iii. molecular symmetry and the symmetry...

Symmetry planes and reflections Center of inversion Proper axes and proper rotations Improper axes and improper rotations

Products and relations of symmetry operations The symmetry point groups

III. III. Molecular symmetry andMolecular symmetry andthe symmetry groupsthe symmetry groups

IIIIII--5. Products and relations 5. Products and relations of symmetry operations of symmetry operations

iyzxyxziyzxzxy

111111

111111

111111

zyxzyxyzzyxzyxxzzyxzyxxy

Reflection on the plane

Inversion through the center

111111 zyxzyxi x

z

y

1 1 1( , , )x y z

Rotation about the proper axis

1

2

2

1

1

1

zyx

zyx

An

1000cossin0sincos

Cn

z

contour clockwise

1000cossin0sincos

Cn

z

clockwise

1000cossin0sincos

Sn

z

1000cossin0sincos

Sn

z

Rotation about the proper axis

zCxyzS nn

n2 x

z

y

1 1 1( , , )x y z

x

z

y

1 1 1( , , )x y z

Example:

100010001

1000cossin0sincos

C2

z

1111112 zyxzyxzC The same way we can derive

1111112 zyxzyxxC

1111112 zyxzyxyC 1112111111211122

1112111111211122

zyxzCzyxzyxxCzyxyCxCzyxzCzyxzyxyCzyxxCyC

Example: 11111111121112

1111111111112

zyxizyxzyxzCzyxxyzCzyxizyxzyxxyzyxzCxy

2S

x

y(x1, y1)

d d

45 45

(-y1, -x1)

(y1, x1)

111111

111111

zxyzyxzxyzyx

d

d

x

y

(x1, y1)

d d

4545

(y1, x1)

(-y1, -x1)

111111

111111

zxyzyxzxyzyx

d

d

Example: clockwise d

d

zCxzxzzC

4

4

1 2

34

x

y

xz

zC 4 4 1

23

x

y

xz

xz 1 4

32

x

y

xz

d

1 2

34

x

y

xzd

GTAchem4-1

Calculate

1 2

3

4

x

y

v 5

6

zC 6

v

?

???

6

6

6

6

zCzC

zCzC

v

v

v

v

and then specify where they are in Figure

Basically we need study Cn and Sn only

As Cn is subgroup of Sm , so the most important is nS

*The symmetry element can be thought of as the whole of space+Note the equivalences and

perpendicular to rotation axis

Rotation by followed by reflection

n-Fold axis ofimproper rotation+

iInversionCenter of inversionReflectionMirror planeRotation by n-Fold symmetry axis

EIdentity*SymbolSymmetry operationSymmetry element

Summary Important Symmetry operations and symmetry elements

2 / n nC

2 / nnS

1 =S 2 =S

Enantiomer

Not superimposed (by proper rotation)

Chiral (different molecule)

Achiral (same molecule)

superimposed

3H COH

H3CHHO

H

3H CH

3CHHH

H

Molecules that are not superimposable on their mirror image

These molecules are called dissymmetric

Theorem: A molecule that has no improper rotation axis must be dissymmetric

If molecule has a Sn axis, then it includes the following collinear axes

oddnfor ,,,evennfor ,,,

22

2

ESSSSESSS

nn

nnnn

nnnn

So if n = odd, there is hnnS

If n = even, iCS h 22

Mirror plane

A BCD

AB

C D

A B

CD

AB

C D

A B

CD

ABC D

i Rotate by

A BCD

AB

C D

superimposed Proper rotationis OK

Example

34

1

23

4

1

23

4

12

C

4

Mirror plane

34

1

2 34

1

2C 4

4S

superimposed

Example

See textbook page 37

1,3,5,7-tetramethylcyclooctatetraene has neither plane symmetry nor center symmetry

