Rotation-reflection axis of symmetry

Chiroptical spectroscopies are based on the concept of chirality, the signals are exactly zero for non-chiral samples. In terms of molecular symmetry, this means that the studied system must not contain a rotation-reflection axis of symmetry. This lapidary definition implies that the more known symmetry elements (symmetry plane - equivalent to the one-fold rotation-reflection axis and the center of symmetry - equivalent to the two fold rotation-reflection axis) must also be absent and that the system must be able to exist at least formally in two mirror image-like forms. At first glance this limitation seems to be a disadvantage, however, this direct relation to molecular geometry gives chiroptical properties their enormous sensitivity to even minor and detailed changes in the three-dimensional structure. 

A symmetry operation transforms an object into a position that is physically indistinguishable from the original position and preserves the distances between all pairs of points in the object. A symmetry element is a geometrical entity with respect to which a symmetry operation is performed. For molecules, the four kinds of symmetry elements are an n-fold axis of symmetry (C ), a plane of symmetry (cr), a center of symmetry (i), and an n-fold rotation-reflection axis of symmetry (5 ). The product of symmetry operations means successive performance of them. We have " = , where E is the identity operation also, 5, = o-, and Si = i, where the inversion operation moves a point at x,y, zto -X, -y, -z.Two symmetry operations may or may not commute. 

A /7-fold rotation-reflection axis of symmetry designated by Sp where rotation through 27t// or 360°/p followed by a reflection at a plane perpendicular to the axis of rotation produces an orientation indistinguishable from the original molecule. 

We can also describe the symmetry of a molecule using the compound rotation-reflection operation, i.e. rotate about an axis by lirln and then reflect in the plane perpendicular to that axis . This type of operation is given the symbol The symmetry element consists of an axis and a plane. Examples of rotation-reflection operations are shown in Figure 2.7. It is crucial to note that the reflection plane must be perpendicular to the rotation axis. Also, a rotation-reflection axis of order 2n wiU be associated with a pure-rotation axis of order n. 

For non-linear molecules with two or three different principal moments of inertia, the origins of the band shapes are more complicated, but the outcome is the same we can distinguish between the IR-active symmetry species. For example, non-linear symmetric tops (that is, molecules with a rotation (or rotation-reflection) axis of order three or more) also have two types of IR band, depending on whether the associated dipole change is parallel (e.g. the a modes of PF5) or perpendicular (the e modes) to the top axis. For the parallel bands, the selection rules lead to an overall structure with distinct P, Q and R branches. For perpendicular bands, the result is a broad band with overlapping P and R sub-branches, and a regular series of Q sub-branches, which are usually distinct peaks (Figure 8.15). 

For information about point groups and symmetry elements, see Jaffd, H. H. Orchin, M. Symmetry in Chemistry Wiley New York, 1965 pp. 8-56. The following symmetry elements and their standard symbols will be used in this chapter An object has a twofold or threefold axis of symmetry (C2 or C3) if it can be superposed upon itself by a rotation through 180° or 120° it has a fourfold or sixfold alternating axis (S4 or Sh) if the superposition is achieved by a rotation through 90° or 60° followed by a reflection in a plane that is perpendicular to the axis of the rotation a point (center) of symmetry (i) is present if every line from a point of the object to the center when prolonged for an equal distance reaches an equivalent point the familiar symmetry plane is indicated by the symbol a. 

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. 

This is the operation of clockwise rotation by 2w/ about an axis followed by reflection in a plane perpendicular to that axis (or vice versa, the order is not important). If this brings the molecule into coincidence with itself, the molecule is said to have a n-fold alternating axis of symmetry (or improper axis, or rotation-reflection axis) as a symmetry element. It is the knight s move of symmetry. It is symbolized by Sn and illustrated for a tetrahedral molecule in Fig. 2-3.3.f... 

We consider four kinds of symmetry elements. For an n fold proper rotation axis of symmetry Cn, rotation by 2n f n radians about the axis is a symmetry operation. For a plane of symmetry a, reflection through the plane is a symmetry operation. For a center of symmetry /, inversion through this center point is a symmetry operation. For an n-fold improper rotation axis Sn, rotation by lir/n radians about the axis followed by reflection in a plane perpendicular to the axis is a symmetry operation. To denote symmetry operations, we add a circumflex to the symbol for the corresponding symmetry element. Thus Cn is a rotation by lit/n radians. Note that since = o, a plane of symmetry is equivalent to an S, axis. It is easy to see that a 180° rotation about an axis followed by reflection in a plane perpendicular to the axis is equivalent to inversion hence S2 = i, and a center of symmetry is equivalent to an S2 axis. 

It may be useful to illustrate this idea with one or two examples. The H2 molecule (or any other homonuclear diatomic) has cylindrical symmetry. An electron that finds itself at a particular point off the internuclear axis experiences exactly the same forces as it would at another point obtained from the first by a rotation through any angle about the axis. The internuclear axis is therefore called an axis of symmetry we have seen in Section 1.2 that such an axis is called an infinite-fold rotation axis, CFigure 10.2 illustrates the Cm symmetry and also some of the other symmetries, namely reflection in a mirror plane, abbreviated internuclear axis and equidistant from the nuclei, and rotation of 180° (twofold axis, C2) about any axis lying in that reflection plane and passing through the internuclear axis. (There are infinitely many of these C2 axes only two are shown.) There are, in addition to those elements of symmetry illustrated, others an infinite number of mirror planes perpendicular to the one illustrated and containing the internuclear axis, and a point of inversion (abbreviated i) on the axis midway between the nuclei. 

Chirality is the geometric property of a rigid object (or spatial arrangement of points or atoms), which is nonsuperposable on its mirror image such an object has no symmetry elements of the second kind (a mirror plane, a center of inversion, a rotation-reflection axis,. ..). If the object is superposable on its mirror image, the object is described as being achiral. 

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... 

Consequently, there are four symmetry elements the n-fold axis of rotation, labeled C the plane of symmetry, labeled o the center of inversion, /, and the n-fold rotation-reflection axis, labeled S . Because of mathematical reasons, it is necessary to include the identity symmetry element, /. 

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