Week 5 M - The
Cn
point
groups:
 • ...

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Unformatted text preview: The
Cn
point
groups:
 •  The
Cn
point
groups
all
have
only
a
single
rota4onal
axis
(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
 rota4onal
axis.

 •  The
Cnh
point
groups
also
have
a
σh
mirror
plane
at
right
angles
to
 the
principal
rota4onal
axis.
 O
 C5 iodine F
 [IF6O]‐
 F
 The
Cn
point
groups:
 •  
 Cn
axis
as
their
only
symmetry
element.

 •  They
generally
resemble
propellers
which
have
the
front
and
back
different.

 C3 triphenyl phosphine viewed down C3 axis C3 C3 triphenyl phosphine viewed from the side C3 C3 Cobalt(III) tris-glycinate viewed down C3 axis C3 Cobalt(III) tris-glycinate viewed from the side Characteris4c
symmetry
elements
of
some
important
classes

 Point
 Characteris1c
symmetry
elements
 Group
 Comments
 C1
 E,
C1
 Cs
 E,
one
σ
plane
 Ci 
 E,
inversion
centre
 Cn
 E,
one
principal
n‐fold
axis
 S n
 Only
Sn
(n
even,
n≠2)
 Cnv
 E,
one
principal
n‐fold
axis,
n
σv
planes
 Cnh
 E,
one
principal
n‐fold
axis,
one
σh
plane
 The
Sn
axis
necessarily
follows
from
the
Cn
 and
one
Sn‐fold
axis,
coincident
with
the
Cn
 axis
and
σh
plane.
For
n=2,
4
or
6
there
is
 axis.
 also
a
centre
of
inversion
 Dnh
 E,
one
principal
n‐fold
axis,
n
C2
axes,
one
 σh
plane,
n
σv
planes

and
one
Sn‐fold
axis.
 The
Sn
axis
necessarily
follows
from
the
Cn
 axis
and
σh
plane.
For
n=2,
4
or
6
there
is
 also
a
centre
of
inversion
 Dnd
 E,
one
principal
n‐fold
axis,
n
C2
axes,
n
σv
 planes

and
one
S2n‐fold
axis.
 For
n=3
or
5
there
is
also
a
centre
of
 inversion
 Td
 tetrahedral
 Oh
 octahedral
 Ih
 Icosahedral
 Some
C2v
molecules:
 C2 σv P
 C2 S
 σv Phosphorus iodotetrafluoride (PF4I) σv C
 σv sulfur tetrafluoride (SF4) C2 σv σv carbonyl chloride (COCl2) Characteris4c
symmetry
elements
of
some
important
classes

 Point
 Characteris1c
symmetry
elements
 Group
 Comments
 C1
 E,
C1
 Cs
 E,
one
σ
plane
 Ci 
 E,
inversion
centre
 Cn
 E,
one
principal
n‐fold
axis
 S n
 Only
Sn
(n
even,
n≠2)
 Cnv
 E,
one
principal
n‐fold
axis,
n
σv
planes
 Cnh
 E,
one
principal
n‐fold
axis,
one
σh
plane
 The
Sn
axis
necessarily
follows
from
the
Cn
 and
one
Sn‐fold
axis,
coincident
with
the
Cn
 axis
and
σh
plane.
For
n=2,
4
or
6
there
is
 axis.
 also
a
centre
of
inversion
 Dnh
 E,
one
principal
n‐fold
axis,
n
C2
axes,
one
 σh
plane,
n
σv
planes

and
one
Sn‐fold
axis.
 The
Sn
axis
necessarily
follows
from
the
Cn
 axis
and
σh
plane.
For
n=2,
4
or
6
there
is
 also
a
centre
of
inversion
 Dnd
 E,
one
principal
n‐fold
axis,
n
C2
axes,
n
σv
 planes

and
one
S2n‐fold
axis.
 For
n=3
or
5
there
is
also
a
centre
of
 inversion
 Td
 tetrahedral
 Oh
 octahedral
 Ih
 Icosahedral
 Examples
of
molecules
belonging
to
Dnh
point
 groups:
 C2 C4 C3 C3 D2h D3h C4 D4h D4h C5 C3 D3h D3h C5 D5h D5h Characteris4c
symmetry
elements
of
some
important
classes

 Point
 Characteris1c
symmetry
elements
 Group
 Comments
 C1
 E,
C1
 Cs
 E,
one
σ
plane
 Ci 
 E,
inversion
centre
 Cn
 E,
one
principal
n‐fold
axis
 S n
 Only
Sn
(n
even,
n≠2)
 Cnv
 E,
one
principal
n‐fold
axis,
n
σv
planes
 Cnh
 E,
one
principal
n‐fold
axis,
one
σh
plane
 The
Sn
axis
necessarily
follows
from
the
Cn
 and
one
Sn‐fold
axis,
coincident
with
the
Cn
 axis
and
σh
plane.
For
n=2,
4
or
6
there
is
 axis.
 also
a
centre
of
inversion
 Dnh
 E,
one
principal
n‐fold
axis,
n
C2
axes,
one
 σh
plane,
n
σv
planes

and
one
Sn‐fold
axis.
 The
Sn
axis
necessarily
follows
from
the
Cn
 axis
and
σh
plane.
For
n=2,
4
or
6
there
is
 also
a
centre
of
inversion
 Dnd
 E,
one
principal
n‐fold
axis,
n
C2
axes,
n
σv
 planes

and
one
S2n‐fold
axis.
 For
n=3
or
5
there
is
also
a
centre
of
 inversion
 Td
 tetrahedral
 Oh
 octahedral
 Ih
 Icosahedral
 C∞v
and
D∞h
point
group
 •  C∞
indicates
an
∞‐fold
axis
of
rota4on
i.e.
that
possessed
 by
a
linear
molecule.
 •  In