Has plane symmetry

Example

4S

4S3CH

3CH

3CH

3H C

In general if molecule has S2m symmetry

Mirror plane

mS 2

C2m

superimposed

IIIIII--6. The symmetry point groups6. The symmetry point groups

For a given molecule, we collect a complete list of symmetry operations

h321 A A A A

Those operations form a Group

This group is called a point group

3C

v

2C

2C 2C h

vv

E

For example: planar AB3 molecule

3C 3S23C23

53 CS h

Totally 12 symmetry operations

53

53

33

hh

vv

vv

vv

22

22

22

23

23

33

S

S S C CC CC CC C

C CE E

S

533hvvv222

233

533hvvv222

233

S S C C C C C E

S S C C C C C E

533hvvv2222333 S S C C C C C E C

2C2C

2C v

v

v

B1

B2

B3

B1

B2

B3

2CB1

B2

B3

3CB1

B2 B3

B1

B2

B3

2C 223 CCC

count clockwise

23C

clockwise

It looks like we can use vCC 23 , but this is not true

If we use coordinate system, we can know that

Rotation * rotation = roatation

321 axisCaxisCaxisC kmn General law

axiscrossCplaneplane kvv 21


2C

2C

2C v

v

v

B1

B2

B3

v

B1

B2

B3

3CB1

B2 B3

B1

B2

B3

v vvC 3



Should not be 2C

count clockwise

23C

clockwise


2C

2C

2C v

v

v

B1

B2

B3

v

B1

B2

B3

v B1

B2B3

B1

B2

B3

23C 2

3Cvv


Should not be 2C

count clockwise

3C clockwise

223 CCC 232 CCC


C C C C C C 2222223 C C C C C C C 2223222 C

C C C C C C 22223222 C 23222222 C C C C C C C

223 CCC 223 CCC

vvC 3 vv C 3

vvvvv3 vC v3vvv vvC

23Cvv 3Cvv

2 ' '' ' '' 53 3 2 2 2 3 3

2 ' '' ' '' 53 3 2 2 2 3 32 '' ' '' ' 5

3 3 3 2 2 2 3 32 2 ' '' ' '' 53 3 3 2 2 2 3 3

' '' 2 5 ' ''2 2 2 2 3 3 3 3' ' '' 2 5 ' ''2 2 2 2 3 3 3 3'2

v v v h

v v v h

v v v h

v v v h

h v v v

h v v v

E C C C C C S SE E C C C C C S SC C C E C C C S SC C E C C C C S SC C C C E C C S SC C C C C E C S SC

' '' ' 2 5 '' '

2 2 2 3 3 3 3' '' 5 2 ' ''

3 3 3 3 2 2 2' ' '' 5 2 ' ''

3 3 3 3 2 2 2'' '' ' 5 2 '' '

3 3 3 3 2 2 25 ' '' ' '' 2

3 3 2 2 2 3 35 '' ' '' ' 2

3 3 3 2 2 2 3 3

h v v v

v v v v h

v v v v h

v v v v h

h h v v v

h v v v

C C C C C E S SS S E C C C C C

S S C E C C C CS S C C E C C C

S S C C C E C CS S S C C C C C E

5 5 ' '' ' '' 23 3 3 2 2 2 3 3h v v vS S S C C C C E C

2 ' ''3 3

2 ' ''3 32 ' ''

3 3 32 2 '' '3 3 3

'' ' 23 3

' ' '' 23 3

'' '' ' 23 3

v v v

v v v

v v v

v v v

v v v v

v v v v

v v v v

E C CE E C CC C C EC C E C

E C CC E CC C E

C3v

subgroup

v

v v

Molecular has no symmetry operations other than E 1CE C1 (1)

Elements Group name (order)

Molecular has only one reflection plane

E2 Cs (2)

Molecular has only one inversion center

Eii 2 Ci (2)

Nomenclature of point group

Molecular has only one proper axis Cn

E CC C CC nnnnnnn 132 Cn (n)


Molecular has only one improper axis Sn (n=even)

hnnS

Eii 2 Ci (2)

