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

2. C3 C3v D4d D4h 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
plus n 2-fold axes at
right angles to it
A ‘C’ indicates only one rotational axis
C C3v D4d D4h
3-fold rotational has σv but 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 σv σh
D4h
D3d
A subscript ‘d’ (or v for C groups) means there is no σh mirror
plane, but only n σv mirror planes containing the principal Cn axis.

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

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

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

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

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

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 C1 F
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
axis, we have a
Dn point group,
If not, it is a Cn
point group. Dn Cn

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. The Cn point groups have no other symmetry elements, the Cnv point groups have also n mirror planes containing the Cn rotational axis, while the Cnh point groups also have a σh mirror plane at right angles to the principal rotational axis. C5 O
iodine F F
[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. σv σv there is no σh plane at right angles to the C2 axis, but
there are two σv
planes containing
the C2 axis. C2 C2 C2 σv σv
The point group of the water molecule is C2v

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

18. Some more C2v molecules: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): C3
triphenyl
phosphine
viewed down
C3 axis
Cobalt(III)
tris-glycinate
viewed down
C3 axis C3 C3 C3 C3 C3
triphenyl
phosphine
viewed from
the side
Cobalt(III)
tris-glycinate
viewed from
the side

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

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

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

24. D2 The Dn point groups: n-fold axis, and n C2 principal axis C2 C2
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
with H-atoms
omitted for clarity.
(en = ethylene
diamine) Cu N D2 C2

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

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

27. Other examples of the D3 point groupC3
principal axis C2 C2 C2 C2 C2 C2 D3 D3
[Co(oxalate)3] [Co(bipyridyl)3]3+

28. Some cobalt(III) complexes belonging to the D3 point group:
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 D3 C3 C2
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. D4d The D4d point group: σv σv σv σv C4 principal axis C2 C2 C2 C2 C2
[ZrF8]4-
Square antiprism
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
right angles to C4, so the point group does not rate an h, and is D4d.

32. [K(18-crown-6)]+, an example of a D3d point group:
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