Euler’s angles

Next, Euler s angles are employed for deriving the outcome of a general rotation of a system of coordinates [86]. It can be shown that R(k, 0) is accordingly presented as... 

This expression for the classical rotational energy of a rigid body will now be developed in terms of Euler s angles. 

Note that the first column in the transformation matrix is just the last column of Table 1, while the second column is the same as the second column of R(x) [Eq. (16)]. Of course, as x is along-the z axis, its coefficient is equal to one. Substitution of the angular velocity components given by Eq. (18) allows the rotational energy JEq. (10)] to be expressed in terms of the velocities with respect to Euler s angles (see problem 7). 

The equations of internal motion defined within the local (molecular) fixed coordinate system have to be transformed to the laboratory fixed coordinate system in which all experiments are performed. Thus, introducing Euler s angles, (b, 0, and X, the coordinates of the electrons (e) and nuclei (n) are transformed in the following... 

The matrix elements for the operator //ges = HIot + Hq, which describes the problem of a rotating molecule containing a quadrupolar nucleus, are obtained by simple addition of the matrix elements of the operators Hrot and Hq. This arises from the fact that Hrot operates only on the coordinates of the one system described by the rotational coordinates, i.e. the Euler s angles. 

Given the vector rsfxi.Xg.Xj) joining the two involved centers, then the Euler s angles (p,0) are computed using the simple algorithm ... 

Fig. 9. Geometrical arrangement of Trp-29 in erabutoxin b. Coordinate system of a spherical protein and tryptop-hanyl residue are shown by (x y z ) and (xyz), respectively. The origin of (x y z ) system is chosen at the CH2 group connecting the peptide chain and indole ring. Internal rotation of the tryptophan from the (x y z ) system to the (xyz) system is expressed with Euler s angles (a6y). The location of the quencher, -NHg of Lys-27, is also illustrated in the Fig. Polar coordinates of the N atom of the quencher in the system (x y z ) are indicated by a and 3q (33). ...

The preceding Cartesian coordinates selected for the diatomic are then randomly rotated through Euler s angles [45] to give a random orientation ... 

The R M molecule is not strongly fixed (PhAC-3) and can freely migrate on the crystalline surface, and change its spatial orientation in the area of the potential surface area minimum. An indication is a fuzzy contour of the R M distribution function histograms and Euler s angles of dispersion in the minimum neighbourhood, and so on. 

As a first example, we shall consider the dipole-dipole mechanism. If two spins I and S in the same molecule interact by a dipolar interaction, the perturbation Hamiltonian is time dependent because the vector joining the two spins is characterized by a random motion (Euler s angles 0 and (j> are time dependent). When I and S are in different molecules, becomes time dependent. If 6 and (j) alone are time dependent, the relaxation mechanism is purely intramolecular. If is time dependent, the mechanism is intermolecular. In the first case relaxation is due to molecular rotation, in the second case it corresponds to translation. The theoretical treatment gives different results if the two spins I and S are identical or not (2, p. 291). For two identical spins, at a fixed r distance (intramolecular interaction), and R2 are given by eqs. 17 and 18. Whereas R ... 

In an attempt to define biaxial orientation factors, using Euler s angles of spherical coordinates, Stein defined six orientation factors for the three crystallographic axes (60). These are equation 3 plus the three expressions... 

Biaxial Orientation. Biaxial orientation in polymers can be represented in several ways, eg, by Euler s angles (60,68,75,76). This involves the angle (p with respect to the first symmetry axis and a second latitude. A different set of angles was proposed by Wilchinsky (77) in 1963 and later expanded (67,78) (Fig. 2). These (80-85) involve the above mentioned angle , denoted here by [Pg.891]

These representations would all seem to be equivalent each is associated with certain advantages. The system of Euler s angles has a long history and forms the basis of the well-known set of orthogonal functions, the associated Legendre polynomials. Mathematical representation of orientation distributions are... 

Orientation factor representations have been developed for the second moments for each of these cases. In fact, two sets have been developed using Euler s angles by Stein (60) and by Nomura, Kawai, and co-workers (68). 

The previous representation of orientation may be generalized for crystalline polymers to represent all three crystallographic axes (79). They may be readily obtained from equations 15 and 16 by introducing angles between appropriate crystallographic axes and the defined laboratory axes. Equations 3-6, and 7-9, represent this for orientation factors based on Euler s angles. For orientation... 

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