The Plane of Symmetry

If a plane exists for which reflection of each atom in a molecule gives an indistinguishable configuration, the molecule is said to have a plane of symmetry, which is given the label a. 

The dihedral plane has been introduced using an example where there is more than one type of vertical plane. However, oa planes are defined by their relationship to the horizontal C2 axes this means that molecules do exist in which oa is the only type of vertical plane. 

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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... 

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... 

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. 

These nuclei lie on the planes of symmetry and there are three nuclei belonging to each set. The three nuclei have nine degrees of freedom of which two belong to the uj species and one to the 2 species. The nuclei have six degenerate degrees of freedom... 

Figure 11.3 illustrates the classification of the MOs of butadiene and cyclobutene. There are two elements of symmetry that are common to both s-cw-butadiene and cyclobutene. These are a plane of symmetry and a twofold axis of rotation. The plane of symmetry is maintained during a disrotatory transformation of butadiene to cyclobutene. In the conrotatory transformation, the axis of rotation is maintained throughout the process. Therefore, to analyze the disrotatory process, the orbitals must be classified with respect to the plane of symmetry, and to analyze the conrotatory process, they must be classified with respect to the axis of rotation. 

The intensity of mass transfer in the flow in the plane of the interacting jets axis is lower than in the plane of symmetry. 

The mirror plane is defined by the dummy atoms. The migrating hydrogen H le is not allowed to move out of the plane of symmetry, and must consequently have the same distance to C4 and C5. A minimization will locate the lowest energy structure within the given Cs symmetry, and a subsequent frequency calculation will reveal that the optimized structure is a TS, with the imaginary frequency belonging to the a" representation (breaking the symmetry). 

Draw the meso form of each of the following molecules, and indicate the plane of symmetry in each ... 

It is seen that the maximum bond strength for set I (Powell s second set) is greater than that for set II (Powell s first set), and also that there is for each of the two kinds of functions great deviation from cylindrical symmetry. The values of the functions in the planes of symmetry are shown in Figures 2 and 3. Comparison with the corresponding cross sections for d2 and d,2-.2 shows that the functions I are qualitatively similar to d 2, and the functions II are more similar to dx2-t2. The values for the two functions are indicated in Figure 1. 

This compound does not possess a plane of symmetry, but it does have a center of inversion. If we invert everything around the center of the molecule, we regenerate the same thing. Therefore, this compound will be superimposable on its mirror image, and the compound is meso. You will rarely see an example like this one, but it is not correct to say that the plane of symmetry is the only symmetry element that makes a compound meso. In fact, there is a whole class of symmetry elements (to which the plane of symmetry and center of inversion belong) called S axes, but we will not get into this, because it is beyond the scope of the course. For our purposes, it is enough to look for planes of symmetry. 

A number of theoretical (5), (19-23). experimental (24-28) and computational (2), (23), (29-32). studies of premixed flames in a stagnation point flow have appeared recently in the literature. In many of these papers it was found that the Lewis number of the deficient reactant played an important role in the behavior of the flames near extinction. In particular, in the absence of downstream heat loss, it was shown that extinction of strained premixed laminar flames can be accomplished via one of the following two mechanisms. If the Lewis number (the ratio of the thermal diffusivity to the mass diffusivity) of the deficient reactant is greater than a critical value, Lee > 1 then extinction can be achieved by flame stretch alone. In such flames (e.g., rich methane-air and lean propane-air flames) extinction occurs at a finite distance from the plane of symmetry. However, if the Lewis number of the deficient reactant is less than this value (e.g., lean hydrogen-air and lean methane-air flames), then extinction occurs from a combination of flame stretch and incomplete chemical reaction. Based upon these results we anticipate that the Lewis number of hydrogen will play an important role in the extinction process. 

Starting from stoichiometric conditions (< = 1) and then proceeding in the lean direction (< < 1), we anticipate that the peak fiame temperature will be reduced gradually. In addition, as the maximum temperature is lowered and the corresponding adiabatic fiame speed of an unstrained a = 0) fiame is reduced, we anticipate that the fiame will move closer to the plane of symmetry. Ultimately, as the fuel to air ratio is lowered below a critical value, radical production in the fiame will be severely restricted and the fiame will extinguish (lean extinction). The arclength continuation procedure will then generate unphysical solutions for additional continuation steps until a maximum value of the... 

