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
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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.