Chem 373- Symmetry

8/3/2019 Chem 373- Symmetry

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Symmetry

Translation

Rotation

Reflection

Slide rotation (Sn)


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Lecture 36: Character Tables

The material in this lecture covers the following in Atkins.

15 Molecular Symmetry

Character tables

15.4 Character tables and symmetry labels(a) The structure of character tables

(b) Character tables and orbital degeneracy

(c) Characters and operators

Lecture on-line

Character Tables (PowerPoint)

Character tables (PDF)

Handout for this lecture


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Audio-visuals on-line

Symmetry (Great site on symmetry in art and science by MargretJ. Geselbracht, Reed College , Portland Oregon)

The World of Escher:

Wallpaper Groups: The 17 plane symmetry groups

3D Exercises in Point Group Symmetry


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We shall now turn our attention away fromthe symmetries of molecules themselves

and direct it towards the symmetry characteristics of :

1. Molecular orbitals 2. Normal modes of vibrations

This discussion will enable us to :

I. Symmetry label molecular orbitals

II. Discuss selection rules in spectroscopy

sageCharacter Table


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A rotation through 180 about the internuclear axis leaves thesign of a orbital unchanged

Simple caseCharacter Table

but the sign of a orbital is changed.

In the language introduced in this lectture:

The characters of the C2 rotation are +1 and -1 for the and

orbitals, respectively.


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A B

Symmetry label C (i.e. rotation by 18 )

1

- 1

2 0

C2

180

C2

180

Simple caseCharacter Table


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C v3 Character Table Structure of character table

Symmetry groupSymmetry Operations

A

C

B

E A

C

B


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C C C E3 3 3 =C C C3 33

1= C C E3

13

=

C31 A

C

B

B

A

C

C3 A

C

B

C

B

A


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A

C

B

A

B

C

v AC

B

C

A

B

v'

A

C

B

B

C

A

v"


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C v3 Character Table Classes of elements

In a group G={E,A,B,C,...},we say that two elements B and C

are conjugate to each other if :

ABA-1 = C,

for some element A in G.

An element and all its conjugatesform a class.

A

B C

v1

A

C B

v1

C3

B

A C

v1

C3v

B

A C

C3

1

B

A C=


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v vC C31

31 =

v vC C' '31

31 =

v vC C'' ''31

31 =

C C C C3 3 3

1

3

=C C C C3

13 3 3

=

EC E C31

3 =

We have in general:

Thus C3 and C3-1 form

a class of dimension 2

The

can

two elements C and C

be related to each other byand

3 3-1

v v' v' , , '

C Cv3 31=


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A

B CC3

1

B

C A

C31v

B

A C

C3

1C3 v

C

B A

=C

B A

v"

E Ev v =1

C Cv v3 31 = "

C Cv v31

3

= ' v v v v

=1

v v v v' ' " =1

v v" =1

In general

Thus

formv

and

a classof dimension

3. The elementsare related by

C and C

v v'

3 3-1

,

"

Elements conjugated to v ?


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v 'v ' 'v

C3 The symmetry operations are

grouped by classes withthe dimension of each classindicated

Also indicated is the dimensionof the group h

h = total number of symmetry

elements


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2

v2

Name of point group

Symmetry elements

E : identity

C2 : Rotation

(xz) mirror plane

'(yz) mirror plane

Number of symmetry

elements

Character Table Structure of character table


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v2

2 Name of irreduciblerepresentations

A1 A2 B1 B2


Characters of irreduciblerepresentations


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The px,py, and pz orbitals

on the central atom of

a C2v molecule and the

symmetry elements of the

group.


+-

+-

C2 v

v'

++

-

+

C2 v

v'


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v2

2E =

2=

v xz( ) =

vyz

' ( ) =

Symmetry is a1

pz

pz

pz

pz

pz

pz

pz

pz


Irrep is A1


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v2

2E =

2=

v xz( ) =

vyz

' ( ) =

Symmetry is b2

py

py

py

py

py

py

-py

-py


Irrep. is B2


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v2

2E =

2=

v xz( ) =

vyz

' ( ) =

Symmetry is b1

px

px

px

px

px

-px

px

-px


Irrep. is B1


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v2

2E =

2=

v xz( ) =

vyz

' ( ) =

Symmetry is ?

