# Group Theory in Chemistry - questions and answers

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GROUP THEORY Q + A Dr. Christoph Sontag Phayao University

1

SYMMETRY ELEMENTS

Please write down all symmetry elements of :

Ammonia

c-axis:

:

Acetone

c-axis:

:

Dimethyl-cyclopentane

c-axis:

:

Ethanediol

c-axis:

:

i :

Propadiene

c-axis:

: S-axis:

2

N

HH

H

O

C

CH3 CH3

C

OH

C

OH

H

H

H

H

POINT GROUPS

Use the flowchart to find the point groups of the 5

molecules before:

Ammonia:

Acetone:

Dimethyl-cyclopentane:

Ethanediol:

Propadiene:

3

Dnd

ANSWERS

Ammonia: C3 (and 2C3), 3 v => C3v

Acetone: C2, v , v => C2v

Dimethyl-cyclopentane: => Cs

Ethanediol: C2, i, h => C2h

Propadiene: C2, C2 , C2, v , v, S4, 3S4 => D2d

4

SYMMETRY ELEMENTS

Can you draw the S-axis into the propadiene molecule

(remember: S contains C-axis and plane !)

Please mark the inversion center in ethanediol:

Mark the S3 axis in

Trihydroxybenzene:

5

ANSWERS

S-axis in propadiene:

Inversion center of ethanediol:

S3 axis in Trihydroxybenzene:

6

CHARACTER TABLES

The water molecule has C2v symmetry:

Which characters does the px orbital of Oxygen have and

what is the name of the representation ?

7

ANSWER

8

=> The px orbital has B1 representation

SYMMETRY OF AN MO OF WATER

The water molecule has C2v symmetry:

One molecular orbital consists

of the px orbital of Oxygen

overlapping with the 2 s-orbitals

of the Hydrogens:

Determine the characters of this MO in the point group

C2v:

This representation consists of which irreducible

representation from the character table ?

9

ANSWERS

The MO of the water molecule has the following

symmetry behaviour for the C2v operations:

This representation equals the B1 symmetry from the

character table.

10

BH3 MOLECULE

BH3 is a planar molecule. Can you draw the structure

and all symmetry elements ?

What is the point group of the molecule ?

What are the characters of the pz orbital of Boron and

what is the representation of it ?

11

ANSWERS

The symmetry elements of BH3 are:

C3, 3x C2, 2x S3, 2x v => belongs to D3h

The pz orbital transforms as follows:

12

Notice that the pz orbital has

the same representation as

the z-axis.

Therefore we can find the

representation directly in the character table under z.

MO DIAGRAM OF BH3

To construct the MOs of the molecule, we can combine

the atomic orbitals of Boron with combination group

orbitals of 3x H:

What is the characters of these 3 group orbitals ?

and have to be taken together which

leads to the representation e

Therefore they match the symmetry of which AO of

Boron ?

13

ANSWERS

The combination of H-orbitals with no node has the

symmetry a1

and the 2 other combinations with one node e

Therefore the first group orbital can overlap with the s-

orbital of Boron (both have a1 symmetry):

and the e orbitals with px and py of Boron:

The pz orbital of Boron has no partner and stays non-

bonding

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COMPLETE MO DIAGRAM BH3

15

From the diagram we can conclude that the molecule is a

LEWIS-ACID because the LUMO is low in energy (its non-

bonding) and if we fill electrons into it will not affect the B-H bonds.

ORBITALS OF CYCLOPROPENYL

The Cyclopropenyl cation is an aromatic ring system

with 3 p orbitals:

Determine the representation of these 3 p orbitals in the

D3h point group:

This representation is the sum of which irreducible representations of the character table ?

16

ANSWERS

The 3 p-orbitals transform under the symmetry

operations of D3h as follows:

This representation is the sum of which representations

of the character table ?

We use the reduction formula to check for example if A2

is in the representation :

With the same method

we find that also e is

in .

17

MOLECULAR ORBITALS

Butadiene has the point group C2h:

Which representation does the HOMO of the molecule

have ?

Draw the C2 axis, inversion center and mirror plane and

check if the MO not the 4 single AOs ! behave.

18

ANSWER

The HOMO has the representation = 1 -1 -1 1

and is therefore Bu.

For example: C2 rotation would reverse the phases of all

orbitals => -1 character

19

REDUCING REPRESENTATIONS

When we have a representation of = 4 1 -2

for a molecule with C3v symmetry, how many times is

the representation A2 in ?

20

ANSWER

21

The representation A2 is 2 times inside

HYBRIDIZATION

Trichlorborane is in the point group D3h.

What are the characters of the 3 -bonds ?

E 2C3 3C2 h 2S3 3v

: 3

is the sum of which basic representation ?

22

Inorganic graphics/Ic3/B/molecules/bf3c.mol

ANSWER

The 3 bonds have the characters:

E 2C3 3C2 h 2S3 3v

3 0 1 3 0 1

Each of the 3 C2 rotations keep one bond at the same

position, but exchanges the 2 other bonds => 1

The same is true for the 3 v mirror planes which are each

on one bond.

The horizontal mirror plane leaves all 3 bonds in their

position => 3

is the sum of the representations A1 and E.

A1 represents also the s-orbital, and E represents the px

and py orbital on the central atom.

From this we can conclude that the 3 B-Cl -bonds are

formed by the combination of s + px + py = sp2 hybrid.

23

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