1. The determination of point groups of
molecules
only one rotational two σv but no σh mirror planes means
axis = C2 point group is C2v
The point group of the water molecule is C2v

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.):
A ‘D’ indicates an n-fold
principal rotation axis
A ‘C’ indicates only plus n 2-fold axes at
one rotational axis right angles to it
C3 C3v D4d D4h
3-fold rotational has σv but 4-fold d = no ‘h’ indicates
axis no σh mirror principal σh mirror a σh mirror
planes in a C group axis plane plane

3. Naming point groups (contd.):
A subscript ‘h’ means that there is a σh mirror plane at
right angles to the n-fold principal axis:
C4 principal axis C3 principal axis
only one
of the three
σv planes
is shown
σh σv
D4h D3d
A subscript ‘d’ (or v for C groups) means there is no σh mirror
plane, but only n σ mirror planes containing the principal C axis.

4. Naming platonic solids:
Platonic solids:
T = tetrahedral = 4 three-fold axes
O = octahedral = 3 four-fold axes
I = icosahedral = 6 five-fold axes
C60
‘bucky-ball’
or ‘Fullerene’
Td Oh Ih

5. Flow chart for determining point groups.

6. The point group of the carbon dioxide
molecule
i We start at the top of the
flow-chart, and can see that
C∞ the CO2 molecule is linear,
and has a center of inversion
(i) so it is D∞h. Note the C∞
D∞h
principal rotation axis.

7. Other linear molecules:
The top row of linear molecules all have a center of
inversion (i) and so are D∞h.
i D∞h i
N2 O2 F2 H2
HC≡N HI C≡O
The bottom row have no C∞v All have a C∞
i and so are C∞v axis

8. The Platonic solids:
tetrahedron octahedron icosahedron
Td Oh Ih C60
‘buckyball’

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

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

11. The C1 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 C1 point group. In other
words, they have an axis of 360º/360º = 1-fold, so have
a C1 axis. Examples are:
I
I N
Cl
C H
Br
Cl
C1 F C1
Bromo-chloro-fluoro-iodo- chloro-iodo-amine
methane

12. The division into Cn and Dn point groups:
After we have
decided that
there is a
principal rotat-
ional axis, we
come to the
red box. If there
are n C2 axes
at right angles
to the principal Dn
axis, we have a
Dn point group,
If not, it is a Cn Cn
point group.

13. The Cn point groups:
The Cn point groups all have only a single rotational
axis, which can theoretically be very high e.g. C5 in
the complex [IF6O]- below. They are further divided
into Cn, Cnv, and Cnh point groups.
C5
•The Cn point groups have no
other symmetry elements
•the Cnv point groups have also n
mirror planes containing the Cn
rotational axis
•the Cnh point groups also have a
σh mirror plane at right angles to
the principal rotational axis.
[IF6O]-

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 (Td),
octahedral (Oh), or icosahedral,
(Ih) so we proceed down the chart

15. C2 Yes, there is a principal Cn axis,
so we proceed down the chart, but
in answer to the next question, there
are no further C2 axes at right angles
to the principal axis, which is the only
axis, so we proceed down the chart

16. C2 C2 C2 there is no σh plane
at right angles to
σv σv the C2 axis, but
there are two σv
planes containing
the C2 axis.
The point group of the water molecule is C2v

17. Other Cnv molecules:
water ammonia
σv
C2v
C3v
σv
C4v V σv
Vanadyl tetrafluoride (VOF4)

18. Some more C2v molecules:
C2 σv C2 σv C2 σv
P S
C
σv σv σv
Phosphorus iodo- sulfur tetra- carbonyl
tetrafluoride (PF4I) fluoride (SF4) chloride (COCl2)

19. The Cn point groups:
These have a Cn axis as their only symmetry element. They
generally resemble propellers which have the front and back
different. Important examples are (hydrogens omitted for clarity):
triphenyl C3 Cobalt(III)
phosphine tris-glycinate
viewed down viewed down
C3 axis C3 C3 axis
C3 C3
C3 C3
triphenyl Cobalt(III)
phosphine tris-glycinate
viewed from viewed from
the side the side

