3.10 Problems
47
Table 3.6 Symmetries of a
tetrahedral molecule in a
uniform magnetic (B)or
electric (E)field
ˆ
E
8 ˆ
C 3
3 ˆ
C 2
6 ˆ
S 4
6 ˆ
σ d
T d ∩ C ∞h
B ˆ
S 4
ˆ
E
ˆ
C 2
2 ˆ
S 4
S 4
B ˆ
C 3
ˆ
E
2 ˆ
C 3
C 3
B ⊥ˆ σ d
ˆ
E
ˆ
σ d
C s
T d ∩ C ∞v
E ˆ
S 4
ˆ
E
ˆ
C 2
2 ˆ
σ d
C 2v
E ˆ
C 3
ˆ
E
2 ˆ
C 3
3 ˆ
σ d
C 3v
E ∈ˆ σ d
ˆ
E
ˆ
σ d
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 intersection 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
electric field : H = G ∩ C ∞v
(3.38)
This intersection group will depend on the orientation of the field in the molecular 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 directions. 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 representative 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
47
Table 3.6 Symmetries of a
tetrahedral molecule in a
uniform magnetic (B)or
electric (E)field
ˆ
E
8 ˆ
C 3
3 ˆ
C 2
6 ˆ
S 4
6 ˆ
σ d
T d ∩ C ∞h
B ˆ
S 4
ˆ
E
ˆ
C 2
2 ˆ
S 4
S 4
B ˆ
C 3
ˆ
E
2 ˆ
C 3
C 3
B ⊥ˆ σ d
ˆ
E
ˆ
σ d
C s
T d ∩ C ∞v
E ˆ
S 4
ˆ
E
ˆ
C 2
2 ˆ
σ d
C 2v
E ˆ
C 3
ˆ
E
2 ˆ
C 3
3 ˆ
σ d
C 3v
E ∈ˆ σ d
ˆ
E
ˆ
σ d
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 intersection 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
electric field : H = G ∩ C ∞v
(3.38)
This intersection group will depend on the orientation of the field in the molecular 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 directions. 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 representative 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