Rotation axes operations

The energy required to adjust the DNA to these receptor sites is given in Table VI. The DNA can kink equally well in both grooves with base pairs held at a distance sufficient for intercalation (Az = 6.76 A, ax = 0°) and for kinks (Az > 6.76 A, ax 0°). These receptor sites are constructed by operations on a pair of initially coincident base pairs. Each is rotated by +ax/2 and -ax/2 about a kink axis. This axis is perpendicular to the helix and dyad axes of the base, and parallel to the Cl (py)-Cl (pu) axis. It lies approximately along the C6(py)-C8(pu) axis. Then each base pair is rotated about the helix axis by +az/2 and -az/2 and separated by Az. The combinations of ax, az, and Az which permit the construction of a phosphate backbone defines families of receptor sites. With this approach, the base pairs adjacent to the BPDE are symmetrically... 

The square matrix A x transforms the vector x into a vector y by the product y=Ax. Multiplication by the matrix A associates two vectors from the Euclidian space fR" and therefore corresponds to a geometric transformation in this space. A is a geometric operator. Non-square matrices would associate vectors from Euclidian spaces with different dimensions. The ordered combination of geometric transformations, such as multiple rotations and projections, can be carried out by multiplying in the right order the vector produced at each stage by the matrix associated with the next transformation. 

For an Abelian group, each element is in a class by itself, since X 1AX = AX IX = A. Since rotations about the same axis commute with each other, the group e is Abelian and has n classes, each class consisting of one symmetry operation. 

Example 12.6-1 The point group C2v = E C2z perpendicular axes, all operations except E are irregular and there is consequently only one doubly degenerate spinor representation, Ei/2. Contrast C 21, = E C2z / [ in which rrh is az = IC2z and thus an improper binary rotation about... 

In order to apply group theory to the physical properties of crystals, we need to study the transformation of tensor components under the symmetry operations of the crystal point group. These tensor components form bases for the irreducible repsensentations (IRs) of the point group, for example x, x2 x3 for 7(1) and the set of infinitesimal rotations Rx Ry Rz for 7(1 )ax. (It should be remarked that although there is no unique way of decomposing a finite rotation R( o n) into the product of three rotations about the coordinate axes, infinitesimal rotations do commute and the vector o n can be resolved uniquely... 

S,R,Q,P,0 branches of the rotational Raman spectrum. The treatment outlined here is a little more general than usual since we have retained the antisymmetric operator contribution (2,.39b), which generates the less familiar selection rules for antisymmetric scattering, namely at = I with AX = O forbidden if K = 0 [18]. 

This character can be zero only for a = n and, hence, for binary rotations with n = 2. To examine whether or not the matrix for a binary rotation can be class-conjugated to minus itself, we may limit ourselves to the study of one orientation of the rotation axis, say C. Indeed, in SU 2) any orientation can always be transformed backward to this standard choice by a unitary transformation. The problem thus reduces to finding a spinor operation X represented by a matrix X with Cayley-Klein parameters ax,bx, which transforms (C ) into minus itself ... 

The only remaining type of operation is S , the rotary refleoliou. Since this operation first rotates the system about an axis, then reflects it through a plane perpendicular to the axis of rotation, xr will be zero unless there is an atom located at the intersection of the axis and the plane. For such an atom, the equations of transformation for Ax and Ay are the same as in (7), but Az = —Az. Therefore the contribution to... 

Figure 3.7. Left the symmetry operations of the two-dimensional square lattice the thick straight lines indicate the reflections (labeled ax,ay,ai,as) and the curves with arrows the rotations (labeled C4, C2, C4). Right the Irreducible Brillouin Zone for the two-dimensional square lattice with full symmetry the special points are labeled F, S, A, M, Z, X.

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