Eulerian angle transformation

Similarly the quadrupole interaction coordinate system can be related to the new one by an Eulerian angle transformation which will enable 1, 1, ly to be expressed in terms of S, Sy. In total, the angles d and Eulerian angles a, fi, and y are required to define the geometry uniquely. 

The unitary matrix which transforms the two right-handed Cartesian bases e and e can be written in terms of Eulerian angles a, p, and y, (Arfken 1970, Steinborn and Ruedenberg 1973, Edmonds 1974) such that... 

In the study of the dynamical problem of SRMs the transformations of eulerian angles induced by isometric transformations of the frame system will be required. This leads in a natural way from the group T(3 X to the group A(3) (X), defined as follows ... 

Transformation Group of the Dynamical Variables. The transformation groups r(NCf) X), r(3) X and A(3) X all refer to the frame system 1 . By means of the relation between the frame and laboratory system Eq. (2.1) they may be used to define the transformations of the eulerian angles as follows ... 

The set of properly orthogonal transformations R1 forms the group SO(3), the reflexion Z1 at the origin of the LS likewise leaves A symmetric, since the eulerian angles remain unaffected by Z1. Therefore, H is symmetric w.r.t. the full rotation group 0(3/. However, in agreement with the usual conventions we will omit the elements Z R1 0(3). As a consequence we will consider hence -forward the group... 

The simplicity of these transformation formulae is to be traced back to the general formulae (3.33), (3.34) and the fact that the operators PH act on both the eulerian angles and the internal coordinates simultaneously, as expressed by the representation F. The analogy of the Eqs. (3.37) to the representation r(NC1) Sffs should be noted. 

To this equation which defines transformations of the eulerian angles... 

The local frame for rigid molecules, once chosen, will always be clearly defined. The necessary transformation of the separation vector from the laboratory to the local frame is usually accomplished by multiplication by the rotation matrix of the central molecule. The construction of this rotation matrix is usually a straightforward task. In fact, it will already be available in any program that describes molecular orientations in terms of quaternions (coordinate-transformed eulerian angles) [3,24]. 

Eulerian angle The three successive angles of rotation needed to transform one set of Cartesian coordinates into another. 

Note that this equation is not summed over (p, v).0(lv are the constant (i.e., x-independent) parameters that define the 10 transformations in the xyi-xv plane of the 10-parameter Poincare group three Eulerian angles of rotation in space, three components of the relative speed between inertial frames, and the four translations in space and time. 

As is well known, the 3x3 matrix Oy can be diagonalized by an appropriate orthogonal coordinate transformation (rotational transformation), provided it is a symmetric matrix generally it is considered to be symmetric because of its physical meaning. If the principal-axes frame of o, where o is expressed by a diagonal matrix, is transformed to the laboratory frame by a rotational transformation R(o, /3, y) which is defined by three Eulerian angles a, /3 and y, then the representations of o in both frames are related to each other by the equation = (5)... 

Therefore, the experimentally observable chemical shift 033 becomes a function of the principal values as well as the Eulerian angles a and P that transform the principal-axes frame to the laboratory frame. As has been mentioned already, the chemical shift o ou, O22, 033) is defined for individual carbon atoms in a molecule depending on the molecular conformation and O33 depends on Eulerian angles a and p. Therefore, the lineshape is decided by the spatial alignment of the molecule to Bq during measurement, i.e. by the molecular alignment in the sample as well as by the alignment of the sample with respect to Bo. 

Consider the transformation of an initial Cartesian coordinate system (x,y, z) to another x, y, z ) by means of three successive rotations over angles (p, 6, and y (the Eulerian angles) performed in the sequence depicted below. 

The orientation of molecules in a mesophase can be specified by a singlet distribution function /(fi), where Q, denotes the Eulerian angles (0,0,-0) that transform between the molecular frame and the director frame. The average of any single-molecule property X(n) over the orientations of all molecules is defined by... 

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