3.7 Overview of the Point Groups
41
Fig. 3.9 Molecular realizations of cylindrical symmetry: (a) D 5h ruthenocene in an eclipsed pentagonal prismatic conformation and D 5d ferrocene in a staggered pentagonal antiprismatic conformation; (b) top view of the pentagonal antiprism with the position of the five twofold axes
perpendicular to the pentagonal direction; (c) staggered conformation of triplet ethylene with D 2d
symmetry and horizontal ˆ
S 4 axis (in the triplet the two carbon 2p-orbitals shown are singly occupied); (d) alternative view of the D 2d symmetry of a bisphenoid, with vertical ˆ
S 4 axis
in-plane ˆ
C 2 axes. The triplet excited state of ethylene adopts a staggered conformation, which has D 2d symmetry (Fig. 3.9(c)). Benzene has prismatic D 6h symmetry.
In this case, there are two classes of perpendicular twofold axes, which are distinguished in the tables by the labels ˆ
C ′
2 and ˆ
C ′′
2 , and, likewise, two classes of vertical
symmetry planes, ˆ
σ v and ˆ
σ d . The standard choice for orienting these elements in the
hexagonal molecular frame is shown in Fig. 3.10. This choice is conventional and
may be changed, but note that the two symmetries are coupled in that the ˆ
σ v planes
contain the ˆ
C ′
2 axes and, similarly, the ˆ
σ d planes contain the ˆ
C ′′
2 axes.
The Rotating Cylinder
Other than geometrical distortions, symmetry breaking of a cylinder may also be
realized by a dynamic effect, as in a rotating cylinder. In this case, only symmetry
elements that will not change the direction of rotation are allowed. As a result, the
twofold rotation axes have to be removed, and the cylindrical symmetry is reduced
to that of the point group, C ∞h . This is the symmetry of an axial vector or pseudovector. It corresponds to the spatial symmetry of a magnetic field. Note that this
group is abelian, and so are its molecular subgroups, with symmetry C nh .Again,the
parity of n is important here. C (2n+1)h groups are cyclic. They have one generator,
41
Fig. 3.9 Molecular realizations of cylindrical symmetry: (a) D 5h ruthenocene in an eclipsed pentagonal prismatic conformation and D 5d ferrocene in a staggered pentagonal antiprismatic conformation; (b) top view of the pentagonal antiprism with the position of the five twofold axes
perpendicular to the pentagonal direction; (c) staggered conformation of triplet ethylene with D 2d
symmetry and horizontal ˆ
S 4 axis (in the triplet the two carbon 2p-orbitals shown are singly occupied); (d) alternative view of the D 2d symmetry of a bisphenoid, with vertical ˆ
S 4 axis
in-plane ˆ
C 2 axes. The triplet excited state of ethylene adopts a staggered conformation, which has D 2d symmetry (Fig. 3.9(c)). Benzene has prismatic D 6h symmetry.
In this case, there are two classes of perpendicular twofold axes, which are distinguished in the tables by the labels ˆ
C ′
2 and ˆ
C ′′
2 , and, likewise, two classes of vertical
symmetry planes, ˆ
σ v and ˆ
σ d . The standard choice for orienting these elements in the
hexagonal molecular frame is shown in Fig. 3.10. This choice is conventional and
may be changed, but note that the two symmetries are coupled in that the ˆ
σ v planes
contain the ˆ
C ′
2 axes and, similarly, the ˆ
σ d planes contain the ˆ
C ′′
2 axes.
The Rotating Cylinder
Other than geometrical distortions, symmetry breaking of a cylinder may also be
realized by a dynamic effect, as in a rotating cylinder. In this case, only symmetry
elements that will not change the direction of rotation are allowed. As a result, the
twofold rotation axes have to be removed, and the cylindrical symmetry is reduced
to that of the point group, C ∞h . This is the symmetry of an axial vector or pseudovector. It corresponds to the spatial symmetry of a magnetic field. Note that this
group is abelian, and so are its molecular subgroups, with symmetry C nh .Again,the
parity of n is important here. C (2n+1)h groups are cyclic. They have one generator,