symmetry translation rotation reflection slide rotation (s n )

Symmetry

QuickTime™ and aGIF decompressor

are needed to see this picture.







Translation

Rotation

Reflection

Slide rotation (Sn)

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 operatorsLecture on-line

Character Tables (PowerPoint) Character tables (PDF)Handout for this lecture

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

We shall now turn our attention away from the 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

UsageCharacter Table

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

Simple case Character 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.

A B

Symmetry label C 2 (i.e. rotation by 180°)

σ 1

π -1

C2

180°

σ

C2

180° π

Simple case Character Table

C3v Character Table Structure of character table

Symmetry group Symmetry Operations

A

C

B

E A

C

B



C3C3C3 =EC3C3 =C3−1 C3

−1C3 =E

C3−1 A

C

B

B

A

C

C3 A

C

B

C

B

A



A

C

B

A

B

C

v A

C

B

C

A

B

v'

A

C

B

B

C

A

v"

C3v 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 conjugates form a class.

A

B C

v−1

A

C B

v−1C3

B

A C

v−1C3σv

B

A C

C3−1

B

A C=


σvC3σv−1=C3

−1

σv'C3σv'−1=C3

−1

σv''C3σv''−1=C3

−1

C3C3C3−1=C3

C3−1C3C3 =C3

EC3E−1=C3

We have in general:

Thus C3 and C3-1 form

a class of dimension 2

The two elements C3 and C3-1

can be related to each other byσv,σv', and σv''

C3 =σvC3−1


A

B CC3

−1

B

C A

C3−1v

B

A C

C3−1C3 v

C

B A

=C

B A

σv"

EσvE−1=σv

C3σvC3−1=σv"

C3−1σvC3 =σv'

σvσvσv−1=σv

σv'σvσv'−1=σv"

σv"σvσv"−1=σv'

In general Thus σv,σv' and σv"form a class of dimension 3. The elements are related by

C3 and C3-1

Elements conjugated to v ?



σv σ'vσ''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 symmetryelements

C2

C2v

Name of point group

Symmetry elements

E : identity

C2 : Rotation

σ(xz) mirror plane

σ'(yz) mirror plane

Number of symmetry elements

Character Table Structure of character table

C2v

C2 Name of irreducible representations

A1 A2 B1 B2


Characters of irreducible representations

The px,py, and pz orbitals on the central atom ofa C2v molecule and thesymmetry elements of the group.


+ -+ -

C2 σv

σv'

++

-

+

C2 σv

σv'

C2v

C2 E =

C2=

σv(xz) =

σv' (yz) =

Symmetry is a1

pz

pz

pz

pz

pz

pz

pz

pz


Irrep is A1

C2v

C2 E =

C2=

σv(xz) =

σv' (yz) =

Symmetry is b2

py

py

py

py

py

py

-py

-py


Irrep. is B2

C2v

C2 E =

C2=

σv(xz) =

σv' (yz) =

Symmetry is b1

px

px

px

px

px

-px

px

-px


Irrep. is B1

C2v

C2 E =

C2=

σv(xz) =

σv' (yz) =

Symmetry is ?

1s1

1s1

1s2

1s2

1s2

1s2

1s2

1s2


C2v

C2 E =

C2=

σv(xz) =

σv' (yz) =

Symmetry is ?

1s1

1s2

1s21s1

1s1

1s1

1s1

1s1


C2vE(1s1 1s2)=(1s1 1s2)

1 00 1

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1s1 1s2C2(1s1 1s2)=(1s1 1s2)

0 11 0

⎛ ⎝ ⎜

⎞ ⎠ ⎟

This representation is not reduced


σv(1s1 1s2)=(1s1 1s2)0 11 0

⎛ ⎝ ⎜

⎞ ⎠ ⎟

σv'(1s1 1s2)=(1s1 1s2)0 11 0

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Character Table Structure of character table C2v

C2 E =

C2=

σv(xz) =

σv' (yz) =

Symmetry is a1

1s+

1s+

1s+

1s+

1s+

1s+

1s+

1s+

Irrep is A1


C2 E =

C2=

σv(xz) =

σv' (yz) =

Symmetry is b1

1s-

1s-

1s-

1s-

1s-

1s-

-1s-

-1s-

Irrep. is B1


C2

1s-

1s+

px

py

Only orbitals with same symmetry label interact

A1 A1 pzA1

B1B1 B1

B2

QuickTime™ and aAnimation decompressor


C2v

C2

Vibrations and normal modes

Structure of character table Character Table

H

O

H

O

H H

H

O

H


C2 E =

C2=

σv(xz) =

σv' (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

C2v

C2 E =

C2=

σv(xz) =

σv' (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


QuickTime™ and aAnimation decompressor


A1

A1

B1

C2v

Vibrations andnormal modes

C3v

We have three classes ofsymmetry elements:

E the identity

Two three fold rotations

C3 and C3-1

Three mirror planesσv,σ'v ,σ''v

Character Table

C3v

Molecular orbitals of NH3

a1 ex eyNormal modes of NH3

Character Table

What you must learn from this lecture

2.. You must understand the different parts of a charactertable 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 grouptheory. However, you are expected to use it as a tool

3. For simple cases you must be able to deduce what irreduciblerepresentation a function or a normal mode belongs to by the help of a character table.

Appendix on C3v

Symmetry operations in the same class are related to oneanother by the symmetry operations of the group. Thus, thethree mirror planes shown here are related by threefold rotations, and the two rotations shown here are related byreflection in v.

Character Table

The dimension is 6 since wehave 6 elements.

We have three different symmetryrepresentations as we have threedifferent classes of symmetry elements

Character Table Appendix on C3v

The pz orbitaldoes not change

with E, C3, C3-1

σv, σ'v ,σ"vThe symmetryrep. is A1

px pydoes not change

with E, C3, C3-1

σv, σ'v ,σ"v

X

Y

X

Y


X

Y

X

Y

X

Y

px p'x p''x

Epx =px ; C3px =p'x; C3-1 px =p"x

X

Y

X

Y Y


px py( )D(C3)= px py( )−

12

32

3

2−

1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

px py( )D(C3−1 )= px py( )

−12

−32

−3

2−

1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

The trace is -1 forboth matrices

px py( )D(E)= px py( )1 0

0 1

⎛

⎝ ⎜

⎞

⎠ ⎟

The trace is 2which is also thedimension ofthe representation


px py( )D(σv)= px py( )−1 0

0 1

⎛

⎝ ⎜

⎞

⎠ ⎟

px py( )D(σv' )= px py( )

12

−32

−3

2−

1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

px py( )D(σv" )= px py( )

12

32

3

2−

1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

The trace is -1 forboth matrices


Typical symmetry-adapted linear combinations of orbitals in aC 3v molecule.


symmetry translation rotation reflection slide rotation (s n )

Documents

lecture slide

point group symmetry

symmetry elements

symmetry operations

typical symmetry

plane symmetry groups

symmetry great site

group g