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Page 1: Group Theory in Chemistry - questions and answers

GROUP THEORY Q + A

Dr. Christoph Sontag Phayao University

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Page 2: Group Theory in Chemistry - questions and answers

SYMMETRY ELEMENTS

Please write down all symmetry elements of :

Ammonia

c-axis:

σ :

Acetone

c-axis:

σ :

Dimethyl-cyclopentane

c-axis:

σ :

Ethanediol

c-axis:

σ :

i :

Propadiene

c-axis:

σ : S-axis:

2

N

HH

H

O

C

CH3 CH3

C

OH

C

OH

H

H

H

H

Page 3: Group Theory in Chemistry - questions and answers

POINT GROUPS

Use the flowchart to find the point groups of the 5

molecules before:

Ammonia:

Acetone:

Dimethyl-cyclopentane:

Ethanediol:

Propadiene:

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Dnd

Page 4: Group Theory in Chemistry - questions and answers

ANSWERS

Ammonia: C3 (and 2C3), 3 σv => C3v

Acetone: C2, σv , σv => C2v

Dimethyl-cyclopentane: σ => Cs

Ethanediol: C2, i, σh => C2h

Propadiene: C2, C2’ , C2’, σv , σv, S4, 3S4 => D2d

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Page 5: Group Theory in Chemistry - questions and answers

SYMMETRY ELEMENTS

Can you draw the S-axis into the propadiene molecule

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

Please mark the inversion center in ethanediol:

Mark the S3 axis in

Trihydroxybenzene:

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Page 6: Group Theory in Chemistry - questions and answers

ANSWERS

S-axis in propadiene:

Inversion center of ethanediol:

S3 axis in Trihydroxybenzene:

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Page 7: Group Theory in Chemistry - questions and answers

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 ?

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Page 8: Group Theory in Chemistry - questions and answers

ANSWER

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=> The px orbital has B1 representation

Page 9: Group Theory in Chemistry - questions and answers

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 Г consists of which irreducible

representation from the character table ?

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Page 10: Group Theory in Chemistry - questions and answers

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.

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Page 11: Group Theory in Chemistry - questions and answers

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 ?

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Page 12: Group Theory in Chemistry - questions and answers

ANSWERS

The symmetry elements of BH3 are:

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

The pz orbital transforms as follows:

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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”.

Page 13: Group Theory in Chemistry - questions and answers

MO DIAGRAM OF BH3

To construct the MO’s 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 ?

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Page 14: Group Theory in Chemistry - questions and answers

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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Page 15: Group Theory in Chemistry - questions and answers

COMPLETE MO DIAGRAM BH3

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From the diagram we can conclude that the molecule is a

LEWIS-ACID because the LUMO is low in energy (it’s non-

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

Page 16: Group Theory in Chemistry - questions and answers

Π ORBITALS OF CYCLOPROPENYL

The Cyclopropenyl cation is an aromatic ring system

with 3 pπ 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 ?

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Page 17: Group Theory in Chemistry - questions and answers

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 Г:

With the same method

we find that also e” is

in Г.

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Page 18: Group Theory in Chemistry - questions and answers

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 AO’s ! – behave.

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Page 19: Group Theory in Chemistry - questions and answers

ANSWER

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

and is therefore Bu.

For example: C2 rotation would reverse the phases of all

orbitals => -1 character

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Page 20: Group Theory in Chemistry - questions and answers

REDUCING REPRESENTATIONS

When we have a representation of Г = 4 1 -2

for a molecule with C3v symmetry, how many times is

the representation A2 in Г ?

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Page 21: Group Theory in Chemistry - questions and answers

ANSWER

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The representation A2 is 2 times inside Г

Page 22: Group Theory in Chemistry - questions and answers

HYBRIDIZATION

Trichlorborane is in the point group D3h.

What are the characters of the 3 σ-bonds ?

E 2C3 3C2 σh 2S3 3σv

Г : 3

Г is the sum of which basic representation ?

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Page 23: Group Theory in Chemistry - questions and answers

ANSWER

The 3 σ bonds have the characters:

E 2C3 3C2 σh 2S3 3σv

Г 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 σv mirror planes which are each

on one bond.

The horizontal mirror plane leaves all 3 bonds in their

position => 3

Г 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 σ-bonds are

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

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