3.7 Overview of the Point Groups
43
Fig. 3.12 The twisted
cylinder. On the right is a
cartoon of the green
fluorescent protein (GFP). It
adopts a β-barrel structure.
Eleven β-sheet strands are
wound in a helix around the
central fluorophore. The
approximate symmetry is
D 11 . In the barrel a cleft is
opened to provide
interactions with the
surrounding solvent
The Twisted Cylinder
An alternative symmetry lowering of the cylinder is attained in the twisted cylinder.
Here, all improper symmetry elements are broken, the remaining group being the
rotational subgroup, D ∞ (see Fig. 3.12). The twist itself can be in two opposite
directions, which are mirror images of each other. Molecules of this type may be
formed by applying a twist to prisms or antiprisms. A simple example is ethane.
In the eclipsed high-energy conformation this is a prism with D 3h symmetry, while
in the staggered conformation it is an antiprism, D 3d . By rotating the two methyl
groups with respect to each other in opposite senses about the threefold axis, we may
interconvert these two conformations. In between the two extremal conformations
the frame adopts the largest common subgroup, which is D 3 .
Cones
A cone can be considered as a deformation of a cylinder, in which the poles of the
uniaxial direction are no longer equivalent. The symmetry group is reduced accordingly to C ∞v , where only the vertical symmetry planes remain. Conical symmetry
is exemplified by hetero-nuclear diatomic molecules, but it is also the symmetry of
a polar vector, such as a translation in a given direction, or a polarized medium or
an electric field, etc. Conical molecules have C nv symmetries, as was the case for
the ammonia model. Again, the smallest trivial member of this series is C 2v , which
is fully anisotropic. This is the point group of the water molecule.
The Rotating Cone
Adding a rotation to the cone will destroy the vertical symmetry planes, since reflection in these planes would alter the sense of the rotation. As a result, only rotations
43
Fig. 3.12 The twisted
cylinder. On the right is a
cartoon of the green
fluorescent protein (GFP). It
adopts a β-barrel structure.
Eleven β-sheet strands are
wound in a helix around the
central fluorophore. The
approximate symmetry is
D 11 . In the barrel a cleft is
opened to provide
interactions with the
surrounding solvent
The Twisted Cylinder
An alternative symmetry lowering of the cylinder is attained in the twisted cylinder.
Here, all improper symmetry elements are broken, the remaining group being the
rotational subgroup, D ∞ (see Fig. 3.12). The twist itself can be in two opposite
directions, which are mirror images of each other. Molecules of this type may be
formed by applying a twist to prisms or antiprisms. A simple example is ethane.
In the eclipsed high-energy conformation this is a prism with D 3h symmetry, while
in the staggered conformation it is an antiprism, D 3d . By rotating the two methyl
groups with respect to each other in opposite senses about the threefold axis, we may
interconvert these two conformations. In between the two extremal conformations
the frame adopts the largest common subgroup, which is D 3 .
Cones
A cone can be considered as a deformation of a cylinder, in which the poles of the
uniaxial direction are no longer equivalent. The symmetry group is reduced accordingly to C ∞v , where only the vertical symmetry planes remain. Conical symmetry
is exemplified by hetero-nuclear diatomic molecules, but it is also the symmetry of
a polar vector, such as a translation in a given direction, or a polarized medium or
an electric field, etc. Conical molecules have C nv symmetries, as was the case for
the ammonia model. Again, the smallest trivial member of this series is C 2v , which
is fully anisotropic. This is the point group of the water molecule.
The Rotating Cone
Adding a rotation to the cone will destroy the vertical symmetry planes, since reflection in these planes would alter the sense of the rotation. As a result, only rotations