Symmetry axis rotation-reflection

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... 

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. 

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, /. 

Note that according to the foregoing definition, chirality occurs only in molecules that do not have a rotation/reflection axis. However, if the molecule has only ( ) an axis of rotation, it is chiral. For example, both trans-1,2-dibromocyclohexane (D in Figure 3.3) and the dibromosuccinic acid E have a two-fold axis of rotation (C2) as the only symmetry element. In spite of that, these compounds are chiral because the presence of an axis of rotation, in contrast to the presence of a rotation/reflection axis, is not a criterion for achirality. 

S An improper rotation or rotation-reflection axis. Clockwise rotation through an angle of Injn radians followed by a reflection in the plane perpendicular to the axis of rotation. Also known as an alternating axis of symmetry. Note that S is equivalent to a/, and S2 is equivalent to i. 

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. 

An object that is chiral is an object that can not be superimposed on its mirror image. Chiral objects don t have a plane of symmetry. An achiral object has a plane of symmetry or a rotation-reflection axis, i.e. reflection gives a rotated version. 

Alternating symmetry axis (S ) An axis about which a rotation by an angle of 360/n, followed by a reflection across a plane perpendicular to the axis results in an entity that is indistinguishable from (superimposable on) the original. Also called a rotation-reflection axis. See also symmetry axis. 

Chiral A geometric figure, or group of points is chiral if it is nonsuperimposable on its mirror image [82]. A chiral object lacks all of the second order (improper) symmetry elements, a mirror plane), i center of symmetry), and S rotation-reflection axis). In chemistry, the term is (properly) only applied to entire molecules, not to parts of molecules. A chiral compound may be either racemic or nonracemic. An object that has any of the second order symmetry elements i.e., that is superimposable on its mirror image) is achiral. It is inappropriate to use the adjective chiral to modify an abstract noun one cannot have a chiral opinion and one cannot execute a chiral resolution or synthesis. 

Rotation-reflection axis See alternating symmetry axis. 

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 rotation by 2x/n about an axis (not necessarily a symmetry axis) followed by reflection in a plane (not necessarily a symmetry plane) perpendicular to the axis of rotation is called a rotation-reflection symmetry operation, the axis is called a rotation-reflection axis and given the symbol iS . Figure 1.10 demonstrates the Ss when neither the 6-fold axis rotation axis nor the horizontal reflection plane are symmetry operations of the system. Whenever a figure has a Cn and a horizontal plane of symmetry, an is automatically implied. The square (Figure 1.6) provides an example of this. 

Any plane through the center of a sphere is a reflection plane, and any axis through the center is a rotation axis, as well as a rotation-reflection axis. In addition, the sphere also is centrosymmetric, which means that the center is a point of inversion. The resulting infinite-dimensional symmetry group of the sphere is usually denoted... 

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