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Group Theory II
Group Theory II
Today
 Repetition
 Block matrices
 Character tables
 The great and little orthogonality theorems
 Irreducible representations
 Basis functions and Mulliken symbols
 How to find the symmetry species
Projection operator
 Applications
Repetition
We already know…
 Symmetry operations obey the laws of group theory.
 Great, we can use the mathematics of group theory.
 A symmetry operation can be represented by a matrix
operating on a base set describing the molecule.
 Different basis sets can be choosen, they are
connected by similarity transformations.
 For different basis sets the matrices describing the
symmetry operations look different. However, their
character (trace) is the same!
Repetition
We already know…
 Matrix representations of symmetry operations can often
be reduced into block matrices. Similarity transformations
may help to reduce representations further. The goal is to
find the irreducible representation, the only representation
that can not be reduced further.
 The same ”type” of operations (rotations, reflections etc)
belong to the same class. Formally R and R’ belong to the
same class if there is a symmetry operation S such that
R’=S-1RS. Symmetry operations of the same class will
always have the same character.
Block Matrices
Block matrices are good
 A' 0 0 0  A' ' 0 0 0  A
 0 B ' 0 0  0 B ' ' 0 0  0



0 0
 0
 0
0
C’  
C’’  

0
0 0
 0
 0
A’A’’=A
B’B’’=B
C’C’’=C
0 0 0
B 0 0

0
C 
0

Block Matrices
If a matrix representing a symmetry operation is
transformed into block diagonal form then each little block
is also a representation of the operation since they obey
the same multiplication laws.
When a matrix can not be reduced further we have
reached the irreducible representation. The number of
reducible representations of symmetry operations is
infinite but there is a small finite number of irreducible
representations.
The number of irreducible representations is always
equal to the number of classes of the symmetry point
group.
Group Theory II
Reducing big matrices to block diagonal form is always
possible but not easy. Fortunately we do not have to do
this ourselves.
As stated before all representations of a certain symmetry
operation have the same character and we will work with
them rather than the matrices themselves. The
characters of different irreducible representations of point
groups are found in character tables. Character tables
can easily be found in textbooks.
Character Tables
The C3v character table
Symmetry operations
C3v
1
2
3
E C3 C32  v  v '  ' 'v
1 1
1
1
1
1
1 1
1 1 1 1
2 1 1 0
0
0
Irreducible
representations
The order h is 6
There are 3 classes
Character Tables
Operations belonging to the same class will have the
same character so we can write:
Classes
C3v
1
2
3
E 2C3 3 v
1
1
1
1
1
1
2 1
0
Irreducible representations
(symmetry species)
The Great Orthogonality Theorem
”Consider a group of order h, and let D(l)(R) be the
representative of the operation R in a dl-dimensional
irreducible representation of symmetry species (l) of the
group. Then
h
D ( R) * D ( R)   ll ' ii' jj'

dl
R
(l )
ij
( l ')
i' j'
Read more about it in section 5.10.
”
The Little Orthogonality Theorem
Here’s a smaller one,
(l )
(l )
c
(
R
)
*
c
( R)  h ll'

