Euler angles function

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. 

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. 

Euler angles Q define the e orientation, and an orientational correlation function of 1th order is introduced in the usual way ... 

As discussed in Section II. A, the adiabatic electronic wave functions, a and / 1,ad depend on the nuclear coordinates R> only through the subset (which in the triatomic case consists of a nuclear coordinate hyperradius p and a set of two internal hyperangles this permits one to relate the 6D vector W(1)ad(Rx) to another one w(1 ad(q J that is 3D. For a triatomic system, let aIX = (a1 -. blk, crx) be the Euler angles that rotate the space-fixed Cartesian frame into the body-fixed principal axis of inertia frame IX, and let be the 6D gradient vector in this rotated frame. The relation between the space-fixed VRi and is given by... 

Performing the integrations over the Euler angles and using the orthogonality of the J, K",Mi +m,p ) functions, we then obtain... 

Here /j,m (0, < ), X) is the original unrotated function evaluated at a point whose angular coordinates are 0, (f>, % 0, (]), % are the Euler angles through which this function is rotated to obtain the rotated function Q V /j,m whose value at the above point is denoted Q V /j,m (0, X)-... 

Second, we need to define a matrix representation between the atomic term functions for each symmetry operation, that depends upon three Euler angles [73]... 

Equation 7.2-22 indicates that the separating force is proportional to the local shear stress (fiy) in the liquid, it is a sensitive function of the Euler angles of orientation, and is proportional to the projection of the cross-sectional are (S = nc2). The angular velocities of rotation of the freely suspended spheroid particle were given by Zia, Cox, and Mason (46)... 

The residual anisotropy A describes the orientation assumed by heavy water molecules. It is expressed as a function of the Saupe order tensor and of the Euler angles describing the orientation of the D20 water molecule with respect to the local director (29) ... 

If one performs a turn of the coordinate system x y z by Euler angles a, 0,7, thus transferring it to the position xyz, then the components Ay and A of the vector A in the new coordinate system may be expressed through the components Ay>, A11 in the initial coordinate system by means of Wigner D-functions... 

Note that, because Euler angles 0 and (f>, there is no contribution from the term in J2. Substituting (2.138) into (2.136), we obtain the wave equation for the rotation vibration wave functions in the Born adiabatic approximation ... 

In conclusion, we have determined that a function of the Euler angles transforms under E as... 

In a case (a) basis set, the electron spin angular momentum is quantised along the linear axis, the quantum number E labelling the allowed components along this axis. Because we have chosen this axis of quantisation, the wave function is an implicit function of the three Euler angles and so is affected by the space-fixed inversion operator E. An electron spin wave function which is quantised in an arbitrary space-fixed axis system,. V. Ms), is not affected by E, however. This is because E operates on functions of coordinates in ordinary three-dimensional space, not on functions in spin space. The analogous operator to E in spin space is the time reversal operator. 

Thus, so far as the nuclei are concerned, l 2 produces the same effect as E and, since the position of the molecule-fixed axes depends on the location of the nuclei only, P 2 has the same effect on a function of the Euler angles as does E ... 

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