C∞v
point
group,
there
are
also
an
infinite
number
of
σv
 planes
but
NO
σh
plane
or
inversion
center.
 •  
For
D∞h
same
symmetry
elements
than
in
C∞v
but
with
a
 CENTRE
OF
INVERSION,
a
σh
plane.
 C∞v
 D∞h
 Characteris4c
symmetry
elements
of
some
important
classes

 Point
 Characteris1c
symmetry
elements
 Group
 Comments
 C1
 E,
C1
 Cs
 E,
one
σ
plane
 Ci 
 E,
inversion
centre
 Cn
 E,
one
principal
n‐fold
axis
 S n
 Only
Sn
(n
even,
n≠2)
 Cnv
 E,
one
principal
n‐fold
axis,
n
σv
planes
 Cnh
 E,
one
principal
n‐fold
axis,
one
σh
plane
 The
Sn
axis
necessarily
follows
from
the
Cn
 and
one
Sn‐fold
axis,
coincident
with
the
Cn
 axis
and
σh
plane.
For
n=2,
4
or
6
there
is
 axis.
 also
a
centre
of
inversion
 Dnh
 E,
one
principal
n‐fold
axis,
n
C2
axes,
one
 σh
plane,
n
σv
planes

and
one
Sn‐fold
axis.
 The
Sn
axis
necessarily
follows
from
the
Cn
 axis
and
σh
plane.
For
n=2,
4
or
6
there
is
 also
a
centre
of
inversion
 Dnd
 E,
one
principal
n‐fold
axis,
n
C2
axes,
n
σv
 planes

and
one
S2n‐fold
axis.
 For
n=3
or
5
there
is
also
a
centre
of
 inversion
 Td
 tetrahedral
 Oh
 octahedral
 Ih
 Icosahedral
 Td,
Oh,
Ih
point
groups
 Td
point
group
 Oh,
Ih
point
groups
 Characteris4c
symmetry
elements
of
some
important
classes

 Point
 Characteris1c
symmetry
elements
 Group
 Comments
 C1
 E,
C1
 Cs
 E,
one
σ
plane
 Ci 
 E,
inversion
centre
 Cn
 E,
one
principal
n‐fold
axis
 S n
 Only
Sn
(n
even,
n≠2)
 Cnv
 E,
one
principal
n‐fold
axis,
n
σv
planes
 Cnh
 E,
one
principal
n‐fold
axis,
one
σh
plane
 The
Sn
axis
necessarily
follows
from
the
Cn
 and
one
Sn‐fold
axis,
coincident
with
the
Cn
 axis
and
σh
plane.
For
n=2,
4
or
6
there
is
 axis.
 also
a
centre
of
inversion
 Dnh
 E,
one
principal
n‐fold
axis,
n
C2
axes,
one
 σh
plane,
n
σv
planes

and
one
Sn‐fold
axis.
 The
Sn
axis
necessarily
follows
from
the
Cn
 axis
and
σh
plane.
For
n=2,
4
or
6
there
is
 also
a
centre
of
inversion
 Dnd
 E,
one
principal
n‐fold
axis,
n
C2
axes,
n
σv
 planes

and
one
S2n‐fold
axis.
 For
n=3
or
5
there
is
also
a
centre
of
 inversion
 Td
 tetrahedral
 Oh
 octahedral
 Ih
 Icosahedral
 How
to
determine
a
Point
group
 Use
flow
chart
 Flow
Chart
 Molecular
Structure
 Is
the
molecule
linear?
 C∞v
 Does
the
molecule
contain
 inversion
centre?
 No
 Does
the
molecule
belong
 to
high
symmetry
group?
 D∞h
 No
 Yes
 Yes
 Yes
 T,
Td,
Th,
I,
Ih,
O,
Oh
point
groups
 Dnh

 Yes
 No
 Yes
 Does
the
molecule
contain
 a
proper
rota4on
axis
(Cn)?
 No
 Iden4fy
the
highest
order
Cn.
 Are
there
n
perpendicular
C2
 axes?
 Yes
 Does
the
 molecule
 contain
a
σh?
 No
 Does
the
molecule
contain
 a
mirror
plane?
 Yes
 Cs
 No
 Does
the
molecule
contain
 a
σh?
 Yes
 No
 Cnh
 Does
the
molecule
 contain
n
σd?
 No
 Does
the
molecule
contain
 an
inversion
center?
 Yes
 Ci 
 Does
the
molecule
 contain
n
σv?
 Yes
 Cnv
 No
 No
 C1
 Does
the
molecule
contain
 a
2n‐fold
improper
rota4on
 axis?
 Yes
 S2n
 No
 Cn
 No
 Dn

 Yes
 Dnd
 ...
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