Euler angle rotation

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). 

From the pole conventions in Section 11.3 it follows that R(— n) = K( (j> n), and this restricts the rotation angle to the range 0 < [Pg.234]

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. 

However, it is more convenient to use curvilinear cartesian coordinates to describe the rotational motion of the nuclei. To this end we relate a set of rotating, molecule-fixed axes to the space-fixed axes by the three Euler rotations. In our experience the Euler angles and the rotations based upon them are not easily visualised Zare [7] has given as good a description as any. Figure 2.2 defines the Euler angles [Pg.45]

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). ...

In dealing with finite rotations one normally does not specify the three Euler angles (a, (3, y), but rather the rotation angle "/ and the two polar angles 0 and rotation axis. The connection between the two sets of angles is ... 

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

Fortunately for us, the problem of assigning rotational angles in unambiguous ways has been mathematically resolved. There are several different sets of rotational angle representations, and we will consider only one of these here—the so-called Euler angles. Other representations can be foimd in the further list Reading Section at the end of this chapter. 

The substitution of the simplified transformation matrices of Eqs. (7.6) into this condition reveals those terms which have been neglected by the associated assumptions. In the case of moderate rotations, Eq. (7.6a), and small rotations, Eq. (7.6b), these are the products of four, respectively of two, rotational angles given in Eq. (7.5). Since the rotational angles in the latter case are thereby decoupled, the outcome is identical to the results of other rotational transformations, for example involving Euler angles. 

To calculate the distances r, Vy and between the different transition monopoles, it is convenient to let the first molecule s center be the origin and its axes define the x, y and z direction of the dimer system. The second molecule may be related to the first by a translation and three rotations, the angles and axes of which are defined by the Euler transformations (21). Thus, the second molecule may be defined by the location of its center, (X, Y, Z), and its three Euler rotations (a, b, c). 

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 ... 

Here and are vectors of the Cartesian coordinates and velocities selected above and R(ft x) s the Euler rotation matrix. The angles 6. , x are chosen randomly according to... 

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