Symmetry, plane

Planes of symmetry. Planes through which there is reflection to an identical point in the pattern. In a lattice there may be a lateral movement parallel to one or more axes (glide plane). 

To completely specify the orientational ordering, the complete set of orientational order parameters, P/,L = 0,2,4.. ., is required. Only the even rank order parameters are non-zero for phases with a symmetry plane perjDendicular to the director (e.g. N and SmA phases). 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of ineitia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes a,b,c). In order to detemiine the parity of the molecule through inversions in SF, we first rotate all the electrons and nuclei by 180° about the c axis (which is peipendicular to the molecular plane) and then reflect all the electrons in the molecular ab plane. The net effect is the inversion of all particles in SF. The first step has no effect on both the electronic and nuclear molecule-fixed coordinates, and has no effect on the electronic wave functions. The second step is a reflection of electronic spatial coordinates in the molecular plane. Note that such a plane is a symmetry plane and the eigenvalues of the corresponding operator then detemiine the parity of the electronic wave function. 

The C2H2CI2 molecule has a ah plane of symmetry (plane of molecule), a C2 axis ( to plane), and inversion symmetry, this results in C2h symmetry. Using C2h symmetry labels... 

Fig. 35. Normal modes of tropolon moleeule partieipating in tunneling tautomerization. Symmetry of modes is given in braekets. For the off-plane vibrations vjj and the symmetry plane is shown. The equilibrium bond lengths are indieated in the leftmost diagram.

This geometry possesses three important elements of symmetry, the molecular plane and two planes that bisect the molecule. All MOs must be either symmetric or antisymmetric with respect to each of these symmetry planes. With the axes defined as in the diagram above, the orbitals arising from carbon 2p have a node in the molecular plane. These are the familiar n and n orbitals. 

FIGURE 10.17 Velocity contours in symmetry plane for a flanged rectangular opening with an aspect ratio of 2 1 by analytical and point sink methods. 

In the previous section we discussed wall functions, which are used to reduce the number of cells. However, we must be aware that this is an approximation that, if the flow near the boundary is important, can be rather crude. In many internal flows—where all boundaries are either walls, symmetry planes, inlets, or outlets—the boundary layer may not be that important, as the flow field is often pressure determined. However, when we are predicting heat transfer, it is generally not a good idea to use wall functions, because the convective heat transfer at the walls may be inaccurately predicted. The reason is that convective heat transfer is extremely sensitive to the near-wall flow and temperature field. 

When a molecule is symmetric, it is often convenient to start the numbering with atoms lying on a rotation axis or in a symmetry plane. If there are no real atoms on a rotation axis or in a mirror plane, dummy atoms can be useful for defining the symmetry element. Consider for example the cyclopropenyl system which has symmetry. Without dummy atoms one of the C-C bond lengths will be given in terms of the two other C-C distances and the C-C-C angle, and it will be complicated to force the three C-C bonds to be identical. By introducing two dummy atoms to define the C3 axis, this becomes easy. 

The question of how to terminate the box is fundamental to all the calculations of interfacial energy in compounds, including the calculation of surface energies. It has been addressed previously for particular cases by Chetty and Martin [11,12]. These authors pointed out that a suitable termination is one which is on a symmetry plane of the crystal, or which follows symmetry planes if it is not parallel to the boundary. However, it may not always be possible to find a symmetry plane. I offer a solution here which is more general. It reconciles the atomistic picture with the thermodynamic limit. 

Figure 9.4 The achiral propanoic acid molecule versus the chiral lactic acid molecule. Propanoic acid has a plane of symmetry that makes one side of the molecule a mirror image of the other side. Lactic acid has no such symmetry plane.

The situation is different for 2-methylcyclohexanone. 2-Methylcvclo-bexanone has no symmetry plane and is chiral because C2 is bonded to four different groups a -CH3 group, an —H atom, a -COCH2- ring bond (C1), and a —CH2CH2— ring bond (C3). 

Figure 9.11 A symmetry plane through the C2-C3 bond of meso-tartaric acid makes the molecule achiral. H ho co2h...

To see whether a chirality center is present, look for a carbon atom bonded to four different groups. To see whether the molecule is chiral, look for the presence or absence of a symmetry plane. Not all molecules with chirality centers are chiral overall—meso compounds are an exception. 

A look at the structure of cis-1,2-dimethylcyclobutane shows that both methyl-bearing ring carbons (Cl and C2) are chirality centers. Overall, though, the compound is achiral because there is a symmetry plane bisecting the ring between Cl and C2. Thus, the molecule is a meso compound. 

Problem 9.17 Does the following structure represent a meso compound If so, indicate the symmetry plane. 

Mcso compound (Section 9.7) A compound that contains chirality centers but is nevertheless achiral by virtue of a symmetry plane. 

Symmetry plane (Section 9.2) A plane that bisects a molecule such that one half of the molecule is the mirror image of the other half. Molecules containing a plane of symmetry are achiral. 

D-Allaric acid has a symmetry plane and is a meso compound, but o-glucaric acid is chiral. 

With respect, to a local symmetry plane, the basic orbitals are either <7,, symmetric or antisymmetric. This is true not only for the... 

Let us now apply these results to the ethylene molecule (Fig. 14), for which we attempt to build the bonding molecular orbitals. Clearly there are three symmetry planes. Two of these are of special interest... 

In molecules with little or no symmetry, it may still be possible to recognize the main localized-orbital component of certain molecular orbitals. It is then convenient to adopt the label of this localized type as the label of the molecular orbital, even though the molecular symmetry does not coincide with the local symmetry. For instance, in methylenimine again, the 5A orbital is clearly built out of the in-plane 7rc 2 group orbital, with a small NH component. We therefore label the orbital t CU2, although the molecule does not have a vertical symmetry plane. Similarly, the orbitals 7A and 8A of propylene are labeled 7TqH3, tt CU2 (111.49).a Other examples where the local symmetry is sufficiently preserved and only weakly perturbed by the molecular environment are hydrazine (111.34) and methylamine (III.31). In some cases we have omitted the label as no unambiguous classification is possible. 

The reason for the weak mixing of and cr orbitals in propylene is that there is a weak pseudo vertical symmetry plane in the molecule. It is the plane which would have existed if the carbon skeleton were linear, with 110 central hydrogen atom. One can also visualize its existence by joining the two local vertical symmetry planes of the CH2 and CH3 groups. 

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