules, we can ascertain whether this is a complete character table for these 6 symmetry operations.

Chimica Inorganica 3

Rule 1 The sum of the squares of the dimensions, , of irreducible representation Γi is equal to the order, h, of the group.

Five important rules govern irreducible representations and their characters:

Since the character under the identity operation is equal to the dimension of Γi (since E is always the unit matrix), the rule can be reformulated as:

!i2 =

i! h

! E( )( )i

2=

i! h

Character under E

With reference to the previous example: ! E( )( )i2=

i! 1( )2 + 2( )2 + 1( )2 =1+ 4 +1= 6

Chimica Inorganica 3

Rule 2 The sum of squares of the characters of irreducible representation Γi equals h

Five important rules govern irreducible representations and their characters:

!iR( )( )2 =

R! h

Character of Γi under operation R

With reference to the previous example:

!A1R( )( )2 =

R! 1( )2

E+ 2 " 1( )2

C3+C32+ 3" 1( )2

" v+" v' +" v

"=1+ 2 + 3 = 6

!A2R( )( )2 =

R! 1( )2

E+ 2 " 1( )2

C3+C32+ 3" #1( )2

" v+" v' +" v

"=1+ 2 + 3 = 6

!E R( )( )2 =R! 2( )2

E+ 2 " #1( )2

C3+C32+ 3" 0( )2

" v+" v' +" v

"= 4 + 2 + 0 = 6

Chimica Inorganica 3

Rule 3 Vectors whose components are characters of two different irreducible representations are orthogonal

Five important rules govern irreducible representations and their characters:

!iR( )!" #$ !

jR( )!

"#$ =

R% 0 for i & j

With reference to the previous example:

!A1R( )!" #$ !A2

R( )!" #$ =R% 1( ) & 1( )

E+ 2 & 1( ) & 1( )

C3+C32

+ 3& 1( ) '1( )" v+" v

' +" v"

=1+ 2 ' 3 = 0

!A1R( )!" #$ !E R( )!" #$ =

R% 1( ) & 2( )

E+ 2 & 1( ) & '1( )

C3+C32

+ 3& 1( ) 0( )" v+" v

' +" v"= 2 ' 2 + 0 = 0

!A2R( )!" #$ !E R( )!" #$ =

R% 1( ) & 2( )

E+ 2 & 1( ) & '1( )

C3+C32

+ 3& '1( ) 0( )" v+" v

' +" v"

= 2 ' 2 + 0 = 0

Chimica Inorganica 3

Rule 4 For a given representation, characters of all matrices belonging to operations in the same class are identical

Five important rules govern irreducible representations and their characters:

Chimica Inorganica 3

Rule 5 The number of Γis of a group is equal to the number of classes in a group.

Five important rules govern irreducible representations and their characters:

Chimica Inorganica 3

With these rules one can algebraically construct a character table. Returning to our example, let’s construct the character table in the absence of an arbitrary basis: Rule 5: (E); (C3, C3

2) (σv, σv’, σv”) … 3 classes ∴ 3 Γis Rule 1: 12 + 22 + 32 = 6 ∴ 1 = 2 = 1, 3 = 2 Rule 3: All character tables have a totally symmetric representation. Thus one of the irreducible representations, Γi, possesses the character set χ1(E) = 1, χ1(C3, C3

2) = 1, χ1(σv, σv’, σv”) = 1. The application of rule 3 implies for the second one-dimensional irreducible representation 1 × χ1(E) × χ2(E) + 2 × χ1(C3) × χ2(C3) + 3 × χ1(σv) × χ2(σv) = 0 1 × 1 × χ2(E) + 2 × 1 × χ2(C3) + 3 × 1 × χ2(σv) = 0 1 × 1 × 1 + 2 × 1 × χ2(C3) + 3 × 1 × χ2(σv) = 0; 2χ2(C3) + 3χ2(σv) = -1 χ2(C3) = 1, χ2(σv) = -1

Chimica Inorganica 3

For the case of Γ3 (3 = 2) there is not a unique solution to Rule 3

1 × 1 × 2 + 2 × 1 × χ3(C3) + 3 × 1 × χ3(σv) = 0

We then need a second independent equation. 1 × 22 + 2 × [χ3(C3)]2 + 3 × [χ3(σv)]2 = 6 (Rule 2) We then obtain [χ3(C3)] = -1 and [χ3(σv)] = 0, i.e., the same result previously obtained

Chimica Inorganica 3

ak= 1h

! R( )!k R( )R=1

h

!

Given a reducible representation Γ, it is straightforward to get the number of times a specific irreducible representaion Γk contribute to Γ.

Let us assume as basis for a reducible representation Γ the 1s orbitals of three hydrogen atoms positioned at the vertices of a regular triangle. These orbitals may be labeled A, B, and C. Matrices corresponding to the different symmetry operations are:

Chimica Inorganica 3

E =1 0 00 1 00 0 1

!

"

###

$

%

&&&

sv" =

0 0 10 1 01 0 0

!

"

###

$

%

&&&

C 32

0 0 11 0 00 1 0

!

"

###

$

%

&&&

C 3 =0 1 00 0 11 0 0

!

"

###

$

%

&&&

sv =1 0 00 0 10 1 0

!

"

###

$

%

&&&

sv' =

0 1 01 0 00 0 1

!

"

###

$

%

&&&

C3v E 2C3 3σv Γ 3 0 1

ak= 1h

! R( )!k R( )R=1

h

!

aA1= 163!1+ 2 !0 !1+ 3!1!1( ) =1 a

E= 163!2+ 2 !0 ! "1( )+ 3!1!0( ) =1

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