Rotation-reflection axis improper

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... 

The Cahn-Ingold-Prelog rules work for inorganic compounds too but coordination complexes often have coordination numbers greater then four and may exhibit helical chirality, for example, denoted A and A (or Pand Min the Cahn-Ingold-Prelog system). The formal condition for chirality is that the molecule should not have an improper axis of rotation (i.e. a rotation + reflection axis, 5n =... 

This denotes an axis about which a rotation-reflection (or improper rotation) operation may be carried out. The rotation-reflection operation Sn involves a... 

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. 

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. 

S , Rotation of the molecule through an angle 360°/n followed by reflection of all atoms through a plane perpendicular to the axis of rotation the combined operation (which may equally follow the sequence reflection then rotation) is called improper rotation-,... 

I Rotation by 2nr/ followed by a reflection in the plane 1 to tbe axis Improper rotation axis of order a... 

A twofold rotation around the molecular axis of cA-7,2-dichloroethylene is not a covering operation, because the rotation exchanges hydrogens and chlorines. However, the compound operation of the twofold rotation followed by a reflection in a plane perpendicular to the molecular axis - a C2 followed by a ah — is a covering operation. The combination of an n—fold rotation and a reflection in a perpendicular plane is called a rotatory-reflection or improper rotation, symboUzed Sn-... 

Improper rotation axis. Rotation about an improper axis is analogous to rotation about a proper synunetry axis, except that upon completion of the rotation operation, the molecule is mirror reflected through a symmetry plane perpendicular to the improper rotation axis. These axes and their associated rotation/reflection operations are usually abbreviated X , where n is the order of the axis as defined above for proper rotational axes. Note that an axis is equivalent to a a plane of symmetry, since the initial rotation operation simply returns every atom to its original location. Note also that the presence of an X2 axis (or indeed any X axis of even order n) implies that for every atom at a position (x,y,z) that is not the origin, there will be an identical atom at position (—x,—y,—z) the origin in such a system is called a point of inversion , since one may regard every atom as having an identical... 

In the Schoenfiies system the improper axis is an axis of rotation-reflection (see page 52). In the Internationa) system the axis of rotatory inversion ( ) is ore of n-fold rotation followed by inversion (see Fig. 3.29). 

An even number of reflections in the same plane send each point from (x,y,z) back to itself, which is equivalent to an identity operation (a proper isometry), while an odd number of reflections in the same plane is equivalent to a single reflection (an improper isometry). The product of two reflections in separate planes is a proper isometry. We distinguish two cases (1) If the two planes are parallel and spaced by a distance d, the isometry is a translation by 2d in a direction perpendicular to the two planes (2) if the two planes intersect at an angle a, the isometry is a proper rotation (or, simply, rotation) through 2a about the line of intersection (the axis of rotation). Translations and rotations are continuous operations because motion can be controlled by continuous changes in the distance or angle between the two mirrors. 

Let us set up a 2D unitary matrix representation for the transformation of the spin functions a and (1 in Civ. So far, we have established only a relation between 0(3)+ and SU(2). The matrix representations of reflections or improper rotations do not belong to 0(3)+ because their determinants have a value of -1. To find out how a and p behave under reflections, we notice that any reflection in a plane can be thought of as a rotation through n about an axis perpendicular to that plane followed by the inversion operation. For instance, 6XZ may be constructed as xz = Cz(y) i. Herein, it is not necessarily required... 

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... 

The combination, in either order, of rotating the molecule about an axis passing through it by 2ttln and reflecting all atoms through a plane that is perpendicular to the axis of rotation is called improper rotation and is symbolized S . 

A rotation-reflection (improper rotation) about the z-axis is represented by a similar matrix, the zz-element changes its sign ... 

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... 

Improper rotation-reflection operation is the rotation about the improper axis S through an angle luk/n, combined with reflection k times in a plane normal to this axis. 

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 . 

A molecule may be chiral, and therefore optically active, only if it does not possess an axis of improper rotation, S . An improper rotation is a rotation followed by a reflection and this combination of operations always converts a right-handed object into a left-handed object and vice versa., hence an S axis guarantees that a molecule cannot exist in chiral forms,... 

Plane Center of inversion Proper axis Improper axis Reflection through plane Inversion of aU atoms through the center One or more rotations about the axis Rotation followed by a reflection in a plane to the rotation axis... 

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