The molecules shown in Fig. 1 are planar thus, the paper on which they are drawn is an element of symmetry and the reflection of all points through the plane yields an equivalent (congruent) structure. The process of carrying out the reflection is referred to as the symmetry operation a. However, as the atoms of these molecules are essentially point masses, the reflection operations are in each case simply the inversion of the coordinate perpendicular to the plane of symmetry. Following certain conventions, the reflection operation corresponds to z + z for BF3 and benzene, as it is the z axis that is chq ep perpendicular to die plane, while it is jc —> —x for water. It should be evident that the symmetry operation has an effect on the chosen coordinate systems, but not on the molecule itself. 

The symmetry of the static complex is at most Cs, requiring one of the g-matrix principal axes to be normal to the plane of symmetry - this is assumed... 

Figure 5.16 The plane of symmetry of ies0-2,3-dibromobutane. This plane divides the molecule into halves that are mirror images of each other.

An MLR, which is illustrated in Fig. 13.le, is a miniature version of a fiber loop resonator created from an MNF. An MNF/microsphere resonator and MNF/ microdisk resonator, which are illustrated in Fig. 13.lg, h, consist of an MNF coupled to a microsphere or a microdisk. The excited WGMs are localized in the neighborhood of the microsphere (microdisk) circumference situated in the plane of symmetry of these devices. The transmission power of an MLR, a microsphere, or a microdisk sensor, P(/1), near the resonance wavelength 20 can be found in the form similar to the transmission power of a ring, a disk, and a microsphere resonator66,67 ... 

The line divides the molecule is such a way that the upper half becomes the mirror image of the lower half. Therefore, the plane of symmetry is that which divides the molecule into two such parts so that each becomes the mirror image of the other. Compounds possessing this plane of symmetry would always be optically inactive. 

Compounds with one asymmetric carbon atom would be active because of a tetrahedral structure. When there are two or more such asymmetric carbon centres, we will have to take into consideration the concepts of the plane of symmetry. 

Another polarimetric method for the accurate determination of KIEs bears a strong resemblance to the isotopic quasi-racemate method, described above. In this method, Bach and co-workers (1991) utilized what they called isotopically engendered chirality to determine the primary deuterium KIE for an elimination reaction. In theory, the method can be used for any reaction where a substrate with a plane of symmetry yields, under normal conditions, a racemic mixture. For instance, if the plane of symmetry in the unlabelled... 

The centre of symmetry (inversion through a point) is represented by 1 and the plane of symmetry (mirror symmetry) by the letter m. The inversion operation... 

Conformational analysis in connection with determinations of ffee-energy differences (AG°) between axial and equatorial conformers is still attracting interest. Schneider and Hoppen (114) discussed A values ( —AG°) and preferred orientations of axial substituents with lone pairs at heteroatoms directly attached to C (e.g., -OR, -NR2, and -N3), as well as of some other nonspherical substituents (X = -NC, -NCS, -CN, -C CH). Phenyl and vinyl groups were investigated by Eliel and Manoharan (277), who found A values of 2.87 0.09 kcal/mol for phenyl and 1.68 0.06 kcal/mol for vinyl. The latter value was essentially confirmed by Buchanan (196) the formyl group A = 0.84 0.08 kcal/mol) in axial position adopts a predominant (93%) conformation (305) with the plane of the axial CHO group nearly perpendicular to the plane of symmetry of the cyclohexyl residue (Scheme 71) (196). 

When we considered trans- and c -1,2-dimethyl-cyclohexane, we found that only three configurational isomers exist, enantiomeric forms of the trans isomer, together with the cis isomer, which is an optically inactive meso compound (see Section 3.4.5). The meso relationship could be deduced from the plane of symmetry in the hexagon representation. 

When we move on to camphor, a ketone derivative of bornane, we find this can exist in two enantiomeric forms because the plane of symmetry has been destroyed. Nevertheless, there are only two configurational isomers despite the presence of two chiral centres bridging does not allow the other two variants to exist. 

Now, you may have noticed that the hydroxyl group in methylecgonine is oriented differently from that in tropine. In methylecgonine it is easy to define the position of the hydroxyl, since this is a chiral centre and we can use the R/S nomenclature. An alternative stereoisomer of tropine exists, and this is called pseudotropine. How can we define the configuration for the hydroxyl when the plane of symmetry of the molecule goes through this centre and means this centre is not chiral but can exist in two different arrangements ... 

Furan ring coinciding with the plane of symmetry of 1,3-dioxane ring. 

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