The determination of point groups

1. The determination of point groups of
molecules
only one rotational
axis = C
2
two σ
v
but no σ
h
mirror planes means
point group is C
2v
The point group of the water molecule is C
2v

2. Naming point groups:
The name of the point group has information about the
symmetry elements present. The letter is the rotational group
and the subscript number after the letter indicates the order
of the principal rotational axis (e.g. 3-fold or 4 fold etc.):
C
3
C
3v
D
4d
D
4h
A ‘C’ indicates only
one rotational axis
A ‘D’ indicates an n-fold
principal rotation axis
plus n 2-fold axes at
right angles to it
3-fold rotational has σ
v
but 4-fold d = no ‘h’ indicates
axis no σ
h
mirror principal σ
h
mirror a σ
h
mirror
planes in a C group axis plane plane

3. A subscript ‘h’ means that there is a σ
h
mirror plane at
right angles to the n-fold principal axis:
Naming point groups (contd.):
D
4h
C
4
principal axis
σ
h
C
3
principal axis
σ
v
A subscript ‘d’ (or v for C groups) means there is no σ
h
mirror
plane, but only n σ
v
mirror planes containing the principal C
n
axis.
only one
of the three
σ
v
planes
is shown
D
3d

4. Naming platonic solids:
Platonic solids:
T = tetrahedral = 4 three-fold axes
O = octahedral = 3 four-fold axes
I = icosahedral = 6 five-fold axes
T
d
O
h
I
h
C
60
‘bucky-ball’
or ‘Fullerene’

5. Flow chart for determining point groups.

6. The point group of the carbon dioxide
molecule
We start at the top of the
flow-chart, and can see that
the CO
2
molecule is linear,
and has a center of inversion
(i) so it is D
∞h
. Note the C
∞

principal rotation axis.
i
C
∞
D
∞h

7. Other linear molecules:
HC≡N HI C≡O
N
2
O
2
F
2
H
2
D
∞h
C
∞v
i i
The top row of linear molecules all have a center of
inversion (i) and so are D
∞h
.
The bottom row have no
i and so are C
∞v
All have a C
∞
axis

8. The Platonic solids:
T
d
O
h
I
h
C
60
‘buckyball’
tetrahedron octahedron icosahedron

9. The C
s
point group:
C
s
σ
chloro-difluoro-iodo-
methane
I
F
Cl
C
F

10. Most land animals have bilateral symmetry,
and belong to the C
s
point group:
Mirror planes (σ)C
s
C
s

11. The C
1
point group:
Molecules that have no symmetry elements at all except
the trivial one where they are rotated through 360º and
remain unchanged, belong to the C
1
point group. In other
words, they have an axis of 360º/360º = 1-fold, so have
a C
1
axis. Examples are:
Bromo-chloro-fluoro-iodo- chloro-iodo-amine
methane
I
Br
F
Cl
C
I
Cl
H
N
C
1
C
1

12. The division into C
n
and D
n
point groups:
After we have
decided that
there is a
principal rotat-
ional axis, we
come to the
red box. If there
are n C
2
axes
at right angles
to the principal
axis, we have a
D
n
point group,
If not, it is a C
n
point group.
D
n
C
n

13. The C
n
point groups:
The C
n
point groups all have only a single rotational
axis, which can theoretically be very high e.g. C
5
in
the complex [IF
6
O]
-
below. They are further divided
into C
n
, C
nv
, and C
nh
point groups.
C
5
[IF
6
O]
-
•The C
n
point groups have no
other symmetry elements
•the C
nv
point groups have also n
mirror planes containing the C
n

rotational axis
•the C
nh
point groups also have a
σ
h
mirror plane at right angles to
the principal rotational axis.

14. The point group of the water molecule
We start at the top of the
flow-chart, and can see that
the water molecule is not
linear, and is not tetrahedral (T
d
),
octahedral (O
h
), or icosahedral,
(I
h
) so we proceed down the chart

15. C
2
Yes, there is a principal C
n
axis,
so we proceed down the chart, but
in answer to the next question, there
are no further C
2
axes at right angles
to the principal axis, which is the only
axis, so we proceed down the chart

16. The point group of the water molecule is C
2v
there is no σ
h
plane
at right angles to
the C
2
axis, but
there are two σ
v
planes containing
the C
2
axis.
C
2
C
2
C
2
σ
v
σ
v

17. Other C
nv
molecules:
C
2v
C
3v
C
4v
ammoniawater
σ
v
σ
v
σ
v
Vanadyl tetrafluoride (VOF
4
)
V

18. Some more C
2v
molecules:
σ
v
σ
v
σ
v
σ
v
σ
v
σ
v
C
2 C
2
C
2
Phosphorus iodo- sulfur tetra- carbonyl
tetrafluoride (PF
4I) fluoride (SF
4) chloride (COCl
2)
C
SP

