Operator improper rotation

Proper rotational operations are represented by the n-fold rotation axes n 1000 (n = 2, 3,4, 6). Rotation-inversion axes such as the 2 axis are improper rotation operations, while screw axes and glide planes are combined rotation-translation operations. 

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

An improper rotation operator, 2 is the same as the inversion operator i, and any Si operator is the same as a reflection operator. 

S is the improper rotation operator, equal to proper rotation... 

An improper rotation operator is a proper rotation followed by a horizontal mirror reflection. 

S total electronic spin angular ipomentum, overlap integral, improper rotation operator (S ), L = 0 atomic state, entropy t time... 

In other words, the even improper rotation operations can be always rewritten through the equivalent operartions. For the improper axis of 6th order rotation, can be written the equivalence ... 

Figure 2.7 Summary of the improper rotation operation for ethane in the staggered conformation.

An improper rotation is equivalent to an ordinary (proper) rotation followed by a reflection through a plane that is perpendicular to the rotation axis. Both the rotation axis and the reflection plane must pass through the origin. The symbol for an improper rotation operator is S , where the subscript n is the number of applications to achieve one fiiU rotation of360°. The symmetry element for an improper... 

A group exists only if certain conditions are met. The first requirement is that there is an operation associated with the elements of a group. We might contemplate a mathematical group composed of all positive integers with the operation being addition. In the case of molecular symmetry, the operation is successive application of the elements, something we may term a "multiplication." We have already seen that the improper rotation operator... 

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-,... 

If a proper rotation is combined with a reflection with respect to the axis of rotation, it is called an improper rotation The matrix representation of silvan operation is found simply by replacing 1 by -1 in Eq. (104), The Scftfllfllies symbol for an improper rotation by y is S /tp- Hence, matrix the representation of an improper counter-clockwise rotation by y is of the form ... 

An electric dipole operator, of importance in electronic (visible and uv) and in vibrational spectroscopy (infrared) has the same symmetry properties as Ta. Magnetic dipoles, of importance in rotational (microwave), nmr (radio frequency) and epr (microwave) spectroscopies, have an operator with symmetry properties of Ra. Raman (visible) spectra relate to polarizability and the operator has the same symmetry properties as terms such as x2, xy, etc. In the study of optically active species, that cause helical movement of charge density, the important symmetry property of a helix to note, is that it corresponds to simultaneous translation and rotation. Optically active molecules must therefore have a symmetry such that Ta and Ra (a = x, y, z) transform as the same i.r. It only occurs for molecules with an alternating or improper rotation axis, Sn. 

As an example, which will also lead us to the concept of qualitative completeness, consider the allene skeleton, as shown in Fig. 8, and for the moment consider only achiral ligands. Besides the unit element, the symmetry group of the skeleton, "Dm, consists of the rotation operations (in permutation group notation) (12)(34), (13)(24), and (14)(23), plus the improper rotations (1)(2)(34), (12)(3)(4), (1324), and (1432). 

S improper rotation of 2n/n radians (n e N) improper rotations are regular rotations followed by a reflection in the plane perpendicular to the axis of rotation i inversion operator (equivalent to 2) a mirror plane... 

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... 

Our convention is that a symmetry operation R changes the locations of points in space, while the coordinate axes remain fixed. In contrast in Section 1.2 we considered a change (proper or improper rotation) of coordinate axes, while, the points in space remained fixed Let x y z be a set of axes derived from the xyz axes by a proper or improper rotation. Consider a point fixed in space. We found that its coordinates in the x y z system are related to its coordinates in the xyz system by (1.120) or (2.29) ... 

Identity element, 387-388 Identity operation, 54, 395 Improper axis of symmetry, 53 Improper rotation, 396 Index of refraction, 132 INDO method, 71, 75-76 and ESR coupling constants, 380 and force constants, 245 and ionization potentials, 318 and NMR coupling constants, 360 Induced dipole moment, 187 Inertial defect, 224-225 Inertia tensor, 201... 

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. 

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. 

It will be shown later that the operations of rotation and reflection in a plane perpendicular to the rotation axis always give the same result regardless of the order in which they are performed. Thus the definition of improper rotation need not specify the order. 

In these examples, where only C2 and C4 rotations and certain kinds of planes are concerned, the transformation of the coordinates [jc, y, z] to [jc, y, z] by a twofold rotation about the x axis, for example, is fairly obvious by inspection. It is also obvious that a fourfold rotation about the x axis will transform the coordinates into [jc, z, y]. It is also easy to see by inspection the effects of the inversion operation, an improper rotation by 2nl2 or 2nl4 and reflection in a plane that is the xy, xz, or yz plane or a plane rotated by... 

All symmetry operations that we wish to consider can be regarded as either proper or improper rotations. 

The final requirement, that every element of the group have an inverse, is also satisfied. For a group composed of symmetry operations, we may define the inverse of a given operation as that second operation, which will exactly undo what the given operation does. In more sophisticated terms, the reciprocal S of an operation R must be such that RS = SR = E. Let us consider each type of symmetry operation. For <7, reflection in a plane, the inverse is clearly a itself a x a = cr = E. For proper rotation, C , the inverse is C" m, for C x C "m = C" = E. For improper rotation, S , the reciprocal depends on whether m and n are even or odd, but a reciprocal exists in each of the four possible cases. When n is even, the reciprocal of S% is S m whether m is even or odd. When n is odd and m is even, S% — C , the reciprocal of which is Q m. For S" with both n and m odd we may write 5 = C a. The reciprocal would be the product Q ma, which is equal to and which... 

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