group theory questions and answers
DESCRIPTION
Typical questions about group theory in chemistry with solutions.TRANSCRIPT
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GROUP THEORY Q + A
Dr. Christoph Sontag Phayao University
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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
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POINT GROUPS
Use the flowchart to find the point groups of the 5
molecules before:
Ammonia:
Acetone:
Dimethyl-cyclopentane:
Ethanediol:
Propadiene:
3
Dnd
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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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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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ANSWERS
S-axis in propadiene:
Inversion center of ethanediol:
S3 axis in Trihydroxybenzene:
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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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ANSWER
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=> The px orbital has B1 representation
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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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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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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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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”.
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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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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
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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.
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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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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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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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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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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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ANSWER
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The representation A2 is 2 times inside Г
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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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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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