19. These have a C
n
axis as their only symmetry element. They
generally resemble propellers which have the front and back
different. Important examples are (hydrogens omitted for clarity):
The C
n
point groups:
C
3
C
3
C
3
C
3
C
3
C
3
triphenyl
phosphine
viewed down
C
3
axis
Cobalt(III)
tris-glycinate
viewed down
C
3
axis
triphenyl
phosphine
viewed from
the side
Cobalt(III)
tris-glycinate
viewed from
the side

21. The D
nh
point groups:
C
2 σ
h
four C
2
axes at
rt. angles
to C
4
axis
C
2
C
2
C
2
C
4

principal
axis
mirror plane
at rt. angles
to C
4
axis
D
4h

22. Examples of molecules belonging to D
nh
point
groups:
D
2h
D
3h
D
3h
D
3h
D
4h
D
4h
D
5h
D
5h
C
2 C
3
C
3
C
3
C
4
C
4
C
5
C
5

23. C
6
principal axis
C
2
C
2
C
2
C
6
C
2
σ
v
σ
v
Benzene, an example of the D
6h
point group:
σ
h
C
6
principal axis
C
6
principal axis
D
6h

24. The D
n
point groups:
C
2
C
2
C
2
principal axis
D
2
these have a principal
n-fold axis, and n
2-fold axes at right
angles to it, but no
mirror planes.
[Cu(en)
2
]
2+
complex
with H-atoms
omitted for clarity.
(en = ethylene
diamine)
Cu
N
N
C

25. Some further views of the symmetry elements
of [Cu(en)
2
]
2+
, point group D
2
:
C
2
[Cu(en)
2
]
2+
complex
with H-atoms
omitted for clarity.
(en = ethylene
diamine)
C
2
C
2
C
2
C
2
C
2
C
2
principal
axis
C
2
principal axis
C
2
principal
axis
C
2
principal
axis
C
2
C
2
D
2

26. C
2
C
2
C
2
C
3
principal axis
Some views of the symmetry elements of
[Co(en)
3
]
3+
, point group D
3
.
C
3
principal axis
C
2
axis
view down the C
3
axis
of [Co(en)
3
]
3+
showing
the three C
2
axes.
D
3
view down one of the
three C
2
axes of [Co(en)
3
]
3+

at right angles to C
3

27. Other examples of the D
3
point group
[Co(oxalate)
3
]
3-
[Co(bipyridyl)
3
]
3+
C
2 C
2
C
2
C
2
C
2
C
2
C
3
principal axis
D
3
D
3

28. Some cobalt(III) complexes belonging to
the D
3
point group:
NH
2
Co
NH
2
H
2N
H
2
N
H
2N N
H
2
Co
N
N
N
N
N
N
Co
O
O
O
O
O
O
CH
3
H
3C
H
3C
CH
3
CH
3
CH
3
3+ 3+
tris(ethylenediamine) tris(2,2’-bipyridyl)tris(acetylacetonato)
cobalt(III) cation cobalt(III) cationcobalt(III)
D
3
C
2
C
2
C
2

29. Comparison of C
3
and D
3
tris(chelates)
NH
2
Co
NH
2
H
2N
H
2
N
H
2N N
H
2
O
Co
NH
2
H
2N
H
2
N
O O
3+
O
O
O
D
3
C
3C
2
no C
2
axis at
this point
tris(ethylenediamine)cobalt(III) tris(glycinato)cobalt(III)

30. Molecules belonging to the D
nd
point groups
These have mirror planes parallel to the principal
axis, but not at right angles to it.
C
3
axis
C
5
axis
Staggered form
of ethane Staggered form of ferrocene
σ
v
planes
contain the
principal
axis
D
3d D
5d

31. The D
4d
point group:
C
2
C
2
C
2
C
2
C
4
principal axis
C
4
principal axis
C
2
σ
v
σ
v
σ
v
σ
v
C
4
principal axis
[ZrF
8
]
4-
Square antiprism
As predicted by VSEPR, the [ZrF
8
]
4-
anion has a square anti-prismatic
structure. At left is seen the C
4
principal axis. It has four C
2
axes at
right angles to it, so it has D
4
symmetry. One C
2
axis is shown side-on
(center). There are four σ
v
mirror planes (right), but no mirror plane at
right angles to C
4
, so the point group does not rate an h, and is D
4d
.
D
4d

32. [K(18-crown-6)]
+
, an example of a D
3d

point group:
The complex cation [K(18-crown-6)]
+
above is an important structure that
has D
3d
symmetry. It has a C
3
principal axis with 3 C
2
axes at right
angles to it, as well as three σ
v
mirror planes that contain the C
3
axis,
but no σ
h
mirror plane (because it’s not flat, as seen at center), so is D
3d
.
D
3d
σ
v
σ
v
K
+
C
3
principal axis
C
3
principal axis
σ
v
C
2
C
2
C
2
C
2
C
2
C
2

33. Some Point groups