Chemistry Reference
In-Depth Information
These observations also imply that, if an improper axis of even order is found, there must
be a rotation axis with half its order and an inversion centre. Also, for a molecule with an
inversion centre, there must be an improper axis of rotation. In the case of a molecule with
an inversion centre but without any simple rotation axes, the improper rotation is an
S
2
axis. However, this
S
2
axis is not usually quoted, since
S
2
1
i
and
S
2
2
=
=
E
, so there are
no unique operations associated with an
S
2
axis.
The improper rotation consists of a rotation and reflection. Even though the axis and
horizontal plane need not be elements themselves, if they are present then the improper
rotation will also be a symmetry element. For example, the planar molecule BF
3
has a
principal
C
3
axis and a horizontal mirror plane
σ
h
, and so there is also an
S
3
axis collinear
with the
C
3
. For planar molecules, the reflection in
σ
h
does not alter any atom positions;
however, if we place a vertical arrow on one of the F atoms, then it will be reversed by the
reflection. In later chapters, the addition of arrows like this will be used in the analysis of
molecular vibrations and is referred to as a
basis
. A basis allows us to study the effect of
symmetry operations not only on the atom positions but also their motion. In Figure 2.9
the idea is simpler: we add the arrow to highlight operations which turn the molecular
S
1
F
F
B
F
B
F
F
F
S
1
2
2
S
3
=
C
3
F
B
F
F
S
1
3
S
3
=
σ
h
F
B
F
F
S
1
4
1
S
3
=
C
3
F
F
B
F
S
1
S
5
F
F
B
F
S
1
6
S
3
=
E
F
F
B
F
Figure 2.9
The complete set of S
3
operations illustrated using BF
3
. Note that to show the
result of each operation fully requires an arrow perpendicular to the molecular plane. Each
operation is labelled by its relation to the common starting point (top left).
