Molecular symmetry and group theory

Dr. Md. Monirul Islam Department of Chemistry

University of Rajshahi

• Effect of performing successive operations

N

H(1)(2)H

(3)H

C3

v(1)

v(2)

v(3)

Rotate by 120

(C3)

N

H(3)(1)H

(2)H

N

H(3)(2)H

(1)H

v(1)

v(2)

Symmetry operation of ammonia

v(1)C31 = v(2)

• C3 – passing through N atom• v(1)- bisects angle of H(2) & H(3)• v(2)- bisects angle of H(1) & H(3)• v(3)- bisects angle of H(1) & H(2)

NH3 has pyramid structure

• Successive symmetry operations on PF5

P

(1)F

(2)F

F(3)

F(4)

F(5)

C3

C2(1)

C2(2)

C2(3)

P

(1)F

(2)F

F(3)

F(4)

F(5) v(1)v(2)

v(3)

h

hC31 = C3

1h =S3

1 C3

1 C2(1) = C2(3)C31=v(2) h =C2(2)

C31 C2(2) = C2(1)C3

1=v(3) h =C2(3)C3

1 C2(3) = C2(2)C31=v(1) h =C2(1)

C2(1) v(1) = h = v(1) C2(1)C2(2) v(1) = S3

1; v(1) C2(2) = S35

C2(3) v(1) = S35; v(1) C2(3) = S3

1 etc

PF5 has trigonal -bipyramid structure

Group multiplication tables

• It is very difficult to understand the effect of performing successive operations by painting or drawing and also difficult to memorize them.

• In order to investigate those, it is helpful to construct complete group multiplication table.

𝐶3𝑣 𝐸𝐶31𝐶3

2𝜎 𝑣𝜎𝑣′ 𝜎𝑣

′ ′

𝐸𝐶31𝐶3

2𝜎 𝑣𝜎𝑣′ 𝜎𝑣

′ ′

𝐶31𝐶3

2𝐸𝜎 𝑣′ 𝜎𝑣

′ ′𝜎 𝑣

𝐶32𝐸𝐶3

1𝜎 𝑣′ ′𝜎𝑣

❑𝜎 𝑣′

𝜎 𝑣𝜎𝑣′ ′𝜎𝑣

′ 𝐸𝐶32𝐶3

1

𝜎 𝑣′ 𝜎𝑣

❑𝜎𝑣′ ′𝐶3

1𝐸𝐶32

𝜎 𝑣′ ′𝜎 𝑣

′ 𝜎𝑣❑𝐶3

2𝐶31𝐸

• Communicative operation Two operations say A and B is said to be

communicative if AB=BA.

For example, in PF5 molecule

C2(1) v(1) = v(1) C2(1) = h

Therefore, two operations C2(1) and v(1) are communicative.

The multiplication always need not to be communicative.

For example, again in PF5 molecule

C31 C2(1) (= C2(2) ) C2(1) C3

1 (= C2(2) )

Therefore, two operations C31 and C2(1) are

non-communicative.

• Inverse operation There is an inverse operation between A and B if

they obey the relationship AB= E (= BA). The effect of the operation B is exactly opposite to

that of the operation A upon the object. B is said to be the inverse of A (and vice versa) and this is expressed algebraically as B = A-1.

Therefore, we can write,

A-1A = AA-1 = E

An operation and its inverse always communicate A center of inversion or any plane of symmetry follow

the relationshipsi2 = E and 2 = E

Therefore, operations i and are their own inverse.

i = i-1 and = -1

• Inverse operations due to Cn and Sn axes

Cn1 clockwise rotation by (360/n)

Cn-1 anti-clockwise rotation by (360/n)

Cn-1 Cn

1 = E, here Cn-1 is inverse operation of Cn

1

For C3 axis, C3

2C31 = E is valid

In general, Cn

n-1Cn1 = E

Cn-1 = Cn

n-1 means that a clockwise rotation by [(n-1)360/n] is exactly equivalent to an anti-clockwise rotation by (360/n).

For Sn axis:

For Cn axis:

Sn-1Sn

1 = (Cn-1h)(Cn

1h) = (Cn-1h)(h Cn

1) = Cn-1Cn

1 = E

Sn-1Sn

1 = (Cn-1h)(Cn

1h), but Cn1h = h Cn

1

When n is even, Snn-1Sn

1 = E and Sn-1 = Sn

n-1

When n is odd, Sn2n-1Sn

1 = E and Sn-1 = Sn

2n-1

Exercises:1. Identify the symmetry elements belonging to the following

molecules: (a) WF5Cl; (b) PtCl42-; (c) SiH3CN; (d) 1-chloro-3,5-difluorobenzene; (e) allene, CH2=C=CH2; (f) Ni(CO)4; (g) CO3

2-; (h) (PNCl2)3 (a planar, six-membered ring).

2. List the symmetry operations generated by the following axes of symmetry: C5, C6, S3, S8. Can any of these operations be expressed in more than one way?

3. To which symmetry operations in NH3 are the following combinations of operations equivalent: (a) C3

2v(1); (b) v(2) v(3); (c) v(3)C3

1; (d) v(1)C31v(3)?

4. To which symmetry operations in PF5 are the following combinations of operations equivalent: (a) C2(1)h; (b) C3

2C2(2); (c) S31v(1); (d) C3

1C2(3)v(2); (e) hv(1)C2(2)?