Group Theory II. Today Repetition Block matrices Character tables The great and little orthogonality theorems Irreducible representations Basis functions

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 speciesProjection 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.

00

00

000

000

00

00

00''0

000''

00

00

00'0

000'

B

A

B

A

B

A

C’ C’’ C

Block Matrices

A’A’’=AB’B’’=BC’C’’=C

Block matrices are good

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

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.

Reducing big matrices to block diagonal form is always possible but not easy. Fortunately we do not have to do this ourselves.

Character Tables

The C3v character table

000112

111111

111111

'''

3

2

1

2333

vvvv CCEC

Irreducible representations

Symmetry operations

The order h is 6There are 3 classes

Character Tables

Operations belonging to the same class will have the same character so we can write:

012

111

111

32

3

2

1

33

vv CEC

Irreducible representations (symmetry species)

Classes

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

”

Read more about it in section 5.10.

''')'(''

)( )(*)( jjiillR l

lji

lij d

hRDRD

Here’s a smaller one,

where (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

since all operations belonging to the same class have the same character.

The Little Orthogonality Theorem

')()( )(*)( ll

R

ll hRR

')()( )(*)()( ll

c

ll hcccg

character Tables

There is a number of useful properties of character tables:

012

111

111

32

3

2

1

33

vv CEC

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.


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!


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

)]3(),3([),,(),(),(),,(00202

11111

)(11111

11111

,11111

222

2222222

22221

2

22221

244

yxyyxxyzxzyzxzRRyxE

xyzxyB

yxzyxB

RA

zzyxzA

CCEC

yx

z

dvv

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

)]3(),3([),,(),(),(),,(00202

11111

)(11111

11111

,11111

222

2222222

22221

2

22221

244

yxyyxxyzxzyzxzRRyxE

xyzxyB

yxzyxB

RA

zzyxzA

CCEC

yx

z

dvv

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

)]3(),3([),,(),(),(),,(00202

11111

)(11111

11111

,11111

222

2222222

22221

2

22221

244

yxyyxxyzxzyzxzRRyxE

xyzxyB

yxzyxB

RA

zzyxzA

CCEC

yx

z

dvv

A2 transforms like a rotation around z.

E+Rz

C4

+Rz

C2

+Rz

v

-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?

...)()()( )()( 21 RDRDRD


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.


How do we find the irreducible representation?Let’s use an old example from two weeks ago:

1 23

NC3v in the basis (Sn, S1, S2, S3)

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: =4 C3: =1 v: =2


1 23

NSo C3v in the basis (Sn, S1, S2, S3) will have the following characters for the different symmetry operations.

214

32 33

red

vv CEC


1 23

NSo C3v in the basis (Sn, S1, S2, S3) will have the following characters for the different symmetry operations.

012

111

111

214

32

2

1

33

E

A

A

CEC

red

vv Let’s add the character table of the irreducible representation

By inspection we findred=2A1+E


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).


From LOT we can derive the expression (see section 5.10)

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, red the character of the operation l in the reducible representation and i the character of l in the irreducible representation.

l

llredi cccg

ha )(*)()(

1 )()(

i

iired a


Let’s go back to our example again.

10231122416

1

01231121416

1

21231121416

1

2

1

E

A

A

a

a

a

So once again we find red=2A1+E

012

111

111

214

32

2

1

33

E

A

A

CEC

red

vv

l

llredi cccg

ha )(*)()(

1 )()(

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.

R

ll RRh

P ˆ)(1ˆ )(

where Pl is the projection operator of the irreducible representation l, (l) is the character of the operation R for the representation l and R means application of R to our 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 alwaysbe 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.


First find a basis for the molecule. Let’s take the cartesian coordinates for each atom. x1

x3

x2

y1

y2

y3

z1

z2

z3

Water belongs to the C2v group which contains the operations E, C2, v(xz) and v’(yz).

The representation becomes E C2 v(xz) v’(yz)

red 9 -1 1 3


Character table for C2v.

yzRyB

xzRxB

xyRA

zyxzA

yzxzCEC

x

y

vvv

,1111

,1111

1111

,,1111

)(')(

2

1

22

2221

22

3119

)(')(22

redvvv yzxzCEC

Now reduce red to a sum of irreducible representations. Use inspection or the formula.


yzRyB

xzRxB

xyRA

zyxzA

yzxzCEC

x

y

z

vvv

,1111

,1111

1111

,,1111

)(')(

2

1

2

2221

22

The representation reduces to red=3A1+A2+2B1+3B2

trans= A1+B1+B2

rot=A2+B1+B2

vib=2A1+B2Modes left for vibrations


yzRyB

xzRxB

xyRA

zyxzA

yzxzCEC

x

y

z

vvv

,1111

,1111

1111

,,1111

)(')(

2

1

2

2221

22

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 modesvib=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.

ki fOf |ˆ|

An integral will be non-zero only if the integrand belongs to the totally symmetric irreducible representation of the molecular point group.

Off ki ˆ

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

Documents

Group Theory II. Today Repetition Block matrices Character tables The great and little orthogonality theorems Irreducible representations Basis functions