Chemistry Reference
In-Depth Information
plane over. The effect of repeated
S
3
improper rotations on the BF
3
molecule can then be
followed more clearly.
After even numbers of operations the arrow returns to the top face of the molecule, and
so there is an equivalent simple rotation that can give the same result. Since the order of
the axis is odd,
S
3
3
does not return the molecule to its start point; although all the atoms
are in the same place as in the original configuration, the arrow is pointing down, and so
S
3
3
=
σ
h
. Only after six applications of the operation do we find that
S
3
6
=
E
. In this case
there are only two unique operations arising from the
S
3
axis:
S
3
1
and
S
3
5
.
In general, for an
odd
ordered improper rotation:
1.
S
n
2
m
=
C
n
p
, where
m
runs from 1 to (
n
−
1) and
p
=
2
m
for 2
m
<
n
or
p
=
2
m
−
n
for
n
; that is, an even number of applications of an odd-order improper rotation is the
same as a related simple rotation. The operations with rotations less than 360
◦
can also
be described as the same number of simple rotations, and improper rotations involving
angles greater than 360
◦
are related to odd numbers of simple rotations.
2.
S
n
n
2
m
>
=
σ
h
; that is, carrying out the improper rotation a number of times equal to its order
results in a simple reflection through the horizontal mirror plane.
3.
S
n
2
n
E
; that is, it requires 2
n
applications of the operation to return the molecule to
its starting configuration.
=
2.4 Properties of Symmetry Operations
2.4.1 Equivalent Operations and Equivalent Atoms
It is worth pausing at this point to think about what the symmetry operations are doing
when used to describe the molecular geometry. As an example, we will return to the case
of ethane, for which the operations
E
,
C
3
1
,
C
3
2
,3
C
2
(three equivalent
C
2
axes),
i
,
S
6
1
,
S
6
5
σ
d
have now been identified, i.e. a total of 12 unique operations. The results
of these operations have been described in terms of the rearrangement of the hydrogen
atoms around the framework of the ethane structure. The two methyl groups that form
the molecule always remain with the same composition, even when the atom labelling
is taken into account, so that C
1
H
1
-
3
forms one methyl and C
2
H
4
-
6
the other. However,
the symmetry operations can change the end of the molecule occupied by each methyl
group and can change the ordering of hydrogen atoms within the group, from clockwise to
anticlockwise. The symmetry operations give a systematic way to rearrange the equivalent
atoms within a molecule so that each atom 'visits' all chemically equivalent positions. This
idea was used in the discussion of NMR spectroscopy in Chapter 1.
We limit the list of operations by identifying equivalences so that there is no redun-
dancy; each arrangement of the labelled atoms in the molecule is generated only once. The
hydrogen atoms of ethane are particularly useful for this. If we chose the carbon atoms,
then there are only ever two arrangements: the carbon atoms sit on the
C
3
axis and in the
three mirror planes and so are unchanged by
C
3
1
or
C
3
2
rotations or any simple reflection.
They are also swapped over by
S
6
1
and
S
6
5
any
C
2
rotation or
i
. The carbon atoms do not
show the full effect of these operations because they are at special symmetry positions.
For the same reason, in planar molecules like BF
3
, no atom sets show the full effects of
symmetry operations; in particular, the mirror plane containing all the atoms will always
and 3
