Chemistry Reference
In-Depth Information
Ta b l e 3 . 6
Symmetries of a
tetrahedral molecule in a
uniform magnetic (
B
)or
electric (
E
)field
E
8
C
3
3
C
2
6
S
4
6
σ
d
T
d
∩
C
∞
h
B
S
4
E
C
2
2
S
4
S
4
B
C
3
E
2
C
3
C
3
E
B
⊥
σ
d
σ
d
C
s
T
d
∩
C
∞
v
E
S
4
E
C
2
2
σ
d
C
2
v
E
C
3
E
2
C
3
3
σ
d
C
3
v
E
E
∈ˆ
σ
d
σ
d
ˆ
C
s
will thus consist only of symmetry elements that are common to both parts. These
elements form the
intersection
of both symmetry groups. The elements of an inter-
section themselves form a group, which is the largest common subgroup of both
symmetry groups. This can be written as follows:
magnetic field
:
H
=
G
∩
C
∞
h
(3.38)
electric field
:
H
=
G
∩
C
∞
v
This intersection group will depend on the orientation of the field in the molecu-
lar frame. In Table
3.6
we work out an example of a tetrahedral molecule. The top
row lists the symmetry elements of
T
d
. The fields can be oriented along several di-
rections. The highest symmetry positions are along the fourfold or threefold axes.
A lower symmetry position is within or perpendicular to a symmetry plane, or finally
along an arbitrary direction with no symmetry at all. In Appendix
B
we list repre-
sentative intersection groups for several point groups and orientations. Note that in
the case of a magnetic field, the resulting intersection group is always abelian. This
is of course a consequence of
C
∞
h
being abelian.
3.10 Problems
3.1 The multiplication table of a set of elements is given below. Does this set form
a group?
ABCD
A CDAB
B DCBA
C ABCD
D BADC
3.2 Use molecular ball and stick models to construct examples of molecules that
have a reflection plane as the only symmetry element. Similarly, for a center of
inversion and for a twofold axis. In each case find the solution with the smallest
