Reflection plane of symmetry

All p and d atomic orbitals are symmetrical about rotation and reflection axis. However, the hybridised orbitals are highly asymmetric because they do not have any reflection plane of symmetry, rotational axis of symmetry. 

We illustrate these ideas by choosing, as an example, naphthalene. In Fig. B1 the x-direction lies perpendicular to the molecular plane the ten atomic 2p -orbitals are therefore of the p.-type. The molecular framework, and hence also the one-electron Hamiltonian for this molecule, has the symmetry D2 - The x-, y- and r-axes are all two-fold rotation-axes, and the three coordinate-planes are all reflection-planes of symmetry. In addition to this we have a centre of inversion at the origin. Each MO must belong to one or other of the irreducible representations of the group (e.g. ref. 31) for the... 

Any plane of symmetry is a plane of halving of the body, but not any plane of halving is a plane of symmetry. This fact is illustrated in the top side of the Figures 2.6a and 2.6b. In the representation Figure 12a the plane, which divides the points 1 and 3, is just a plane of halving in the representation Figure 12b, it is also a reflection plane, of symmetry. 

In this discussion, we consider molecules of high symmetry, so that reflection planes of symmetry exist. For molecules of lower symmetry, one must resort to actual calculation of the cfia Eq. (12-88). 

FIGURE 13.2 A rectangle has reflection planes of symmetry, labeled a. Upon reflection of all points of the rectangle through the plane of symmetry, the original object is reproduced. The reflection of only one point on the rectangle is shown. Can you find two other reflection planes of symmetry for the rectangle ... 

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). 

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. 

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. 

Although the ultimate criterion is, of course, nonsuperimposability on the mirror image (chirality), other tests may be used that are simpler to apply but not always accurate. One such test is the presence of a plane of symmetry A plane of symmetry (also called a mirror plane) is a plane passing through an object such that the part on one side of the plane is the exact reflection of the part on the other side (the plane acting as a mirror). Compounds possessing such a plane are always optically inactive, but there are a few cases known in which compounds lack a plane of symmetry and are nevertheless inactive. Such compounds possess a center of symmetry, such as in a-truxillic acid, or an alternating axis of symmetry as in 1. A... 

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 29Si NMR spectra are of particular interest because they reflect the configuration of the polymer chain.(33) Some 29Si spectra of alkylpolysilanes are shown in Figure 5. Symmetrically-substituted polymers such as (n-hexyl2Si)a have no chirality since there can be a plane of symmetry through each silicon atom. 

For classes with fewer than four sites, the assertion is trivial. For chiral classes with four or more sites, there is at least one triple of sites which does not lie in a symmetry plane of the skeleton. For, if all sites lie in a common symmetry plane, molecules of the class with the ligands all different would possess planes of symmetry, i.e., the class would not be chiral. On the other hand, suppose that the sites do not lie all in a common mirror plane, but that nevertheless every triple of sites lies in a symmetry plane. It follows that every pair of sites lies on the intersection of two different symmetry planes, therefore on an axis of symmetry of the skeleton. But if more than four sites all lie pairwise on an axis of symmetry of a finite figure, they must all lie on a common axis, and the class is again achiral. For chiral classes, then, there is at least one triple of sites which does not lie on a plane of symmetry of the skeleton. Now consider a molecule in which the sites of this triple are occupied by ligands of three different kinds, the other sites by ligands different from these three, but identical with each other. Such a molecule is chiral, since the only improper operation which leaves the three different ligands invariant is a reflection in the plane of the triple, and this changes the rest of the molecule. The assertion follows immediately. 

The two ring ligands of C(4 ) in 6 have the same constitution but cannot be superposed even after a reflection. They are, therefore, diastereomorphic, and as both have a plane of symmetry, the center is of the type Cghij. As expected,... 

The mirror symmetry, the reflection in a plane perpendicular to the molecular plane, is denoted by the symbol All mirror reflections are denoted by the Greek letter a the subscript v indicates that the plane of symmetry would be vertical if the molecule were drawn on a blackboard in the usual way. 

A molecule has a plane of symmetry, or mirror plane, if reflection of all atoms in the plane is a covering operation. 

A body has a plane of symmetry if the shape is unchanged by reflection in the plane. Orthotropic particles have three mutually perpendicular planes of symmetry. An axisymmetric particle is symmetric with respect to all planes containing its axis, so that it is orthotropic if it has a plane of symmetry normal to the axis, i.e., if it has fore-and-aft symmetry. 

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. 

The two a bonds define a local plane of symmetry, assumed to be the xy plane. The p orbital (pz) will always be perpendicular to the local plane of the molecule and be antisymmetric w.r.t. reflection in the plane. 

Orbital Interaction Analysis. An orbital interaction diagram for the Diels-Alder reaction is shown in Figure 12.5a. The geometry of approach of the two reagents which ensures a maximum favorable interaction between the frontier MOs (dashed lines) preserves a plane of symmetry at all separations. The MOs are labeled according to whether they are symmetric (S) or antisymmetric (A) with respect to reflection in the plane. Simultaneous overlap of both HOMO-LUMO pairs is a necessary feature of all peri-... 

Plane of symmetry. If a plane can be placed in space such that for every atom of the molecule not in the plane there is an identical atom (which is to say, the same atomic number and isotope) on the other side of the plane at equal distance from it (i.e., a mirror image ), the molecule is said to possess a plane of symmetry. The Greek letter o is often used to represent both the plane of symmetry and the operation of mirror reflection that it performs. An example of a molecule possessing a plane of symmetry is methylcyclobutane, as illustrated in Figure B.l. Note that a planar molecule always has at least one ct, since tire plane of tire molecule satisfies the above symmetry criterion in a trivial way (the set of reflected atoms is the empty set). Note also that if we choose a Cartesian coordinate system in such a way tliat two of the Cartesian axes lie in the symmetry plane, say x and y, then for every atom found at position (x,y,z) where z there must be an identical atom at position (x,y,—z). 

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