Molecular symmetry and group theory

Dr. Md. Monirul Islam Department of Chemistry

University of Rajshahi

Groups and their basic properties

Point groupThe collection of symmetry operations upon a molecule that satisfy all the characteristics of a group is defined as point group for that molecule.

For example,The symmetry operations on PF5 molecule are

Point group of PF5

The symmetry operations on NH3 molecule are

Point group of NH3

𝐸 ,𝐶31 ,𝐶3

2 ,𝜎𝑣 (1 ) ,𝜎𝑣 (2 ) ,𝜎𝑣 (3)

Classification of molecules into point groups

C

H F

Cl Br

Symmetry elements: E

Point group: C1

C

Br F

O

Symmetry elements: E,

Point group: Cs

O𝜎

C C

Br Cl

BrClH

H

i

C

H

H

Br

Br

Cl

Cl

Symmetry elements: E, i

Point group: Ci

O O

1H H2

C2

Classification of molecules into point groups

C2

O

H H

C2

𝜎 𝑣𝜎 𝑣′

C2v

N

H(1)(2)H

(3)H

C3

v(1)

v(2)

v(3) C3v

Cnv Cn

P

C3

C3

Cnh

Classification of molecules into point groups

NN

H

H

C2

𝜎 h

C2h

Trans-N2H2

H

C3

𝜎 h

C3h

B(OH)3

BO

H

OH

O

Classification of molecules into point groups group

P

Cl

ClN P

Cl

Cl

NP

Cl

Cl

N

N

P

Cl

Cl

i.e.,(+)P

P

P(+)

(-)P(-)N

(+)N

N(-)

N(+)

(-)

• is coincident with axis

• A molecule belonging to the point group :

Classification of molecules into point groups

Dihedral point group belonging to axes

C

HHH

H

H H

𝐶2

𝐶2 𝐶2

C C

H

H

H

H

HH

𝐶3

Gauche : a molecule belonging to the point group

• and axes point group

Classification of molecules into point groups

Dihedral point group belonging to axes

• , and point group

C C C

H

H

H

H

𝐶2

𝐶2

𝐶2

𝜎 𝑑

𝜎 𝑑

Allene : a molecule belonging to the point group

C C

H H

HHH

H

C

H

H

H

H

H

H

Classification of molecules into point groups

Dihedral point group belonging to axes

• , and point group

𝜎 𝑑𝜎 𝑑

𝜎 𝑑

𝐶3𝐶2

𝐶2

𝐶2

Staggered : belonging to the point group

Classification of molecules into point groups

Dihedral point group belonging to axes

• , and point group

CC

H

H

H

H

𝜎 h

𝐶2

𝐶2

Ethene:

C

O

O

O𝜎 h

𝐶2 𝐶2

𝐶2

:

More examples

: Benzene :

Cubic Point Groups: Td

Point groups of higher symmetry: Cn more than one

Symmetry operations

Cubic Point Groups: Td

Symmetry operations

Infinite point group

C NH C∞

C OO C∞

∞𝜎 𝑣

∞𝜎 𝑣

∞𝐶2

𝜎 h

HCN molecule (linear) CO2 molecule (linear)

Symmetry elements: Symmetry elements

Point group : Point group :

Molecules

Linear?

? 𝐷∞h𝐶∞ h

YNTwo or more

with n>2

YN

Y

?N

?

N𝑇

Y?

𝑇 𝑑

𝜎 𝑑

𝜎 h 𝑇 h

Y

Point group flow chart

?Y𝐼 h

?Y𝑂h

𝑂N N

N?

?𝐶 𝑠Y

N

?𝐶𝑖Y

𝐶1

N

?N

Y

N

?𝐶 h𝑛Y

N

?𝐶𝑛𝑣Y

N

?𝑆2𝑛Y

? N ? N 𝐷𝑛

Y

𝐷 h𝑛

Y

𝐷𝑛𝑑

Y

𝐶𝑛N

Assignments of Points Groups

Example H2O

1. It is a Non-linear molecule2. It has only one Cn(C2) axis3. It does not belong Cn ┴ C2 4. It has v

5. Point group C2V

Example NH3

1. It is a Non-linear molecule2. It has only one Cn(C3) axis3. It does not belong Cn ┴ C2 4. It has v

5. Point group C3V

Example Allene

C C C

H

H

H

H

C2, S4

C2’

C2’