SYMMETRY AND GROUP THEORY

Rotation Angle Symmetry Operation

600 C6

1200 C3 (= C62)

1800 C2(= C63)

2400 C32 (= C6

4)

3000 C65

3600 E (C66)

Examples

Determine the point group where the following molecules belong:

1. Water

2. Staggered ethane

3. Eclipsed ethane

4. Trans-1,2-Dichloroethane

Notes:

Properties of Characters of IrreducibleRepresentations in Point Group

1. The total number of symmetry operations in the group is known as the order (h) which is the same as the total number of symmetry operations in a given point group.

Example: C2V pt group: h = 4

2. Symmetry operations are arranged in classes which is represented by the column in the character table

Example: C2V pt group: no. columns = classes=4

3. The number of irreducible representations equals the number of classes (they are square)

4. The sums of the squares of the dimensions (characters under E) of each of the irreducible representations equals the order of the group.

h = [ i (E) ]2

5. For any irreducible representation, the sum ofthe squares of the characters multiplied by thenumber of operations in the class, equals theorder of the group

h = [ i (R) ]2

6. Irreducible representations are orthogonalto each other. The sum of the products ofthe characters (multiplied together for eachclass) for any pair of irreduciblerepresentation is 0.

i (R) j (R) = 0, when i not equal to j

7. A totally symmetry representation isincluded in all groups, with characters of 1for all operations

Example: C2V has A1 which has allcharacters = 1

Examples:

Determine the Irreducible representation of the following:

1. Water molecule: = 9 -1 3 1

2. NH3 molecule: = 12 0 2

Degrees of Freedom (3N)

• Translational modes

• Rotational modes

• Vibrational modes

– For linear molecule: 3N – 5

– For non-linear molecule: 3N – 6

where: N = no. of atoms

Example

• Consider the irreducible representation of thewater molecule determined in the previousexample,

(a) how many vibrational mode(s) do thiscompound contain?

(b) what irreducible representation(s)represent(s) the vibrational modes of water?

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