Reflection through a plane

Figure 3.50 Carbon-carbon NBOs of CH2=CHNH2 at

Finally, the symbols + and written as superscripts to the A symbol, refer to two types of sigma states, 2+ and 2 . If the wavefunction is unaltered by reflection through a plane passing through the two nuclei, the state is positive (+) if it changes sign, it is negative ( —). 

The equivalent symmetry element in the Schoenflies notation is the improper axis of symmetry, S which is a combination of rotation and reflection. The symmetry element consists of a rotation by n of a revolution about the axis, followed by reflection through a plane at right angles to the axis. Figure 1.14 thus presents an S4 axis, where the Fi rotates to the dotted position and then reflects to F2. The equivalent inversion axes and improper symmetry axes for the two systems are shown in Table 1.1. 

An improper rotation may be thought of as taking place in two steps first a proper rotation and then a reflection through a plane perpendicular to the rotation axis. The axis about which this occurs is called an axis of improper rotation or, more briefly, an improper axis, and is denoted by the symbol where again n indicates the order. The operation of improper rotation by 2nln is also denoted by the symbol Sn. Obviously, if an axis C and a perpendicular plane exist independently, then S exists. More important, however, is that an S may exist when neither the Cn nor the perpendicular a exist separately. 

There are three symmetry operations each involving a symmetry element rotation about a simple axis of symmetry (C ), reflection through a plane of symmetry (a), and inversion through a center of symmetry (i). More rigorously, symmetry operations may be described under two headings Cn and Sn. The latter is rotation... 

Roto-reflection, that is, a rotation followed by a reflection through a plane perpendicular to the axis of rotation... 

A rotation-reflection operation (S ) (sometimes called improper rotation) requires rotation of 360°/n, followed by reflection through a plane perpendicular to the axis of rotation. In methane, for example, a line through the carbon and bisecting the... 

The S 2n groups (n = 1, 2, 3), with additional rotations jt/n about the main axis, followed by a reflection through a plane perpendicular to the main axis ( n or S n-1 rotation-reflections). For n = 1, this corresponds to inversion /. The Sn operations are called improper rotations, by comparison with the proper rotations Gn. The only element of group S2 (besides E) is / so that this group is also noted G . 

Reflection through a plane of symmetry (mirror plane)... 

Rotation about an axis, followed by reflection through a plane perpendicular to this axis... 

Reflect through a plane that is perpendicular to the original rotation axis... 

An improper rotation is equivalent to a proper rotation followed by a reflection through a plane that is perpendicular to the rotation axis. For this to be a point symmetry operation, both the rotation axis and the reflection plane must pass through... 

The four-site spectrum is plotted in Figure 1 where the + and the - denote states which are symmetric and antisymmetric with respect to a reflection through a plane at the midpoint and perpendicular to the axis of the fiber. We note that for x < 1, the linear... 

The second symmetry operation is that of reflection through a plane it is denoted by the symbol a. Using Cartesian coordinates, the effect of reflection is to change the sign of the coordinates perpendicular to the plane while leaving unchanged the coordinates parallel to the plane. The third symmetry operation is that of inversion through a point... 

We abbreviate reflections with a. In particular, a y x,y,z) means a reflection through a plane that includes the x-axis and the y-axis. We easily see that a y x, y, z) (x, y, -z). If we apply two times, or more general In times, the reflection by the same plane, then we arrive at the identity, e.g., = 7. 

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