Reflection axis

This is the operation of clockwise rotation by 2w/ about an axis followed by reflection in a plane perpendicular to that axis (or vice versa, the order is not important). If this brings the molecule into coincidence with itself, the molecule is said to have a n-fold alternating axis of symmetry (or improper axis, or rotation-reflection axis) as a symmetry element. It is the knight s move of symmetry. It is symbolized by Sn and illustrated for a tetrahedral molecule in Fig. 2-3.3.f... 

The Cahn-Ingold-Prelog rules work for inorganic compounds too but coordination complexes often have coordination numbers greater then four and may exhibit helical chirality, for example, denoted A and A (or Pand Min the Cahn-Ingold-Prelog system). The formal condition for chirality is that the molecule should not have an improper axis of rotation (i.e. a rotation + reflection axis, 5n =... 

Chirality is the geometric property of a rigid object (or spatial arrangement of points or atoms), which is nonsuperposable on its mirror image such an object has no symmetry elements of the second kind (a mirror plane, a center of inversion, a rotation-reflection axis,. ..). If the object is superposable on its mirror image, the object is described as being achiral. 

Let us also visualize the important concept of chirality. Only molecules that differ from their mirror image have enantiomers. Molecules of this type are called chiral. For anything to be chiral, that is, non-superimposable on its mirror—there is a necessary and sufficient condition it is the absence of an intramolecular rotation/reflection axis. What s that, you say See the next paragraph. 

The most frequently occurring rotation/reflection axis in organic chemistry is, S the intramolecular mirror plane. Its presence makes ci.v-1,2-dibromocyclohexane (structure A in Figure 33) as well as ma o-2,3-dibromosuccinic acid (structure B) achiral. This is true even though there are two stereocenters in each of these compounds. But why is the dibromocy-clohexane dicarboxylic acid C (Figure 3.3) achiral The answer is that it contains an S2 axis, an inversion center, which occurs rarely in organic chemistry. This compound is thus achiral although it contains four stereocenters. 

According to the foregoing definition, chirality occurs only in molecules that do not have a rotation/reflection axis. However, if the molecule only has an axis ofrotation, it is chiral. For... 

Fig. 3.3. Molecules that contain several stereocenters as well as one (A-C) or no (D, E) rotation/reflection axis and consequently are achiral (A-C) or chiral (D, E).

Consequently, there are four symmetry elements the n-fold axis of rotation, labeled C the plane of symmetry, labeled o the center of inversion, /, and the n-fold rotation-reflection axis, labeled S . Because of mathematical reasons, it is necessary to include the identity symmetry element, /. 

Rotation reflection axis (S ) Rotation about axis by 2n/n followed by reflection in a plane perpendicular to the axis... 

Note that according to the foregoing definition, chirality occurs only in molecules that do not have a rotation/reflection axis. However, if the molecule has only ( ) an axis of rotation, it is chiral. For example, both trans-1,2-dibromocyclohexane (D in Figure 3.3) and the dibromosuccinic acid E have a two-fold axis of rotation (C2) as the only symmetry element. In spite of that, these compounds are chiral because the presence of an axis of rotation, in contrast to the presence of a rotation/reflection axis, is not a criterion for achirality. 

One n-Fold Roto-Reflection Axis S , (Rotation by C Plus One Mirror Perpendicular to Cj)... 

In Section VIII we described a method for finding the most probable rotationally symmetric shape given measurements of point location. The solution for mirror symmetry is similar. In this case, given m measurements (where m - 2q), the unknown parameters are fyjpj, (0 and 0 where 0 is the angle of the reflection axis. However these parameters are redundant and we reduce the dimensionality of the problem by replacing two-dimensional (0 with the one dimensional x0 representing the x-coordinate at which the reflection axis intersects the x-axis. Additionally we replace Rt, the rotation matrix with ... 

Improper Rotations A rotation by 360/n about an axis followed by a reflection in a plane perpendicular to the axis is called rotation-reflection symmetry operation. A combined operation of this kind is called a rotation-reflection or an improper rotation and is denoted by the symbol Sn standing for the combination of a rotation through an angle 2%/n about some axis and reflection in a plane perpendicular to the axis. C4 operation followed by reflection through the plane of molecule gives S4 axis. If we use the symbol oh to denote the reflection in the plane perpendicular to rotatory-reflection axis we can write... 

All p and d atomic orbitals are symmetrical about rotation and reflection axis. However, the hybridised orbitals are highly asymmetric because they do not have any reflection plane of symmetry, rotational axis of symmetry. 

Ibe mbol Sn (CyO means that the element (/t-fold rotational-reflection axis) k rotationally coincident with C . 

Rotation-reflection, S4 Rotation-reflection axis Rotation by 360°/n, followed by CH, (... 

S An improper rotation or rotation-reflection axis. Clockwise rotation through an angle of Injn radians followed by a reflection in the plane perpendicular to the axis of rotation. Also known as an alternating axis of symmetry. Note that S is equivalent to a/, and S2 is equivalent to i. 

---