Chemistry Reference
In-Depth Information
3.6 Axial Groups Containing Mirror Planes:
C
n
h
and
C
n
v
In these groups there is only one axis and the subscript
n
in the group label is used to indi-
cate its order. The single axis must be the principal axis and defines the vertical direction
for the molecule. The
C
n
h
point groups contain a single horizontal plane, while the
C
n
v
groups have
n
vertical planes as discussed below.
We begin with an idealized
C
4v
object created from four generating points. Two of
these are placed on the axis and two off axis, as shown in the two alternative settings
in Figure 3.12a and b. The points are positioned so that one of the vertical mirror planes is
the plane of the paper, as shown by the dotted rectangle labelled
σ
v
. In Figure 3.12a, the
mirror plane is valid for both sets of off-axis points, since the generating points at either
end of the molecule are also in one of the two
σ
v
planes; this is reminiscent of a molecule
in an eclipsed conformation.
(a)
(b)
1
σ
v
σ
v
1
C
4v
C
4v
1
1
σ
d
σ
d
σ
v
σ
v
σ
d
σ
d
σ
v
σ
v
Figure 3.12
Objects in the C
4v
point group generated to give (a) eclipsed and (b) staggered
geometry. The points marked '1' are suggested start points for generating the figures. The
lower diagrams are plan views with shading used to give a sense of depth.
The alternative arrangement, shown in Figure 3.12b, is to have one generating point in
the
σ
d
plane. Figure 3.12b shows that this implies a
staggered conformation for the imaginary molecule, with, in this
C
4v
example, an angle of
45
◦
between the
σ
v
mirror plane and one placed in the
σ
d
. Any other relation between the positions of the sets of points at
either end of the object would not be consistent with this arrangement of the mirror planes,
and so the symmetry of the object would reduce to
C
4
.
From Figure 3.12a and b we can also see that any atom which is away from the principal
axis of a
C
n
v
molecule will have
n
symmetry-related atoms by virtue of the
n
rotational
σ
v
and
