44
3 Groups
Fig. 3.13 The rotating cone. The molecule is a subporphyrin and consists of a central boron in
a tri-pyrrole macrocycle. The subporphyrin itself has the shape of a trigonal dome and exhibits
C 3v symmetry. Three phenyl substituents at the meso-positions are arranged like a propeller and
reduce the symmetry to C 3 . A further symmetry lowering to C 1 is caused by an apical hydroxyl
substituent at the boron position, with its hydrogen pointing in the direction of the upper phenyl
group
around the axis of the cone are retained, limiting the symmetry group to C ∞ .I t s
molecular point groups are the cyclic groups, C n . These symmetries are encountered in propeller-like molecules. An example of a subporphyrin [7]i ss h o w ni n
Fig. 3.13. The smallest nontrivial C n group is found for n = 2. This is the symmetry
of the Möbius strip, which may also be attained in Möbius-type annulenes.
3.8 Rotational Groups and Chiral Molecules
The symmetry operations that we have encountered are either proper or improper.
Proper symmetry elements are rotations, also including the unit element. The improper rotations comprise planes of symmetry, rotation–reflection axes, and spatial
inversion. All improper elements can be written as the product of spatial inversion
and a proper rotation (see, e.g., Fig. 1.1). The difference between the two kinds of
symmetry elements is that proper rotations can be carried out in real space, while
improper elements require the inversion of space and thus a mapping of every point
onto its antipode. This can only be done in a virtual way by looking at the structure via a mirror. From a mathematical point of view, this difference is manifested
3 Groups
Fig. 3.13 The rotating cone. The molecule is a subporphyrin and consists of a central boron in
a tri-pyrrole macrocycle. The subporphyrin itself has the shape of a trigonal dome and exhibits
C 3v symmetry. Three phenyl substituents at the meso-positions are arranged like a propeller and
reduce the symmetry to C 3 . A further symmetry lowering to C 1 is caused by an apical hydroxyl
substituent at the boron position, with its hydrogen pointing in the direction of the upper phenyl
group
around the axis of the cone are retained, limiting the symmetry group to C ∞ .I t s
molecular point groups are the cyclic groups, C n . These symmetries are encountered in propeller-like molecules. An example of a subporphyrin [7]i ss h o w ni n
Fig. 3.13. The smallest nontrivial C n group is found for n = 2. This is the symmetry
of the Möbius strip, which may also be attained in Möbius-type annulenes.
3.8 Rotational Groups and Chiral Molecules
The symmetry operations that we have encountered are either proper or improper.
Proper symmetry elements are rotations, also including the unit element. The improper rotations comprise planes of symmetry, rotation–reflection axes, and spatial
inversion. All improper elements can be written as the product of spatial inversion
and a proper rotation (see, e.g., Fig. 1.1). The difference between the two kinds of
symmetry elements is that proper rotations can be carried out in real space, while
improper elements require the inversion of space and thus a mapping of every point
onto its antipode. This can only be done in a virtual way by looking at the structure via a mirror. From a mathematical point of view, this difference is manifested