Reflection plane operation

Subscripts of 1 or 2 distinguish 1 or -1 eigenvalues, respectively, for a reflection plane operator that is perpendicular to the molecular plane if there is one. 

If there is neither an inversion operator nor a perpendicular reflection plane operator, a prime ( ) or a double-prime (") can be added to the representation s letter designation to distinguish whether the eigenvalue with respect to some other type of reflection plane is 1 or -1, respectively. 

A geometric object can have several symmetry elements simultaneously. However, symmetry elements cannot be combined arbitrarily. For example, if there is only one reflection plane, it cannot be inclined to a symmetry axis (the axis has to be in the plane or perpendicular to it). Possible combinations of symmetry operations excluding translations are called point groups. This term expresses the fact that any allowed combination has one unique... 

If an atom is situated on a center of symmetry, on a rotation axis or on a reflection plane, then it occupies a special position. On execution of the corresponding symmetry operation, the atom is mapped onto itself. Any other site is a general position. A special position is connected with a specific site symmetry which is higher than 1. The site symmetry at a general position is always 1. 

Figure 7.2 The different symmetry elements of the center ABg. (a) A trigonal axis, C3 (b) A binary axis, C2. (c) A symmetry axis belonging to both the 6C4 and SCj classes, (d) A symmetry plane, au- (e, f) Two of the six aj, reflection planes, (g) A view down the C3 axis in (a) to show a roto-reflection operation, S. ...

Note the reflection plane labels do not move. That is, although we start with Hi in the ov plane, H2 in Ov", and H3 in ov", if Hi moves due to the first symmetry operation, ov remains fixed and a different H atom lies in the ov plane. 

For the octahedral set of o orbitals, the operation E leaves all of them unshifted, and thus x = 6. All the rotation operations that do not coincide with any bonds shift all of them and thus give / = 0. The rotations about the x, y, and z axes, the bond axes, leave two bonds unshifted but move all the others (as does a ad reflection) and for them / = 2. The oh type reflection plane contains four bonds, which are thus not shifted, and hence for crA, X = 4. 

Glide-Reflection. This operation is referred to a symmetry element called a glide plane. We have already employed a glide line (its 2D equivalent) in developing the 2D space groups. 

Fig. 3.30 A glide-plane Operation The molecule moves a distance a]2 along the X axis and then is reflected by the xy plane. Note that the chirality of the molecule changes.

Figure 2.10. Effect of the set of symmetry operators G = E C3" Cy ad ae crf on the triangular-based pyramid shown in (a). The C3 principal axis is along z. The symmetry planes

Example 3.2-4 Nevertheless it is convenient to have the MR of a(d y), the operator that produces reflection in a plane whose normal m makes an angle 6 with y (Figure 3.4) so that the reflecting plane makes an angle 6 with the zx plane. 

Figure 2 (a) The polyphenantrene polymer with its glide-reflection plane. Here the A and B sublattices are invariant under this symmetry operation, (b) Polyaceacene polymer with its glide-reflection plane. Notice that here the A and B sublattices transform one into each other under this symmetry operation. 

Improper Rotations A rotation by 360/n about an axis followed by a reflection in a plane perpendicular to the axis is called rotation-reflection symmetry operation. A combined operation of this kind is called a rotation-reflection or an improper rotation and is denoted by the symbol Sn standing for the combination of a rotation through an angle 2%/n about some axis and reflection in a plane perpendicular to the axis. C4 operation followed by reflection through the plane of molecule gives S4 axis. If we use the symbol oh to denote the reflection in the plane perpendicular to rotatory-reflection axis we can write... 

There are six reflection planes, only one of which is shown in Fig. A5-6, giving rise to six ad operations. 

In the same way, it is cleu that the configuration (3), obtained by reflection in a plane perpendicular to the molecular plane is isoenergetic., i.e., the two rotors are equivalent by reflection. The operator corresponding to this internal motion is the double-switch-exchange operator ... 

Figure 18 itht orbitals in a homonuclear diatomic molecule. Both orbitals are unchanged upon operation by the a xz) reflection plane 7t is antisymmetric upon operation by an inversion center, tt is S5mimetric upon operation by an inversion center... 

Does the object has an even-order improper rotation axis S2 but no planes of symmetry or any proper rotation axis other than one collinear with the improper rotation axis The presence of an improper rotation axis of even order S2 without any noncollinear proper rotation axes or any reflection planes indicates the symmetry point group S2 with 2n operations. 

A symmetry element is defined as an operation that when performed on an object, results in a new orientation of that object which is indistinguishable from and superimposable on the original. There are five main classes of symmetry operations (a) the identity operation (an operation that places the object back into its original orientation), (b) proper rotation (rotation of an object about an axis by some angle), (c) reflection plane (reflection of each part of an object through a plane bisecting the object), (d) center of inversion (reflection of every part of an object through a point at the center of the object), and (e) improper rotation (a proper rotation combined with either an inversion center or a reflection plane) [18]. Every object possesses some element or elements of symmetry, even if this is only the identity operation. 

The rigorous group theoretical requirement for the existence of chirality in a crystal or a molecule is that no improper rotation elements be present. This definition is often trivialized to require the absence of either a reflection plane or a center of inversion in an object, but these two operations are actually the two simplest improper rotation symmetry elements. It is important to note that a chiral object need not be totally devoid of symmetry (i.e., be asymmetric), but that it merely be diss)nn-metric (i.e., containing no improper rotation symmetry elements). The tetrahedral carbon atom bound to four different substituents may be asymmetric, but the reason it represents a site of chirality is by virtue of dissymmetry. 

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