Euler angles, orientational averaging

If we know the components of the electronically averaged dipole moment (fix, Py, Pz) in the space-fixed axis system XYZ, and we know the values of the Euler angles (d, f, x) that define the orientation of the xyz axes relative to the XYZ axes (see, for example, Ref. [3]), we can compute the xyz components (p, ff, pf) of the dipole moment from the relations... 

A sufficient number of relative initial orientations must be considered in order to achieve a convenient grid in the space of Euler angles so that an accurate average can be taken to yield a cross section for a particular process /, e.g., direct scattering or electron charge exchange... 

In the presence of rapid molecular motion, the NMR line shape is modified by the reduced EFG tensor depending on the rate of reorientation and the orientations of the principal axes of the EFG tensor relative to the rotation axis. A calculation of the averaged tensor is performed by transformation of the EFG tensor in motion to a final reference frame in terms of Euler angles a, and y [151,154,155]. The spectral frequency in the fast... 

The polarizability of the free oriented interacting molecules CH4 and N2 can be easily calculated if the analytical expressions for polarizabilities oc are averaged over Euler angles Xa a, 4>A and polarizability tensor has only two independent components ... 

To get the hyperpolarizability of free oriented interacting molecules CH4 and N2 (the case of large intermolecular distances) the hyperpolarizability should be averaged over Euler angles using Eqs. (5.1.15) and (5.2.1). In this way, taking into account the Kleinman symmetry rules [64] the following nonzero components are... 

Orientational averages such as those in Eq. (10.81) can be evaluated by a general procedure involving the use of Euler angles [43]. Here are the results for several cases of interest. 

The scattering characteristics depend on the type of the orientation distribution function. By convention, the uniform distribution function is called complete if the Euler angles Op, / p and 7p are uniformly distributed in the intervals (0,360°), (0,180°) and (0,360°), respectively. The normalization constant is 47t for axisymmetric particles and for nonaxisymmetric particles. The uniform distribution function is called incomplete if the Euler angles Op, /3p and 7p are uniformly distributed in the intervals (apmin, pmax), (/ pmin,/ pmax), and (7pmin, 7pmax)> respectively. For axisymmetric particles, the orientational average is performed over Op and / p, and the normalization constant is... 

Here, S(z, p) (l//9)2r m exp -r/m[27+Ei(/92/4Az) ] ) is the layer-layer correlation function,where R2=z2+p2 is Euler s constant, Ex(x) is the exponential integral function, and A = 7(K/B). In Eq. (16), the exponential term incorporates a finite size effect because of the observed finite lamellar domain sizes typically between -2000 and 10,000 A (L is the domain volume). Also because the samples consist of randomly oriented domains, we perform an exact powder average over all solid angles in reciprocal space. The precise steps leading to Eq. (16) have been discussed elsewhere.2,3 analysis consists of... 

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