42
3 Groups
Fig. 3.10 Hexagonal D 6h
symmetry of benzene:
Orientation of twofold
symmetry elements. ˆ
C ′
2 and
ˆ
σ v pass through opposite
atoms; ˆ
C ′′
2 and ˆ
σ d bisect
opposite bonds
Fig. 3.11 The rotating
cylinder has the symmetry of
a magnetic field, B, along the
cylinder axis. (a)showsa
tetra-imine macrocycle, with
the bridging CH 2 groups
below and above the
molecular plane; the resulting
symmetry group is C 2h ;
(b) shows stereographic
projections of C 2h and S 4
which is a reflection axis, ˆ
S 2n+1 , of order 4n + 2. For rotations of even order, two
kinds of subgroups arise: the C 2nh groups and the S 2n groups. The latter are not to
be confused with the symmetric groups but designate cyclic groups, generated by
a2 n-fold rotation–reflection axis. Figure 3.11 shows the stereographic projections
for C 2h and S 4 , and a molecular realization of C 2h . Note that the latter symmetry
group is a combination of the three different kinds of binary symmetry elements of
the point groups: a twofold rotation, the reflection plane, and the spatial inversion.
It reminds us of the famous Euler identity e iπ =−1, which brings together three
special numbers: the base of natural logarithms (e), the square-root of −1(i), and
the ratio of the circumference to the diameter of a circle (π ).
3 Groups
Fig. 3.10 Hexagonal D 6h
symmetry of benzene:
Orientation of twofold
symmetry elements. ˆ
C ′
2 and
ˆ
σ v pass through opposite
atoms; ˆ
C ′′
2 and ˆ
σ d bisect
opposite bonds
Fig. 3.11 The rotating
cylinder has the symmetry of
a magnetic field, B, along the
cylinder axis. (a)showsa
tetra-imine macrocycle, with
the bridging CH 2 groups
below and above the
molecular plane; the resulting
symmetry group is C 2h ;
(b) shows stereographic
projections of C 2h and S 4
which is a reflection axis, ˆ
S 2n+1 , of order 4n + 2. For rotations of even order, two
kinds of subgroups arise: the C 2nh groups and the S 2n groups. The latter are not to
be confused with the symmetric groups but designate cyclic groups, generated by
a2 n-fold rotation–reflection axis. Figure 3.11 shows the stereographic projections
for C 2h and S 4 , and a molecular realization of C 2h . Note that the latter symmetry
group is a combination of the three different kinds of binary symmetry elements of
the point groups: a twofold rotation, the reflection plane, and the spatial inversion.
It reminds us of the famous Euler identity e iπ =−1, which brings together three
special numbers: the base of natural logarithms (e), the square-root of −1(i), and
the ratio of the circumference to the diameter of a circle (π ).