Angle, Euler

Figure Al.4.4. The definition of the Euler angles (0, ( ), x) that relate the orientation of the molecule fixed (x, y, z) axes to the (X, Y, Z) axes. The origin of both axis systems is at the nuclear centre of mass O, and the node line ON is directed so that a right handed screw is driven along ON in its positive direction by twisting it from Z to z through 9 where 0 < 9 < n. ( ) and x have the ranges 0 to In. x is measured from the node line.

We consider rotations of the molecule about space-fixed axes in the active picture. Such a rotation causes the (x, y, z) axis system to rotate so that the Euler angles change... 

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

Here/(9,(p, i ) is the probability distribution of finding a molecule oriented at (0,cp, li) within an element dQ of solid angle with the molecular orientation defined in tenus of the usual Euler angles (figure B 1.5.10). 

FigureBl.5.10 Euler angles and reference frames for the discussion of molecular orientation laboratory frame (v, y, z) and molecular frame (x y, z).

P, Jy, and J , are the components of the total orbital angular momentum J of the nuclei in the IX frame. The Euler angles a%, b, cx appear only in the P, P and P angular momentum operators. Since the results of their operation on Wigner rotation functions are known, we do not need then explicit expressions in temis of the partial derivatives of those Euler angles. 

Figure 1. The space-fixed (ATZ) and body-fixed (xyz) frames. Any rotation of the coordinate system XYZ) to (xyz) may be performed by three successive rotations, denoted by the Euler angles (a, 3, y), about the coordinate axes as follows a) rotation about the Z axis through an angle a(0 < a < 2n), (b) rotation about the new yi axis through an angle P(0 < P < 7i), (c) rotation about the new zi axis through an angle y(0 Y < 2n). The relative orientations of the initial and final coordinate axes are shown in panel (d).

The Euler angles are often used to describe the orientations of a molecule. There are thre Euler cmgles d and ip. is a rotation about the Cartesian z axis this has the effec of moving the x and y axes, d is a rotation about the new x axis. Finally, ip is rotation about the new z axis (Figure 8.4). If the Euler angles are randomly changed b small amounts S[Pg.437]

It is important to note that simply sampling displacements of the three Euler angles does nc lead to a uniform distribution it is necessary to sample from cos 6 rather than 6 (Figure 8.5... 

The angles 0, (j), and x are the Euler angles needed to specify the orientation of the rigid molecule relative to a laboratory-fixed coordinate system. The corresponding square of the total angular momentum operator fl can be obtained as... 

Here /j,m (0, X) is the original unrotated funetion evaluated at a point whose angular eoordinates are 0, (j), x 0, ( ), x are the Euler angles through whieh this funetion is rotated to obtain the rotated funetion Q /j,m whose value at the above point is denoted Q /j,m (0, X)-... 

As a prelude to the discussion it is necessary to consider the definition of orientation in terms of the Euler angles, and the definition ofan orientation distribution function in terms ofan expansion ofLegendre functions. These definitions set the scene for examining the information which can be obtained from different spectroscopic techniques. In this review, infra-red and Raman spectroscopy and nuclear magnetic resonance, will be considered. 

Fig. 1. Definition of Euler angles. Reproduced from Polymer by permission of the publishers, Butterworth Co. (Publishers) Ltd. (C)...

Euler angles, definition of 84 Exchange rate O 31 Extrusion 142... 

It depends on both the angle a of the angular momentum vector rotation and other Euler angles F and q, which determine the molecule s axis shift. Besides, the angle F is also the azimuth of the change in angular momentum A/ = J(t + 0) — J(t — 0), which is the result of collision. 

Here D(Q) = D(a,f, y), Euler angles a, (5 and y being chosen so that the first two coincide with the spherical angles determining orientation e = e(j], a). Using the theorem about transformation of irreducible tensor operators during rotation [23], we find... 

---