Group theory questions and answers

Typical questions about group theory in chemistry with solutions.

1. GROUP THEORY
Q + A
Dr. Christoph Sontag
Phayao University
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2. 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
H
H
H O
C
CH
3
CH
3 C
OH
C
OH
H
H
H
H

3. POINT GROUPS
Use the flowchart to find the point groups of the 5
molecules before:










Ammonia:

Acetone:

Dimethyl-cyclopentane:

Ethanediol:

Propadiene:

3
D
nd

4. ANSWERS
Ammonia: C
3 (and
2
C
3), 3 σ
v => C
3v

Acetone: C
2, σ
v , σ
v => C
2v

Dimethyl-cyclopentane: σ => C
s

Ethanediol: C
2, i, σ
h => C
2h

Propadiene: C
2, C
2’ , C
2’, σ
v , σ
v, S
4,
3
S
4 => D
2d









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5. 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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6. ANSWERS
S-axis in propadiene:




Inversion center of ethanediol:




S3 axis in Trihydroxybenzene:
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7. 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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8. ANSWER
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=> The px orbital has B1 representation

9. 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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10. 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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11. BH
3 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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12. ANSWERS
The symmetry elements of BH
3 are:

C
3, 3x C
2, 2x S
3, 2x σ
v => 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”.

13. MO DIAGRAM OF BH
3
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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14. ANSWERS
The combination of H-orbitals with no node has the
symmetry a
1’



and the 2 other combinations with one node e’



Therefore the first group orbital can overlap with the s-
orbital of Boron (both have a
1’ 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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15. COMPLETE MO DIAGRAM BH
3
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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.

16. 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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17. 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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18. 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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19. 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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20. 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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21. ANSWER
21
The representation A2 is 2 times inside Г

22. HYBRIDIZATION
Trichlorborane is in the point group D
3h.
What are the characters of the 3 σ-bonds ?

E 2C
3 3C
2 σ
h 2S
3 3σ
v
Г : 3









Г is the sum of which basic representation ?


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23. ANSWER
The 3 σ bonds have the characters:
E 2C
3 3C
2 σ
h 2S
3 3σ
v
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 σ
v 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 = sp
2
hybrid.



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