Eulerian angle

In Chapter 4 (Sections 4.7 and 4.8) several examples were presented to illustrate the effects of non-coincident g- and -matrices on the ESR of transition metal complexes. Analysis of such spectra requires the introduction of a set of Eulerian angles, a, jS, and y, relating the orientations of the two coordinate systems. Here is presented a detailed description of how the spin Hamiltonian is modified, to second-order in perturbation theory, to incorporate these new parameters in a systematic way. Most of the calculations in this chapter were first executed by Janice DeGray.1 Some of the details, in the notation used here, have also been published in ref. 8. 

Fig. 9.9 a) G eneral molecular structure of 4-substituted-4 -mercaptobiphenyl. b) Eulerian angles defining the orientation of the molecule with respect to the surface normal, c)... 

Another idea is to use different runs of the same program with slightly different models it is most aptly described as an application of a consistency principle, namely it is required that the solution should appear consistently in all runs, even with a rather low score. Special algorithms have been developed to cluster similar solutions in eulerian angles space and convincing results have shown that it is indeed possible to increase the signal-to-noise ratio of the rotation function in this way (Urzhumtsev and Urzhumtseva, 2002). 

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... 

Fig. 5.2 The Eulerian angles. ON is the line of intersection of the XY and ab planes and is therefore perpendicular to both OZ and Oc. 9 is the angle of rotation about ON

The line of intersection of the XY and ab planes is called the line of nodes , in Fig. 5.2, this is line ON. (Unfortunately, there is no uniformity in the definition of the Eulerian angles.)... 

Let i//, be an asymmetric-top wave function. A convenient complete orthonormal set to use here is the symmetric-top wave functions, which are functions of the same coordinates (the Eulerian angles) and satisfy the same boundary conditions as the asymmetric-top functions ... 

The principal-axis dipole-moment components da, db, and dc depend on the configuration of the nuclei relative to one another, but are independent of the spatial orientation of the molecule they thus depend on the vibrational coordinates, but not on the Eulerian angles. We have... 

In Section 5.1, we noted that to a good approximation the nuclear motion of a polyatomic molecule can be separated into translational, vibrational, and rotational motions. If the molecule has N nuclei, then the nuclear wave function is a function of 3/V coordinates. The translational wave function depends on the three coordinates of the molecular center of mass in a space-fixed coordinate system. For a nonlinear molecule, the rotational wave function depends on the three Eulerian angles 9, principal axes a, b, and c with respect to a nonrotating set of axes with origin at the center of mass. For a linear molecule, the rotational quantum number K must be zero, and the wave function (5.68) is a function of 6 and only only two angles are needed to specify the orientation of a linear molecule. Thus the vibrational wave function will depend on 3N — 5 or 3N — 6 coordinates, according to whether the molecule is linear or nonlinear we say there are 3N — 5 or 3N — 6 vibrational degrees of freedom. 

The selection rules for pure-rotation transitions are found by evaluation of the nine integrals IXOa,/ZOc these involve the nine direction cosines6 cos(XOa),..., cos(ZOc). The volume element in Eulerian angles can be shown to be... 

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 ... 

Representation of Jr by subsitutions of eulerian angles and internal coordinates ... 

From this representation the substitution group of the eulerian angles... 

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. 

Edmonds, A. R. Angular momentum in quantum mechanics. Princeton University Press 1957, p. 7. The definitions for the eulerian angles defined in this book will be used throughout this paper... 

The angle gives the angle of rotation of the J vector around the BC bond. The Cartesian components of the rotational momentum vector are given by the three Eulerian angles 0, , and r/ as... 

On returning to the I l-J one-electron calculation, the assumed linear vibrating structure, shown in Figure 2.18, has to be abandoned in favour of a quantized nuclear framework, and the calculated cylindrically symmetrical structure as suggested by contour maps of electron density should be rotated about all Eulerian angles to reveal the full spherical symmetry of the... 

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