R
where c(l)(R) is the character of the operation (R). Or
even more simple if the number of symmetry
operations in a class c is g(c). Then
 g ( c) c
(l )
(c) * c (c)  h ll'
(l )
c
since all operations belonging to the same class have
the same character.
character Tables
There is a number of useful properties of character tables:
1. The sum of the squares of the dimensionality of all the irreducible
representations is equal to the order of the group
2. The sum of the squares of the absolute values of characters of any
irreducible representation is equal to the order of the group.
3. The sum of the products of the corresponding characters of any two
different irreducible representations of the same group is zero.
4. The characters of all matrices belonging to the operations in the
same class are identical in a given irreducible representation.
5. The number of irreducible representations in a
group is equal to the number of classes of that
group.
C3v
1
2
3
E 2C3 3 v
1
1
1
1
1
1
2 1
0
Irreducible representations
Each irreducible representation of a group has a label called a
symmetry species, generally noted . When the type of
irreducible representation is determined it is assigned a
Mulliken symbol:
One-dimensional irreducible representations are called A or B.
Two-dimensional irreducible representations are called E.
Three-dimensional irreducible representations are called T (F).
The basis for an irreducible representation is said to span the
irreducible representation.
Don’t mistake the operation E for the Mulliken symbol E!
Irreducible representations
The difference between A and B is that the character for a
rotation Cn is always 1 for A and -1 for B.
The subscripts 1, 2, 3 etc. are arbitrary labels.
Subscripts g and u stands for gerade and ungerade,
meaning symmetric or antisymmetric with respect to
inversion.
Superscripts ’ and ’’ denotes symmetry or antisymmetry
with respect to reflection through a horizontal mirror plane.
character Tables
Example: The complete C4v character table
C4 v
E 2C4
C2
2 v
2 d
A1
1
1
1
1
1
z
A2
1
1
1
1
1
Rz
B1
1
1
1
1
B2
1
1
1
1
E
2
0
2 0
x2  y2 , z 2
z2
1
x2  y2
z( x 2  y 2 )
1
xy
xyz
0
( x, y ), ( Rx , R y ) ( xz, yz)
( xz 2 , yz 2 ),[ x( x 2  3 y 2 ), y (3x 2  y 2 )]
These are basis functions for the irreducible
representations. They have the same symmetry
properties as the atomic orbitals with the same
names.
character Tables
Example: The complete C4v character table
C4 v
E 2C4
C2
2 v
2 d
A1
1
1
1
1
1
z
A2
1
1
1
1
1
Rz
B1
1
1
1
1
B2
1
1
1
1
E
2
0
2 0
x2  y2 , z 2
z2
1
x2  y2
z( x 2  y 2 )
1
xy
xyz
0
( x, y ), ( Rx , R y ) ( xz, yz)
( xz 2 , yz 2 ),[ x( x 2  3 y 2 ), y (3x 2  y 2 )]
A1 transforms like z.
E does nothing, C4 rotates 90o about the z-axis, C2 rotates
180o about the z-axis, v reflects in vertical plane and d in a
diagonal plane.
character Tables
C4 v
E 2C4
C2
2 v
2 d
A1
1
1
1
1
1
z
A2
1
1
1
1
1
Rz
B1
1
1
1
1
B2
1
1
1
1
E
2
0
2 0
x2  y2 , z 2
z2
1
x2  y2
z( x 2  y 2 )
1
xy
xyz
0
( x, y ), ( Rx , R y ) ( xz, yz)
( xz 2 , yz 2 ),[ x( x 2  3 y 2 ), y (3x 2  y 2 )]
A2 transforms like a rotation around z.
E
+Rz
C4
+Rz
v
C2
+Rz
-Rz
d
-Rz
Reducible to Irreducible representation
Given a general set of basis functions describing a
molecule, how do we find the symmetry species of the
irreducible representations they span?
D( R)  D
( 1 )
( R)  D
( 2 )
( R)  ...
Reducible to Irreducible representation
If we have an interesting molecule there is often a natural
choice of basis. It could be cartesian coordinates or
something more clever.
From the basis we can construct the matrix
representations of the symmetry operations of the point
group of the molecule and calculate the characters of the
representations.
Reducible to Irreducible representation
How do we find the irreducible representation?
Let’s use an old example from two weeks ago:
N
C3v in the basis (Sn, S1, S2, S3)
1
3
To find the characters of the symmetry operations we look
at how many basis elements ”fall onto themselves” (or
their negative self) after the symmetry operation.
E: c=4
C3: c=1
v: c=2
2
Reducible to Irreducible representation
So C3v in the basis (Sn, S1, S2, S3) will
have the following characters for the
different symmetry operations.
C3v
red
E 2C3 3 v
4
1
2
N
1
3
2
Reducible to Irreducible representation
So C3v in the basis (Sn, S1, S2, S3) will
have the following characters for the
different symmetry operations.
Let’s add the
character table of
the irreducible
representation
N
1
3
2
C3v