1s1

1s1

1s2

1s2

1s2

1s2

1s2

1s2



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v2

2E =

2 =

v xz( ) =

vyz

' ( ) =

Symmetry is ?

1s1

1s2

1s21s1

1s1

1s1

1s1

1s1



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v2

E s s s s( ) ( )1 1 1 11 00 11 2 1 2=

1s1 1s2C s s s s2 1 2 1 21 1 1 1

0 11 0

( ) ( )=

This representation is not reduced


v s s s s( ) ( )1 1 1 10 11 01 2 1 2

=

v s s s s'( ) ( )1 1 1 1 0 11 01 2 1 2=


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Character Table Structure of character tableC v2

2E =

2 =

v xz( ) =

vyz

' ( ) =

Symmetry is a1

1s+

1s+

1s+

1s+

1s+

1s+

1s+

1s+

Irrep is A1


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2

1s

1s+

px

py

Only orbitals with samesymmetry label interact

A1 A1 pz A1

B1

B1 B1

B2


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C v2

2

Vibrations andnormal modes

Structure of character tableCharacter Table

H

O

H

O

H H

H

O

H


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2E =

2 =

v xz( ) =

v

yz' ( ) =

Symmetry is a1

H

O

H

H

O

H

H

O

H

H

O

H

H

O

H

H

O

H

H

O

H

H

O

H


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v2

2E =

2 =

v xz( ) =

v

yz' ( ) =

Symmetry is b1

O

H H

O

H H

O

H H

O

H H

O

H H

O

H H

O

H H

O

H H



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A1

A1

B1

v2

Vibrations andnormal modes

C


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v3

We have three classes ofsymmetry elements :

E the identity

Two three fold rotations

C and C3 3-1

Three mirror planes

v v v , ' , ' '

Character Table

Ch T bl


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v3

Molecular orbitals of NH3

a1 ex ey

Normal modes of NH3

Character Table


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What you must learn from this lecture

2.. You must understand the different parts of a character

table for a symmetry group: (a) Name of symmetry group;

(b)Classes of symmetry operators; (c) Names of irreducible

symmetry representations. (d) The irreducible characters

1. You are not expected to derive any of the theorem of group

theory. However, you are expected to use it as a tool

3. For simple cases you must be able to deduce what irreducible

representation a function or a normal mode belongs to by the

help of a character table.

Ch t T bl


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Appendix C von 3

Symmetry operations in

the same class are related to one

another by the symmetry operationsof the group. Thus, the

three mirror planes shown here

are related by threefold

rotations, and the two rotations

shown here are related by

reflection in v.

Character Table

Character Table


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The dimension is 6 since wehave 6 elements.

We have three different symmetryrepresentations as we have three

different classes of symmetry elements

Character TableAppendix C von 3

Character Table


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The p orbital

does not change

with E, C C

z

3 3-1

,, ' , " v v v

The symmetry

rep. is A1

px pydoes not change

with E, C C3 3

-1,

, ' , " v v v

X

Y

X

Y


Character Table Appendix Con


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X

Y

X

Y

X

Y

px p'x p' 'x

Ep = p ; C p p' ; C p = p"x x 3 x x 3-1

x x=

X

Y

X

Y Y

Character Table Appendix C von 3

Character Table


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p p p px y x y( ) = ( )

D C( )3

1

2

3

23

2

1

2

p p p px y x y( ) = ( )

D C( )3

1

1

2

3

23

2

1

2

The trace is - 1 forboth matrices

p p p px y x y( ) = ( )

D E( )1 0

0 1The trace is 2

which is also thedimension ofthe representation


Character TableC


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p p p px y x y( ) = ( )

D v( )1 0

0 1

p p p px y x y( ) = ( )

D v( )'

12

32

3

2

1

2

p p p px y x y( ) = ( )

D v( )"

1

2

3

23

2

1

2

The trace is -1 for

both matrices


Character TableA di C


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Typical symmetry

-adapted

linear combinationsof

orbitals in a

C 3v molecule.


Chem 373- Symmetry

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