Eulerian rotation angles

To do a full rotation in three-dimensional space, Eulerian rotation angles can be defined, as three successive rotations, each in two dimensions. 

Test initial values of the complex index of refraction (nx — ikx, ny — iky, / - ikx) and of three Eulerian rotation angles for the crystal face. Thus there are nine parameters to be determined, for 96 data points The system is mathematically overdetermined. 

For diffraction studies with monocliromatic radiation, the crystal is connnonly mounted on an Eulerian cradle, which can rotate the crystal so that the nonnal to any set of planes bisects the angle between the incident and reflected beams, which is set for reflection from planes with a particular value of the interplanar spacing, d. 

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). 

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

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... 

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 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... 

Let us consider the electron-vibrational matrix element. As is usually done, we consider two coordinate systems, the origins of which are located at the center of mass of the molecule. The first coordinate system is fixed in space, while the second system (the rotational one) is fixed to the molecule. For describing the orientation of the rotational system with respect to the fixed frame we use the Eulerian angles 6 = a, / , y. In the Born-Oppenheimer approximation, the motion of nuclei may be decomposed into the vibrations of the nuclei about their equilibrium position and the rotation of the molecule as a whole. Accordingly, the wave function of the nuclei X (R) is presented as a product of the vibrational wave function A V(Q) and the rotational wave function... 

Four-circle with Euler geometry. The sample is placed at the center of a Eulerian cradle and the detector rotates in the equatorial plane. Data can be recorded from full spheres in the reciprocal lattice except if blind regions are introduced on the Euler angles (o>, x, <1>) by, for example, the sample environment assembly. 

If the structure is elongated the quadrupole terms (d-functions) have to be considered. Out of the five terms three can be elimated by rotation of the structure by the Eulerian angles a, P and y. Thus we are left with... 

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. 

By comparison with Eq. (2.6) it is seen that rag depends on the internal velocities exclusively, whereas eg depends on the rotational velocities. The time derivatives of the Eulerian angles are replaced, however, by the components of the total angular velocity vector, to, of the molecular system, defined by... 

Now the adsorption potential is independent of the coordinates (x,y) parallel to the adsorbent surface, and one Eulerian angle (say, tp) can be chosen in such a way that the rotation of the molecule through it does not affect the z coordinates of the atoms of the molecule. Thus the integrations over (x,y,cp) can be performed analytically, and instead of equation (1) one obtains ... 

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. 

This Lagrangian should be thought of as dependent on 3Ne + 6 generalized coordinates, qt, and velocities, g, respectively. These are the 3Ng coordinates be, Ce which describe the relative positions of the Ne electrons with respect to the nuclear frame three coordinates Xo, Vo and Zo which describe the position of the molecular center of mass as referred to the laboratory coordinate system, and three Eulerian angles 6, and x which describe the instantaneous orientation of the molecular coordinate system with respect to the space fixed X-, Y- and Z-axes. There are numerous ways of specifying Eulerian angles. Because of later reference we will follow the choice used by Wilson et where and 6 are the ordinary polar coordinates of the molecular c-axis O d n 0 < < 2n) and x is the angle between the nodal line N and the positive b axis as is illustrated in Fig. IV.2. x is positive for clockwise rotation about the c axis. 

---