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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

RS-1RS. Symmetry operations of the same class

will always have the same character.

Block Matrices

Block matrices are good

C

C

C

AAA BBB CCC

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

The order h is 6 There are 3 classes

Irreducible representations

Character Tables

Operations belonging to the same class will have

the same character so we can write

Classes

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 G(l) of the group. Then

Read more about it

in section 5.10.

The Little Orthogonality Theorem

Heres a smaller one, 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 since all operations

belonging to the same class have the same

character.

character Tables

There is a number of useful properties of

character tables

- The sum of the squares of the dimensionality of

all the irreducible representations is equal to

the order of the group

- The sum of the squares of the absolute values of

characters of any irreducible representation is

equal to the order of the group.

- The sum of the products of the corresponding

characters of any two different irreducible

representations of the same group is zero.

- The characters of all matrices belonging to the

operations in the same class are identical in a

given irreducible representation.

- The number of irreducible representations in a

group is equal to the number of classes of that

group.

Irreducible representations

Each irreducible representation of a group has a

label called a symmetry species, generally noted

G. 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. Dont 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

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

A1 transforms like z. E does nothing, C4 rotates

90o about the z-axis, C2 rotates 180o about the

z-axis, sv reflects in vertical plane and sd in a

diagonal plane.

character Tables

A2 transforms like a rotation around z.

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?

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

C3v 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.

C3 c1

sv c2

E c4

Reducible to Irreducible representation

So C3v in the basis (Sn, S1, S2, S3) will have

the following characters for the different

symmetry operations.

Reducible to Irreducible representation

So C3v in the basis (Sn, S1, S2, S3) will have

the following characters for the different

symmetry operations.

Lets add the character table of the irreducible

representation

By inspection we find Gred2A1E

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) where ai is the number of times

the irreducible representation Gi appears in

Gred, 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.

Reducible to Irreducible representation

Lets go back to our example again.

So once again we find Gred2A1E

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.

where Pl is the projection operator of the

irreducible representation l, c(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

Lets 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

First find a basis for the molecule. Lets take

the cartesian coordinates for each atom.

Water belongs to the C2v group which contains

the operations E, C2, sv(xz) and sv(yz).

The representation becomes E C2

sv(xz) sv(yz) Gred

9

-1

1

3

Molecular Vibrations

Character table for C2v.

Now reduce Gred to a sum of irreducible

representations. Use inspection or the formula.

Molecular Vibrations

The representation reduces to Gred3A1A22B13B2

Gtrans A1B1B2

GrotA2B1B2

Modes left for vibrations

Gvib2A1B2

Molecular Vibrations

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

Gvib2A1B2 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.

An integral will be non-zero only if the

integrand belongs to the totally symmetric

irreducible representation of the molecular point

group.

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

(No Transcript)

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