E SS S SS nnnnnnn 132

Sn (n)

Molecular has only one improper axis Sn (n=odd)

ES SS SS nnnn

nnnn

212

Cnh (2n) E CC C CC

E CC C CCnn

nnnnnh

nn

nnnnn

132

132

as

Where S2

Find all symmetry operations of AB3 non-planar moleculeand make multiplication table (give detailed procedures)

GTAchem4-2

Molecular has one proper axis Cn and one C2 axis ( Cn C2 )

E CC C CC nnnnnnn 132

Dn (2n)


All n twofold axes in the same plane and perpendicular to the prAll n twofold axes in the same plane and perpendicular to the principal axis incipal axis CCnn

Cnv (2n) vn

vv vWhen n = odd , n the same type of planes

12222 n CC C C

Molecular has one proper axis Cn and one reflection plane that contain this axis


d

ndd

vn

vv

2/

d

2/v

When n = even , n/2 the same another n/2 same

All n vertical reflection planes intersect at one line which is Cn axis


Cnv (2n)

v

nvv vn = odd , n the same planes


d

ndd

vn

vv

2/

d

2/v

n = even , n/2another n/2

h = Dnh (4n)

Horizontal reflection plane

n improper axes n improper axes E CC C CC nnnnnnnh 132 n proper axes n proper axes

2v nCvnvvh 22/d

22/

v

)2/(

)2/(

Cn

Cnn

dh

nvh

n vertical planes n vertical planes

n Cn C22 axes axes

nhh D

zC 2

111111 zyxzyxxyxyh

yCxC

hvvh

hvvh

2

2

Blackboard

1111112

1111112

zyxzyxyCzyxzyxxC

111111

111111

zyxzyxyzyz

zyxzyxxzxz

v

v

x

y

h

yCyC

xCxC

hhv

hhv

22

22

Multiplied by h

vvvv zC if 2

xCv

2

yCv

2

3C

v

2C

2C2C

v v

In general two planes intersectin a line that is Cn axis

If two planes has dihedral angle is

22

nC nvv

vv

33223/ Cn

Example


Dn (2n) ndh Dd = Dnd (4n)

Vertical reflection plane

n improper axes n improper axes collinear with collinear with CCnn

n proper axes n proper axes

n vertical planes n vertical planes

n Cn C22 axes axes


22222 C CC C C

n

122523222222 n-nnnnnd S S SSC CC C

dndnnnnnnnd E CC C CC d132

22

2C

If principal axis is C2(z) , then there are two C2 in xy-plane

xC 2

yC 2

2

So two C2 in xy-plane has angle andthen dihedral angle (, C2(x)) is /2

d

32

3C

If principal axis is C3(z) , then there are three C2 in xy-plane

xC 2

2C

2 So two C2 in xy-plane has angle and

then dihedral angle (, C2(x)) is /2d

2C

11 yx

2211 yx yx d

dx 22 yx

1

n2 with nC

2cos2sin2sin2sin2cos2cos

1112

1112

yxryyxrx

10002cos2sin02sin2cos

d

1

1

1

2

2

2

zyx

zyx

d

xC2

2C

This Cn(z) with n C2

100010001

2 xC

2


S


d

4

nn 222222

nd SxC 22

Dnh (4n) Dnd (4n)

nhh D ndh DBig difference

Example: ethane

4

5

6

3

1

2

C3(z) S6(z) symmetry2C

3

5

26

1

42C

3

5

2

6

1

4

There areThe other two C2

There is no h symmetry D3d (4*3)

d

For example: planar AB3 molecule

There is h symmetry D3h (4*3)

3C

v

2C

2C 2C h

vv

E

3C 3S23C23

53 CS h

EXERCISES Set A. pages 61-62

A3.1, A3.2, A3.3

EXERCISES Set C. page 64

(1), (3), (6), (9)

GTAchem4-3

GTAchem4-4

A3.2(e) excluded

iii. molecular symmetry and the symmetry...

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