Point Group = the set of symmetry operations for a molecule
Group Theory = mathematical treatment of the properties of the group which
can be used to find properties of the molecule
Assigning the Point Group of a Molecule
1. Determine if the molecule is of high or low symmetry by inspection
A. Low Symmetry Groups
Point Groups
B. High Symmetry Groups
2. If not, find the principle axis
3. If there are C2 axes
perpendicular to Cn the
molecule is in D If not, the
molecule will be in C or S
4. If h perpendicular to Cn then
Dnh or Cnh If not, go to the next
step
5. If contains Cn then Cnv or Dnd
If not, Dn or Cn or S2n
6. If S2n along Cn then S2n
7. If not Cn
The determination of point groups of
molecules
only one rotational
axis = C2
two σv but no σh mirror planes means
point group is C2v
The point group of the water molecule is C2v
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.):
C3 C3v D4d D4h
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 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
A subscript ‘h’ means that there is a σh mirror plane at
right angles to the n-fold principal axis:
Naming point groups (contd.):
D4h
C4 principal axis
σh
C3 principal axis
σv
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.
only one
of the three
σv planes
is shown
D3d
Naming platonic solids:
Platonic solids: Five special polyhedra: Tetrahedron, Cube,
Octahedron, Dodecahedron, and
Icosahedron
Their faces are all exactly the same.
The same number of faces meet at each vertex.
T = tetrahedral = 4 three-fold axes
O = octahedral = 3 four-fold axes
I = icosahedral = 6 five-fold axes
Td Oh Ih
C60 ‘bucky-ball’
or ‘Fullerene’
Flow chart for determining point groups.
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.
i
C∞
D∞h
Other linear molecules:
HC≡N HI C≡O
N2 O2 F2 H2
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
The Platonic solids:
Td Oh Ih C60
‘buckyball’
tetrahedron octahedron icosahedron
The Cs point group:
Cs
σ
chloro-difluoro-iodo-
methane
I
F Cl
C
F
Most land animals have bilateral symmetry,
and belong to the Cs point group:
Mirror planes (σ) Cs
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:
Bromo-chloro-fluoro-iodo- chloro-iodo-amine
methane
I
Br
F
Cl
C
I
Cl
H
N
C1 C1
The division into Cn and Dn point groups:
After we have decided
that there is a principal
rotation 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
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
iodine
[IF6O]-
O
F
F
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
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
The point group of the water molecule is C2v
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
Other Cnv molecules:
C2v
C3v
C4v
ammonia water
σv
σv
σv
Vanadyl tetrafluoride (VOF4)
V
Some more C2v molecules:
σv
σv
σv σv
σv σv
C2 C2 C2
Phosphorus iodo- sulfur tetra- carbonyl
tetrafluoride (PF4I) fluoride (SF4) chloride (COCl2)
C S P
These have a Cn axis as their only symmetry element. Important
examples are (hydrogens omitted for clarity):
The Cn point groups:
C3 C3
C3 C3
C3
C3 triphenyl
phosphine
viewed down
C3 axis
Cobalt(III)
tris-glycinate
viewed down
C3 axis
triphenyl
phosphine
viewed from
the side
Cobalt(III)
tris-glycinate
viewed from
the side
The Dnh point groups:
C2 σh
four C2
axes at
rt. angles
to C4 axis
C2 C2
C2
C4
principal
axis
mirror plane
at rt. angles
to C4 axis
D4h
Examples of molecules belonging to Dnh point
groups:
D2h D3h D3h D3h
D4h D4h D5h D5h
C2 C3
C3 C3
C4
C4
C5 C5
C6
principal axis C2
C2
C2 C6
C2
σv σv
Benzene, an example of the D6h point group:
σh
C6
principal axis
C6
principal axis
D6h
The Dn point groups:
C2
C2
C2 principal axis
D2
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
Some further views of the symmetry elements
of [Cu(en)2]2+, point group D2 :
C2
[Cu(en)2]2+ complex
with H-atoms
omitted for clarity.
(en = ethylene
diamine)
C2
C2
C2
C2
C2
C2 principal
axis
C2 principal axis
C2 principal
axis
C2 principal
axis
C2 C2
D2
C2
C2
C2
C3 principal axis
Some views of the symmetry elements of
[Co(en)3]3+, point group D3.
C3
principal axis
C2
axis
view down the C3 axis
of [Co(en)3]3+ showing
the three C2 axes.
D3
view down one of the
three C2 axes of [Co(en)3]3+
at right angles to C3
Other examples of the D3 point group
[Co(oxalate)3]3- [Co(bipyridyl)3]
3+
C2 C2
C2
C2
C2
C2 C3
principal axis
D3 D3
Molecules belonging to the Dnd point groups
These have mirror planes parallel to the principal
axis, but not at right angles to it.
C3 axis C5 axis
Staggered form
of ethane Staggered form of ferrocene
σv planes
contain the
principal
axis
D3d D5d
The D4d point group:
C2 C2
C2 C2
C4
principal axis
C4 principal axis
C2
σv
σv
σv
σv
C4
principal axis [ZrF8]4-
Square antiprism
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.
D4d
[K(18-crown-6)]+, an example of a D3d
point group:
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.
D3d
σv
σv
K+
C3
principal axis C3 principal axis
σv
C2 C2
C2
C2
C2
C2
Some Point groups
Recommended