Chemistry Reference
In-Depth Information
molecule. Remember that, once the point group is assigned, the full list of symmetry
operations can be read from the top of the character table in Appendix 12.
3.8 Special Groups for Linear Molecules:
C
∞
v
and
D
∞
h
To construct general objects for the axial groups, we have used points on the principal axis
to show that operations involving this axis leave these points unaffected. In groups with an
axis of order infinity, any off-axis points would generate an infinite number of new atom
positions under the rotation operation. This means that any molecule belonging to these
groups must be linear. The principal axis will then contain all the atoms in the molecule,
and so any rotation will leave the atom coordinates unaffected. The order of the axis is
infinite, since there are an infinite number of possible angles of rotation which leave the
molecule in an identical state. In a similar way, any plane containing the axis must be a
valid symmetry element for the object, since it will contain all the points belonging to the
structure. So, there are also an infinite number of vertical mirror planes present.
Figure 3.23 shows the two point groups that arise for linear molecules. If the two ends
of the molecule are different, then the only symmetry elements are
C
∞
and the vertical
mirror planes, so the point group is
C
∞
v
by analogy with
C
2v
,
C
3v
, etc. If the molecule has
equivalent points at either end of the axis then it will also have a horizontal mirror plane
σ
h
and an infinite number of
C
2
axes perpendicular to the principal axis. In this case the point
group will be
D
∞
h
, since the same types of element are present as for
D
n
h
point groups.
The character table titles for these two groups are shown in Figure 3.24. The presentation
(a)
(b)
D
∞h
σ
v
σ
v
σ
v
σ
v
C
2
C
∞
v
C
2
σ
v
σ
v
i
C
∞
C
∞
σ
h
C
2
Figure 3.23
Objects constructed for (a) the C
∞v
and (b) D
∞h
point groups. In each case,
only a small selection of the infinite number of possible
σ
v
planes is shown.
C
∞
v
E
2
C
∞
...
∞
σ
v
(a)
D
∞
h
E
2
C
∞
...
∞
σ
v
i
2
S
∞
...
∞
C
2
(b)
Figure 3.24
The headings used for the character tables for linear molecules: (a) C
∞v
;
(b) D
∞h
. In each case, the column heading '
' indicates an infinite number of similar oper-
ations to the preceding column and the superscript
...
refers to the angle of rotation for the
operation.