E 2C3 3 v
red
4
1
2
A1
1
1
A2
E
1
2
1
1
1 By inspection we find
red=2A1+E
1
0
Reducible to Irreducible representation
The decomposition of any reducible representation into
irreducible ones is uniqe, so if you find combination that
works it is right.
If decomposition by inspection does not work we have to
use results from the great and little orthogonality
theorems (unless we have an infinite group).
Reducible to Irreducible representation
From LOT we can derive the expression (see section 5.10)
1
(l )
ai   g (c) c red (c) * c (l ) (c)
h l
where ai is the number of times the irreducible
representation i appears in red, h the order of the group, l
an operation of the group, g(c) the number of symmetry
operations in the class of l, cred the character of the
operation l in the reducible representation and ci the
character of l in the irreducible representation.
red   ai i
i
Reducible to Irreducible representation
1
(l )
ai   g (c) c red (c) * c (l ) (c)
h l
Let’s go back to our example again.
1
a A1  1 4 1  2 11  3  2 1  2
6
1
a A2  1 4 1  2 11  3  2 1  0
6
1
aE  1 4  2  2 11  3  2  0  1
6
So once again we find red=2A1+E
C3v
E 2C3 3 v
red
4
1
2
A1
1
1
1
A2
E
1
2
1
1
1
0
Projection Operator
Symmetry-adapted bases
The projection operator takes non-symmetry-adapted
basis of a representation and and projects it along new
directions so that it belongs to a specific irreducible
representation of the group.
1
l
ˆ
P   c (l ) ( R)  Rˆ
h R
where P^l is the projection operator of the irreducible
representation l, c(l) is the character of the operation R for
^ means application of R to our
the representation l and R
original basis component.
Applications?
Can all of this actually be useful?
Yes, in many areas for example when studying electronic
structure of atoms and molecules, chemical reactions,
crystallography, string theory (Lie-algebra) etc…
Let’s look at one simple example concering molecular
vibrations. Martin Jönsson will tell you a lot more in a
couple of weeks.
Molecular Vibrations
Water
Molecular vibrations can always
be decomposed into quite simple
components called normal modes.
Water has 9 normal modes of which 3 are translational, 3
are rotational and 3 are the actual vibrations.
Each normal mode forms a basis for an irreducible
representation of the molecule.
Molecular Vibrations
z3
First find a basis for the
molecule. Let’s take the
cartesian coordinates for each
atom.
x3
z1
y3
x2
z2
Water belongs to the C2v group which
contains the operations E, C2, v(xz) and v’(yz).
The representation becomes
E
C2
v(xz)
red
9
-1
1
v’(yz)
3
y2
x1
y1
Molecular Vibrations
Character table for C2v.
C2 v
A1
E C2  v ( xz)  'v ( yz)
1 1
1
1
z
x2 , y2 , z 2
A2
B1
1
1
1
1
1
1
1
1
R2
x, R y
xy
xz
B2
1
1
1
1
y , Rx
yz
C2v
red
E C2  v ( xz)  v ' ( yz)
9 1
1
3
Now reduce red to a sum of irreducible representations.
Use inspection or the formula.
Molecular Vibrations
C2 v
A1
E
1
C2  v ( xz)  'v ( yz)
1
1
1
A2
B1
1
1
1
1
1
1
B2
1
1
1
z
x2 , y2 , z2
1
1
Rz
x, R y
xy
xz
1
y , Rx
yz
The representation reduces to red=3A1+A2+2B1+3B2
trans= A1+B1+B2
rot=A2+B1+B2
vib=2A1+B2
Modes left for vibrations
Molecular Vibrations
C2 v
A1
E
1
C2  v ( xz)  'v ( yz)
1
1
1
A2
B1
1
1
1
1
1
1
B2
1
1
1
z
x2 , y2 , z2
1
1
Rz
x, R y
xy
xz
1
y , Rx
yz
Modes with translational symmetry will be infrared active
while modes with x2, y2 or z2 symmetry are Raman
active.
Thus water which has the vibrational modes
vib=2A1+B2 will be both IR and Raman active.
Integrals
A last example…
Integrals of product functions often appear in for
example quantum mechanics and symmetry analysis
can be helpful with them to.
fi | Oˆ | f k
An integral will be non-zero only if the integrand
belongs to the totally symmetric irreducible
representation of the molecular point group.
fi fk  Oˆ
Summary
 Molecules (and their electronic orbitals, vibrations etc)
are invariant under certain symmetry operations.
 The symmetry operations can be described by a
representation determined by the basis we choose to
describe the molecule.
 The representation can be broken up into its
symmetry species (irreducible representations).
 In character tables we find information about the
symmetry properties of the irreducible
representations.
More (and better) reading
The group theory chapter in Atkins is not very good (in my
opinion). More understandable descriptions can be found in:
Harris and Bertolucci, Symmetry and spectroscopy
Hargittai and Hargittai, Symmetry through the eyes of a
chemist