Proper rotation operator

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. 

Dissymmetric molecules either have no symmetry at all, or they belong to one of the groups consisting only of proper rotation operations, that is, the CR or D groups. (Groups T, O, and 1 are, in practice, not encountered, though molecules in these groups must also be dissymmetric.) Important examples are the bischelate and trischelate octahedral complexes (A5-VIII), (A5-X), and (A5-XIX). 

C is the proper rotation operator. It rotates the system by an angle iTrjn about a particular axis. For example, Q is rotation by Ittjl = tt (or 180°), and Qfx) is rotation by 2-77/4 (or 90°, as shown in Fig. 6.2b). If the rotation is a symmetry element of the molecule, the axis is called a symmetry axis. The symmetry axis with the greatest value of n for a given molecule is assigned to z and is the principal rotation axis (for the rest of this chapter, we shall simply call it the principal axis ). In a few cases, the results are easily expressed in Cartesian coordinates. For example, proper rotation by ir about the z axis transforms a function as follows ... 

There are three distinct C2 proper rotation operators, one for each of the axes that contains the P atom and one of the three equatorial F atoms. A single molecular geometry may have several different proper rotation axes. 

A proper rotation operator turns the molecule by a fixed angle about a particular axis, if proper rotation is a symmetry element of a molecule, the axis of that proper rotation is a symmetry axis of the molecule. 

C proper rotation operator (C ), non-linear molecule rotational constant, heat capacity, constant of integration, coordination nrnnber (C)... 

However, the Yj m are not the eorresponding eigenfunetions beeause the operator P now eontains eontributions from rotations about three (no longer two) axes (i.e., the three prineipal axes). The proper rotational eigenfunetions are the DJm,k funetions... 

Here /, are the three moments of inertia. The symmetry index a is the order of the rotational subgroup in the molecular point group (i.e. the number of proper symmetry operations), for H2O it is 2, for NH3 it is 3, for benzene it is 12 etc. The rotational partition function requires only information about the atomic masses and positions (eq. (12.14)), i.e. the molecular geometry. 

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 ... 

A special position in the crystal is repeated in itself by at least one symmetry element, that is, r = r. According to Eq. (B.2), this means that s must be zero if a symmetry element is to give rise to a special position. It follows that translations, screw operations, and glide planes do not generate special positions. On the other hand, positions located on proper rotation axes or centers of symmetry have lower multiplicity than general positions in the unit cell. 

Proper rotation axis. If a molecule can be rotated about some axis so that the positions originally occupied by eveiy atom are subsequently occupied by identical atoms, the molecule is said to possess a proper rotation axis. The axis and the rotation operation perfonned about it are typically represented by the notation C , where n is the order of the rotation. The order is the largest value of n for which it is true that a rotation of In/n radians about the axis reproduces the original structure this is also referred to as a n-fold rotation axis. 

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... 

The characters of Trot are closely related to those of Ttrans. Any symmetry operation is either a Cn or an Sn operation. Consider first the effect on, say, Rx of a Cn rotation about some axis, not necessarily the x axis. This Cn rotation will move the rotation displacement vectors in such a manner as to transform Rr into a vector R where R r is the vector obtained by applying C directly to Rx an example is shown in Fig. 9.7. Thus for proper rotations, the matrices describing how Rx, R, and Rr transform are exactly those matrices that describe how ordinary (polar)... 

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. 

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... 

Thus far we have addressed the symmetry of crystalline arrays only in terms of the proper rotations and the rotation-inversion operations (the latter including simple inversion, as 1, and reflection, as 2) that occur in point symmetries, along with the lattice translation operations. However, for a complete discussion of symmetry in crystalline solids, we require two more types of operation in which translation is combined with either reflection or rotation. These are, respectively, glide-reflections (or, as commonly called, glides) and screw-rotations. 

Figure 2.5. Symbols used to show an n-fold proper axis. For improper axes the same geometrical symbols are used but they are not filled in. Also shown are the corresponding rotation operator and the angle of rotation .

Example 3.2-2 Consider a basis of three orthogonal unit vectors with e3 (along OZ) normal to the plane of the paper, and consider the proper rotation of this basis about OZ through an angle o by the operator R ( z) (see Figure 3.3). Any vector v may be expressed as the sum of its projections along the basis vectors ... 

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