Mirror planes simple

Prochirality Planar molecules possessing a double bond such as alkenes, imines, and ketones, which do not contain a chiral carbon in one of the side chains, are not chiral. When these molecules coordinate to a metal a chiral complex is formed, unless the alkene etc. has C2V symmetry. In other words, even a simple alkene such as propene will form a chiral complex with a transition metal. So will trans-2-butene, but cis-2-butene won t. If a bare metal atom coordinates to cis-2-butene the complex has a mirror plane, and hence the complex is not chiral. It is useful to give some thought to this and find out whether or not alkenes and hetero-alkenes form chiral complexes. One can also formulate it as follows complexation of a metal to the one face of the alkene gives rise to a certain enantiomer, and complexation to the other face gives rise to the other enantiomer. 

The Mirror Plane, cr Most flowers, cut gems, pairs of gloves and shoes, and simple molecules have a plane of symmetry. A single hand, a quartz crystal, an optically active molecule, and certain cats at certain times4 do not possess such a plane. The symmetry element is a mirror plane, and the symmetry operation is the reflection of the molecule in the mirror plane. Some examples of molecules with and without mirror planes are shown in Fig. 3.1... 

It in addition to the C axis, there is a horizontal plane perpendicular to that axis) the molecule is said to have symmetry. An example of this relatively unimportant group is t/vms-dichloroethene (Fig. 3.13a). If there are r mirror planes containing the rotation axis, Cn, the planes are designated vertical planes, and the molecule has Cm symmetry. Many simple inorganic... 

For the isotopes of water, the subgroup is Cs, consisting in the identity and mirror-plane operations. The deuterium isotopes can also be used in calculating force constants for simple molecules. However, even for such simple molecules as HCN and DCN, the use of isotopes does not lead to a unique solution of the vibrational problem. It was emphasized23 that a certain chemical intuition and a feel for the relative magnitude of force constants is involved. Additional information could be taken from other isotopes (13C, 15N, I70), and this helps in determination of a unique solution. However, such isotopes cause only small frequency shifts, so that frequency measurements must be extremely precise. It appears, then, that the use of isotopic substitution leads to some uncertainties in determination of force constants. 

To illustrate the use of symmetry, let us examine a simple example. Consider the possibility of an interaction between the vacant p orbital in a methyl cation and a hydrogen 1 j orbital. Figure 10.5 shows that a mirror plane is a symmetry element of the molecule. The p orbital is antisymmetric with respect to the mirror plane, since it changes sign on reflection, whereas the s orbital is symmetric because it is unchanged on reflection. Interaction between the s and the p orbitals will be possible only if their overlap is nonzero. Recall from Section 1.2 that the overlap... 

Symmetry elements include axes of twofold, threefold, fourfold, and sixfold rotational symmetry and mirror planes. There are also axes of rotational inversion symmetry. With these, there are rotations that cause mirror images. For example, a simple cube has three <100> axes of fourfold symmetry, four axes of <111>... 

These layers of condensed antiprisms are well separated from each other by two-dimensional layers formed by the four crystallographically independent iron atoms. The shortest Pr-Pr distance between the layers is at 498 pm. The motif of the iron layer is also simple. The Fe4 atoms on the mirror planes at z = 0 and 1/2 have a tetragonally distorted icosahedral iron coordination (CN 12) at Fe-Fe distances between 241 and 257 pm, close to the Fe-Fe distance of 248 pm in bcc iron (Donohue, 1974). The remaining Fel, Fe2, and Fe3 atoms build up the ligands for the icosahedra around Fe4. The Fe4Fei2 icosahedra are connected with each other via many Fe-Fe bonds with Fe-Fe distances in the range of 249-276 pm. 

Consider a simple example of a cycloaddition reaction of two molecules of ethene to form cyclobutane. Let us classify all the MOs of reactants and the product as symmetric (S) or antisymmetric (A) with respect to symmetry planes m and C2. Once these symmetries are noted, correlations of reactants and product orbitals may be drawn so that orbitals of like symmetry are connected. It is assumed that ethene molecules attack each other in parallel planes (i.e. vertically). There are two symmetry planes (the mirror planes), one bisecting the TT-system of the molecules (plane 1, vertical) and the other between the interacting molecules (plane 2, horizontal), as shown in Fig. 8.20. 

Such reactions are used to prepare bridging ligands having one main symmetry element (mirror plane or Cj axis) passing between two (carbon-) bonded atoms. Except for some metal-catalyzed reactions, redox dimerizations are generally interpreted as the recombination of two radical species—followed, if needed, by a dehydrogenation step—hence generating a simple bond. 

Water is a simple triatomic bent molecule with a C2 axis through the oxygen and two mirror planes that intersect in this axis, as shown in Figure 5-27. The point group is therefore Qv... 

Point group A group of symmetry operations that leave unmoved at least one point within the object to which they apply. Symmetry elements include simple rotation and rotatory-inversion axes the latter include the center of symmetry and the mirror plane. Since one point remains invariant, all rotation axes must pass through this point and all mirror planes must contain it. A point group is used to describe isolated objects, such as single molecules or real crystals. 

Naturally chiral surfaces can be created from achiral crystalline materials. The bulk structures of many crystalline materials such as metals are highly symmetric, contain one or more mirror symmetry elements and thus, are not chiral. Although it may seem counterintuitive, such achiral bulk structures can, nonetheless, expose surfaces with chiral atomic structures. These are planes whose normals do not lie in one of the bulk mirror planes. The classification of the symmetry of surfaces of a variety of bulk crystal structures has recently been reviewed by Jenkins et al. and they have identified all planes in those crystal structures that are chiral [9,10]. As a simple example consider the two surfaces illustrated in Fig. 4.1. These are the two enantiomers of the (643) surfaces of a face centered cubic lattice. 

Another classification is based on the presence or absence of translation in a symmetry element or operation. Symmetry elements containing a translational component, such as a simple translation, screw axis or glide plane, produce infinite numbers of symmetrically equivalent objects, and therefore, these are called infinite symmetry elements. For example, the lattice is infinite because of the presence of translations. All other symmetry elements that do not contain translations always produce a finite number of objects and they are called finite symmetry elements. Center of inversion, mirror plane, rotation and roto-inversion axes are all finite symmetry elements. Finite symmetry elements and operations are used to describe the symmetry of finite objects, e.g. molecules, clusters, polyhedra, crystal forms, unit cell shape, and any non-crystallographic finite objects, for example, the human body. Both finite and infinite symmetry elements are necessary to describe the symmetry of infinite or continuous structures, such as a crystal structure, two-dimensional wall patterns, and others. We will begin the detailed analysis of crystallographic symmetry from simpler finite symmetry elements, followed by the consideration of more complex infinite symmetry elements. 

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