The determination of point groups of molecules

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The determination of point groups of molecules
two sv but no sh mirror planes means
point group is C2v
only one rotational axis C2
The point group of the water molecule is C2v
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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
C3 C3v D4d D4h
3-fold rotational has sv but
4-fold d no h indicates
axis no sh mirror
principal sh mirror a sh mirror
planes in a C group axis
plane plane
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Naming point groups (contd.)

• A subscript h means that there is a sh mirror
  plane at right angles to the n-fold principal
  axis

C4 principal axis
C3 principal axis
only one of the three sv planes is shown
sv
sh
D4h
D3d
A subscript d (or v for C groups) means there
is no sh mirror plane, but only n sv mirror
planes containing the principal Cn axis.
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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
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Flow chart for determining point groups.
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The point group of the carbon dioxide molecule
i
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 D8h. Note the
C8 principal rotation axis.
C8
D8h
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Other linear molecules
The top row of linear molecules all have a center
of inversion (i) and so are D8h.
D8h
i
i
N2 O2
F2 H2
HCN HI
CO
The bottom row have no i and so are C8v
C8v
All have a C8 axis
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The Platonic solids
tetrahedron octahedron icosahedron
C60 buckyball
Td Oh Ih
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The Cs point group
s
I
Cs
C
Cl
chloro-difluoro-iodo- methane
F
F
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Most land animals have bilateral symmetry, and
belong to the Cs point group
Cs
Cs
Mirror planes (s)
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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
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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
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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 sh mirror plane at
  right angles to the principal rotational
  axis.

C5
O
iodine
F
F
IF6O-
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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
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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
C2
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there is no sh plane at right angles to the C2
axis, but there are two sv planes containing the
C2 axis.
C2
C2
C2
sv
sv
The point group of the water molecule is C2v
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Other Cnv molecules
ammonia
water
sv
C2v
C3v
sv
C4v
sv
V
Vanadyl tetrafluoride (VOF4)
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Some more C2v molecules
sv
sv
sv
C2
C2
C2
S
P
C
sv
sv
sv
Phosphorus iodo- sulfur tetra-
carbonyl tetrafluoride (PF4I)
fluoride (SF4) chloride (COCl2)
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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
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(No Transcript)
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The Dnh point groups
four C2 axes at rt. angles to C4 axis
C4 principal axis
mirror plane at rt. angles to C4 axis
C2
C2
sh
C2
C2
D4h
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Examples of molecules belonging to Dnh point
groups
C3
C3
C2
C3
D2h
D3h
D3h
D3h
C4
C5
C4
C5
D4h
D4h
D5h
D5h
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Benzene, an example of the D6h point group
C6 principal axis
C2
C2
C2
C2
C6
sh
sv
sv
D6h
C6 principal axis
C6 principal axis
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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)22 complex with H-atoms omitted for
clarity. (en ethylene diamine)
Cu
N
D2
C2
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Some further views of the symmetry elements of
Cu(en)22, point group D2
C2 principal axis
Cu(en)22 complex with H-atoms omitted for
clarity. (en ethylene diamine)
C2
D2
C2 principal axis
C2
C2 principal axis
C2 principal axis
C2
C2
C2
C2
C2
C2
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Some views of the symmetry elements of
Co(en)33, point group D3.
C2
C3 principal axis
C2
C2
D3
C3 principal axis
C2 axis
view down the C3 axis of Co(en)33 showing the
three C2 axes.
view down one of the three C2 axes of
Co(en)33 at right angles to C3
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Other examples of the D3 point group
C2
C3 principal axis
C2
C2
C2
C2
C2
D3
D3
Co(oxalate)33-
Co(bipyridyl)33
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Some cobalt(III) complexes belonging to the D3
point group
C2
C2
C2
tris(ethylenediamine) tris(2,2-bipyridyl) tr
is(acetylacetonato) cobalt(III) cation
cobalt(III) cation cobalt(III)
D3
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Comparison of C3 and D3 tris(chelates)
no C2 axis at this point
D3
C3
C2
tris(ethylenediamine)cobalt(III)
tris(glycinato)cobalt(III)
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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
sv planes contain the principal axis
D3d
D5d
Staggered form of ethane
Staggered form of ferrocene
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The D4d point group
sv
C4 principal axis
C2
C2
sv
sv
C2
C2
sv
C2
C4 principal axis
C4 principal axis
ZrF84- Square antiprism
D4d
As predicted by VSEPR, the ZrF84- 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
sv mirror planes (right), but no mirror plane
at right angles to C4, so the point group does
not rate an h, and is D4d.
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K(18-crown-6), an example of a D3d point group
C3 principal axis
C3 principal axis
sv
C2
C2
K
sv
C2
C2
C2
sv
C2
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 sv mirror
planes that contain the C3 axis, but no sh mirror
plane (because its not flat, as seen at center),
so is D3d.
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Some Point groups