Planes of symmetry

The symmetry operation u is the operation of reflecting the nuclei across the plane. 

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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). 

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. 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

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... 

Certain structural features can sometimes help us determine by inspection whether a mol ecule IS chiral or achiral For example a molecule that has a plane of symmetry or a cen ter of symmetry is superimposable on its mirror image and is achiral... 

A plane of symmetry bisects a molecule so that one half of the molecule is the mirror image of the other half The achiral molecule chlorodifluoromethane for exam pie has the plane of symmetry shown m Figure 7 3... 

A point m a molecule is a center of symmetry if any line drawn from it to some element of the structure will when extended an equal distance m the opposite direction encounter an identical element The cyclobutane derivative m Figure 7 4 lacks a plane of symmetry yet is achiral because it possesses a center of symmetry... 

Locate any planes of symmetry or centers of symmetry in each of... 

Any molecule with a plane of symmetry or a center of symmetry is achiral but their absence is not sufficient for a molecule to be chiral A molecule lacking a center of symmetry or a plane of symmetry is likely to be chiral but the supenmposability test should be applied to be certain... 

One way to demonstrate that meso 2 3 butanediol is achiral is to recognize that its eclipsed conformation has a plane of symmetry that passes through and is perpendicular to the C 2-C 3 bond as illustrated m Eigure 7 11a The anti conformation is achiral as... 

In the same way that a Fis cher formula is a projection of the eclipsed conformation onto the page the line drawn through its center is a projection of the plane of symmetry that is present in the eclipsed conformation of meso 2 3 butanediol... 

Turning to cyclic compounds we see that there are three not four stereoisomeric 1 2 dibromocyclopropanes Of these two are enantiomeric trans 1 2 dibromocyclo propanes The cis diastereomer is a meso form it has a plane of symmetry... 

Disubstituted cyclohexanes present us with a challenging exercise in stereochemistry Con sider the seven possible dichlorocyclohexanes 1 1 as and trans 1 2 as and trans 1 3 and as and trans 1 4 Which are chiral Which are achiral Four isomers—the ones that are achiral be cause they have a plane of symmetry—are relatively easy to identify... 

Section 7 3 A molecule that has a plane of symmetry or a center of symmetry is achi ral as 4 Methylcyclohexanol (Table 7 2) has a plane of symmetry that bisects the molecule into two mirror image halves and is achiral The same can be said for trans 4 methylcyclohexanol... 

Merrifield method See solid phase peptide synthesis Meso stereoisomer (Section 7 11) An achiral molecule that has chirality centers The most common kind of meso com pound IS a molecule with two chirality centers and a plane of symmetry... 

Many transition states of chemical reactions contain symmetry elements not present in the reactants and products. For example, in the umbrella inversion of ammonia, a plane of symmetry exists only in the transition state. 

Chirality and Optical Activity. A compound is chiral (the term dissymmetric was formerly used) if it is not superimposable on its mirror image. A chiral compound does not have a plane of symmetry. Each chiral compound possesses one (or more) of three types of chiral element, namely, a chiral center, a chiral axis, or a chiral plane. 

Multiple Chiral Centers. The number of stereoisomers increases rapidly with an increase in the number of chiral centers in a molecule. A molecule possessing two chiral atoms should have four optical isomers, that is, four structures consisting of two pairs of enantiomers. However, if a compound has two chiral centers but both centers have the same four substituents attached, the total number of isomers is three rather than four. One isomer of such a compound is not chiral because it is identical with its mirror image it has an internal mirror plane. This is an example of a diaster-eomer. The achiral structure is denoted as a meso compound. Diastereomers have different physical and chemical properties from the optically active enantiomers. Recognition of a plane of symmetry is usually the easiest way to detect a meso compound. The stereoisomers of tartaric acid are examples of compounds with multiple chiral centers (see Fig. 1.14), and one of its isomers is a meso compound. 

If a molecule has a plane of symmetry, for which the symbol is a, reflection of all the nuclei through the plane to an equal distance on the opposite side produces a configuration indistinguishable from the initial one. Figure 4.3(a) shows the two planes of symmetry, (7 (xz) and (yfyz), of H2O using conventional axis notation. Just as theyz plane, the plane of the molecule, is a plane of symmetry so any planar molecule has at least one plane of symmetry. The subscript u stands for vertical and implies that the plane is vertical with respect to the highest-fold axis, C2 in this case, which defines the vertical direction. 

In the planar molecule BF3, in Figure 4.3(b), the C3 axis through B and perpendicular to the figure is the highest-fold axis and, therefore, the three planes of symmetry, perpendicular to the figure and through each of the B-F bonds, are labelled (t . The plane of the molecule is also a plane of symmetry and is labelled u , where /z stands for horizontal with respect to C3. 

We have seen in Section 4.1.4 that = n and that S2 = i, so we can immediately exclude from chirality any molecule having a plane of symmetry or a centre of inversion. The condition that a chiral molecule may not have a plane of symmetry or a centre of inversion is sufficient in nearly all cases to decide whether a molecule is chiral. We have to go to a rather unusual molecule, such as the tetrafluorospiropentane, shown in Figure 4.8, to find a case where there is no a or i element of symmetry but there is a higher-fold S element. In this molecule the two three-membered carbon rings are mutually perpendicular, and the pairs of fluorine atoms on each end of the molecule are trans to each other. There is an 54 axis, as shown in Figure 4.8, but no a or i element, and therefore the molecule is not chiral. 

A C point group contains a C axis of symmetry and n a planes of symmetry, all of which contain the C axis. It also contains other elements which may be generated from these. 

The point group Cj , contains only a plane of symmetry, in addition to I. It is therefore the same as Ci and is usually given the symbol Q. 

The point group is the same as 2 - Ethylene (Figure 4.1a) and naphthalene (Figure 4.3c) belong to the >2 point group in which, because of the equivalence of the three mutually perpendicular C2 axes, no subscripts are used for the planes of symmetry. 

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