Mirror plane of symmetry

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

The water molecule possesses two mirror planes of symmetry, as shown in Fig. 6-3. One mirror plane lies in the plane of the diagram through which the whole molecule reflects into itself across the plane. The other, through the oxygen nucleus in the yz plane of the figure, and shown by the dotted line, reflects Ha into Hb and vice versa. 

Figure 8.4 Illustration showing layer normal (z), director (n), and other parts of the SmC structure. Twofold rotation axis of symmetry of SmC phase for singular point in center of layer is also illustrated. There is also mirror plane of symmetry parallel to plane of page, leading to C2h designation for the symmetry of phase. This phase is nonpolar and achiral.

Each line in this diagram divides the starfish into two halves that are mirror images of each other. There is one more point here. If we rotate the starfish by 360°/5 = 72° around an axis that penetrates the center of the starfish, we will get a structure indistinguishable from the original. We can summarize by saying that the starfish has five mirror planes of symmetry and a fivefold rotation axis through its center. 

There are two types of Cs-symmetric metallocenes, XXX and XXXI (Table 8-5). Both types contain a mirror plane of symmetry—a horizontal plane in XXX, a vertical plane in XXXI. Both are achiral molecules, but they differ very significantly in stereoselectivity. XXX produces atactic polymer, while XXXI usually forms syndiotactic polymer. 

A proper MO - a linear combination of the two localized or-fy bitals which is A with respect to reflection in the plane of the molecule, A w.r.t rotation about the C2 axis and S w.r.t. reflection in the bisecting mirror plane of symmetry. 

Figure 14.2. a) Conrotatory electrocyclic reaction showing preservation of a C2 axis of symmetry. (b) Disrotatory ring closure showing preservation of a mirror plane of symmetry. 

Answer to 2(d). This question illustrates that it is the number of electrons, not the number of nuclei, that is important. The orbital correlation diagram is shown in Figure 14.2. In disrotatory opening, a mirror plane of symmetry is preserved. This correlation is with bold symmetry labels and solid correlation lines. Italic symmetry labels and dotted correlation lines denote the preserved rotational axis of symmetry for conrotatory ring opening. For the cation, the disrotatory mode is the thermally allowed mode. It corresponds to a a2s + 05 pericyclic reaction. 

Wilson (1950) has also shown that other symmetry elements can be detected by statistical methods. If a crystal has a mirror plane of symmetry perpendicular to the b axis, the b projection will show atoms overlapping in pairs, and the scattering for the hOl reflections will be equal to that of half the number of atoms, each with twice the scattering factor. The local average F2 (i.e. for a given range of diffraction angle)... 

The basis of the application of group theory to the classification of the normal vibrations of a molecule lies in the fact that the potential and kinetic energies of a molecule are invariant to symmetry operations. A symmetry operation is a physical transformation of the molecule, such as reflection in a mirror plane of symmetry or rotation through 120° about... 

Chiral ear, glove, bolt and femur. Nonchiral soda can and human body (because each has a mirror plane of symmetry). 

Internal mirror plane, cis-1,2-Dichlorocyclopentane has an internal mirror plane of symmetry. 

Notice in the following figure that the chiral trans isomer of 1,2-dichlorocyclopentane does not have a mirror plane of symmetry. The chlorine atoms do not reflect into each other across our hypothetical mirror plane. One of them is directed up, the other down. 

The converse is not true, however. When we cannot find a mirror plane of symmetry, that does not necessarily mean that the molecule must be chiral. The following example has no internal mirror plane of symmetry, yet the mirror image is superimposable on the original molecule. You may need to make models to show that these mirror images are just two drawings of the same compound. 

Using what we know about mirror planes of symmetry, we can see why a chiral (asymmetric) carbon atom is special. Figure 5-4 showed that an asymmetric carbon has a mirror image that is nonsuperimposable on the original structure it has no internal mirror plane of symmetry. If a carbon atom has only three different kinds of substituents, however, it has an internal mirror plane of symmetry (Figure 5-9). Therefore, it cannot contribute to chirality in a molecule. 

For each compound, determine whether the molecule has an internal mirror plane of symmetry. If it does, draw the mirror plane on a three-dimensional drawing of the molecule. If the molecule does not have an internal mirror plane, determine whether or not the structure is chiral, (a) methane (b) cis-1,2-dibromocyclobutane... 

Let s consider whether cis-l, 2-dibromocyclohexane is chiral. If we did not know about chair conformations, we might draw a flat cyclohexane ring. With a flat ring, the molecule has an internal mirror plane of symmetry (cr), and it is achiral. 

Mirror planes of symmetry are particularly easy to identify from the Fischer projection because this projection is normally the most symmetric conformation. In the first preceding example (propan-2-ol) and in the following example [(2S,3f )-2,3-dibromobutane], the symmetry planes are indicated in red these molecules with symmetry planes cannot be chiral. 

Draw any mirror planes of symmetry that are apparent from the Fischer projections. 

Cis-trans isomerism is also possible when there is a ring present. Cis- and trans-1,2-dimethylcyclopentane are geometric isomers, and they are also diastereomers. The trans diastereomer has an enantiomer, but the cis diastereomer has an internal mirror plane of symmetry, so it is achiral. 

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