21. The Dnh point groups:
C4 four C2 mirror plane
principal axes at
at rt. angles
rt. angles
axis to C4 axis to C4 axis
C2
C2 σh
C2 C2
D4h

22. Examples of molecules belonging to Dnh point
groups:
C2 C3 C3 C3
D2h D3h D3h D3h
C4
C5
C4 C5
D4h D4h D5h D5h

23. Benzene, an example of the D6h point group:
C6 C2
principal axis C2
C2
C6 σh C2
σv σv
D6h
C6
principal axis
C6
principal axis

24. The Dn point groups:
these have a principal
n-fold axis, and n
2-fold axes at right
angles to it, but no
mirror planes.
C2 principal axis
C
N
C2 [Cu(en)2]2+ complex
Cu
N with H-atoms
omitted for clarity.
(en = ethylene
diamine)
C2 D2

25. Some further views of the symmetry elements
of [Cu(en)2]2+, point group D2 :
C2 principal axis
[Cu(en)2]2+ complex
C2
D2 with H-atoms
omitted for clarity.
(en = ethylene
diamine)
C2 principal C2
axis C2 principal C2 principal
C2
axis axis
C2 C2 C2
C2 C2

26. Some views of the symmetry elements of
[Co(en)3]3+, point group D3.
C2
C3 principal axis
C2
C2
C3 D3 C2
principal axis axis
view down the C3 axis view down one of the
of [Co(en)3]3+ showing three C2 axes of [Co(en)3]3+
the three C2 axes. at right angles to C3

27. Other examples of the D3 point group
C3 C2
C2 C2 principal axis
C2
C2
C2
D3 D3
[Co(oxalate)3]3- [Co(bipyridyl)3]3+

28. Some cobalt(III) complexes belonging to
the D3 point group:
C2 C2 C2
CH3
3+ 3+
NH2 H H3C O CH3
N
2
H 2N N O O
N N
Co Co Co
O O
H 2N N N N
H2 H3C O CH3
NH2 N
CH3
tris(ethylenediamine) tris(2,2’-bipyridyl) tris(acetylacetonato)
cobalt(III) cation cobalt(III) cation cobalt(III)
D3

29. Comparison of C3 and D3 tris(chelates)
no C2 axis at
this point
C2
D3 O C3
3+
O H2
NH2 H
2 H 2N N
H2N N
Co Co
O O
H2N N O
H2 NH2
NH2 O
tris(ethylenediamine)cobalt(III) tris(glycinato)cobalt(III)

30. Molecules belonging to the Dnd point groups
These have mirror planes parallel to the principal
axis, but not at right angles to it.
C5 axis
C3 axis σv planes
contain the
principal
axis
D3d D5d
Staggered form
of ethane Staggered form of ferrocene

31. The D4d point group:
C4 principal axis σv
C2 C2 σv σv
C2 C2
C2 σv
C4 C4
[ZrF8] 4-
principal axis
Square antiprism
principal axis
D4d
As predicted by VSEPR, the [ZrF8]4- anion has a square anti-prismatic
structure. At left is seen the C4 principal axis. It has four C2 axes at
right angles to it, so it has D4 symmetry. One C2 axis is shown side-on
(center). There are four σv mirror planes (right), but no mirror plane at

32. [K(18-crown-6)]+, an example of a D3d
point group:
C3
principal axis
C3 principal axis σv
C2 C2
K+
σv
C2
C2 C2
C2 σv
D3d
The complex cation [K(18-crown-6)]+ above is an important structure that
has D3d symmetry. It has a C3 principal axis with 3 C2 axes at right
angles to it, as well as three σv mirror planes that contain the C3 axis,
but no σh mirror plane (because it’s not flat, as seen at center), so is D3d.

33. Some Point groups