topic 1(introduction, symmetry elements and symmetry operations)

Molecular symmetry and group theory

Dr. Md. Monirul Islam Department of Chemistry

University of Rajshahi

What is symmetry?

o Mutual relation of the parts of something in respect of magnitude and position; relative measurement and arrangement of parts; proportion.

o Due or just proportion; harmony of parts with each other and the whole; fitting, regular, or balanced arrangement and relation of parts or elements; the condition or quality of being well proportioned or well balanced.

Symmetry in everyday life

Role of symmetry in chemistry

Symmetry in everyday life

• Bilateral symmetry that is a single plane of symmetry, which divides the object into parts which are mirror images of each other

Cymothoe Human symmetry

Symmetry in everyday life

Ivy leafIris

One plane of symmetry

Three planes of symmetry

Ice crystal

Symmetry in everyday life

• Six fold rotation axes each rotated by 2/6

• Six planes of symmetry

2/6

Role of symmetry in chemistry

Pyrene

Importanceo Molecular structureso Crystal structureso Infra-red spectrao Ultra-violet spectrao Dipole momentso Optical activitieso Computation in quantum

chemistryThree monosubstituted pyrenes

Symmetry elements and symmetry operation

Symmetry element is the geometric element (e.g. point, line, plane) through which symmetry operation (inversion, rotation, reflection) takes place.

A symmetry operation is an operation which when applied to a molecule move it in such a way that its final position is physically indistinguishable from its initial position.

1. Identity, E E is always a symmetry element as doing nothing to an

object always leaves it looking just the same as it originally did.

E(x1 ,y1 ,z1) =(x1 ,y1 ,z1)

HO

H HO

H

360

3601 1

2 3 2 3

N

HH

H

N

HH

H

2. Rotation axis, Cn

A line which is applied to rotate a molecule through an angle 2/n and after rotation it leaves molecule unchanged.

1 2 2 1

180

Two-fold (C2) rotation axis in H2O Three-fold (C3) rotation axis in NH3

C2(z)(x1,y1,z1)=(-x1, -y1, z1)

N

HH

H C31

C31

C3-1

**

*

The leveling the operation of rotation

• The rotation by (360/n) about Cn axis Cn1

• The rotation by 2×(360/n) about Cn axis Cn2

Shortly,• The rotation by m×(360/n) about Cn axis Cn

m

B

F(3) F(2)

F(1)

B

F(2) F(1)

F(3)

B

F(1) F(3)

F(2)

B

F(3) F(2)

F(1)(360/3)

C31

2×(360/3) C32

3×(360/3) C33

(360/3) (360/3)

• Cnn = E and Cn axis generates n operations of which one is Cn

n = E .

N N

HH ba

C2

N N

HH ab

Two-fold rotational axis in cis-dinitrogen difluoride

Symmetry elements of a planar hexagon

Pentacarbonylion

C Fe

C

C

C

C

O

O

O

O

O

C3

C2

WC

C C

C

C

C

O

O

O

O

OO

C4

Hexacarbonyltungsten

C2, C3, C6

C2C2

Molecules containing five-fold rotational axes: (a) eclisped ferrocene, side and top view; (b) staggered ferrocene, side and top view. Each molecule has five C2 axes, only one of which shown. Upon rotation about the C2 axis, the atoms interchange: 11, etc.

3. Mirror plane A plane which bisects a molecule into two haves which

are mirror image to each other.

Plane of symmetry in water

σ(xz)(x1, y1, z1) = (x1, -y1, z1)

Reflection operation

O

H(1) H(2)

1 O

H(2) H(1)

O

H(1) H(2)

2 = E

• Every reflection plane has only one operation

O

H(2) H(1)

2

• Classification of mirror plane

Some of the symmetry elements of benzene ring. There is one h and two sets of vertical reflection plane (v, d).

• If more than one v, h or d, these are denoted by v, v, v…. and so on

4. Inversion center, i A center through which the inversion of a molecule

takes place so that each point moves out to the same distance on the other side of the molecule and leaves the molecule indistinguishable.

(x1, y1)

(x2, y2) = (-x1, -y1)

Inversion, i

i (x1,y1,z1) =( −x1,−y1,−z1)

Examples of inversion

i2 = E

The center of symmetry of 1,2-dimethyl-1,2-diphenyl diphosphinedisulfide

Inversion rotation is a composite operation

• It is consisted of rotation by 180 (C2) and reflection in a plane perpendicular to C2 axis.

H(1)

H(2)

H(3)H(4)

H(5)

H(6)

iInversion

at i

H(4)

H(5)

H(6)H(1)

H(2)

H(3)

i

H(3)

H(2)

H(1)H(6)

H(5)

H(4)

i

C2

180

Reflection

H(4)

H(5)

H(6)H(1)

H(2)

H(3)

i

However, inversion may not always be composite operation of rotation by 180 (C2) and reflection in a plane perpendicular to C2 axis.

H(1)

H(2)

H(3)H(4)

H(5)

H(6)

iInversion

at i

H(4)

H(5)

H(6)H(1)

H(2)

H(3)

i

H(2)

H(1)

H(6)H(5)

H(4)

H(3)

i

C2

180

Reflection

H(5)

H(6)

H(1)H(2)

H(3)

H(4)

i

(a) An inversion operation with (b) a two-fold rotation. Although the two operation may sometimes appear to have the same effect that is not the case in general.

5. Improper axis of rotation, Sn

An axis about which the rotation of a molecule takes place an angle 2/n followed by reflection in a plane perpendicular to the axis and leaves the molecule indistinguishable .

A four-fold axis of improper rotation S4 in the CH4 molecule

• The operation of rotation by m×(360/n) around Sn axis is denoted by Sn

m

Examples

An axis of improper rotation (S4)

S1

360

S1

and

S1

h

S1

Reflection

• S1 Operation

S1 operation is equivalent to h

S2

180and

S2

h

S2

Reflection

• S2 Operation

S2 operation is equivalent to i

S2

S2

i

Inversion at i

S2

i

• Number of operations around Sn axis

Number of operations depends on whether n is odd or

even

- If n is even, then total operation is n of which one is Snn

=E (Cnn = E and h

n = E) and operations are Sn1, Sn

2,

………… Snn

- If n is odd, then total operation is 2n and these are Sn1,

Sn2, ………….Sn

n, ………………..Sn2n of which Sn

n = h

(because Cnn = E and h

n = h) and Sn2n =E (because Cn

2n

= E and h2n = E)

• Several of the operations Snn can be expressed by

other operations

S4 axis generates the operations S41, S4

2, S43 and S4

4

• S42 is equivalent to C4

2 and h2. Since h

2 = E, S42 = C4

2 =

C2

• S44 = E

• Therefore, series of operations generated by S4 are S41,

C2, S43, E.

A similar series exists for S5 which must be continued up to

S510

• The operations are S51, S5

2 = C52, S5

3, S54 = C5

4, S55 = h

(because C55 = E and h

5 =h), S56 = C5 (because C5

6 =

C55 C5 = EC5 = C5), S5

7, S58 = C5

3, S59, S5

10 = E.