Spherical polar coordinates symmetry element

For a linear molecule, the position of the symmetry axis (the molecule-fixed. z-axis) in space is specified by only two Euler angles, / and 7, which are respectively identical to the spherical polar coordinates 6 and (see Fig. 2.4). The third Euler angle, a, which specifies the orientation of the molecule-fixed x- and y-axes, is unaffected by molecular rotation but appears explicitly as an O- dependent phase factor in the rotational basis functions [Eq. (2.3.41)]. Cartesian coordinates in space- and molecule-fixed systems are related by the geometrical transformation represented by the 3x3 direction cosine matrix (Wilson et al., 1980, p. 286). The direction cosine matrix a given by Hougen (1970, p. 18) is obtained by setting a = 7t/2 (notation of Wilson et al, 1980 6 fi,4)=, x = oi 7t/2). The direction cosine matrix is expressed in terms of sines and cosines of 9 and 4>. Matrix elements (J M O la JMQ), evaluated in the JMQ) basis, of the direction cosines, are expressed in terms of the J, M, and quantum numbers. The direction cosine matrix elements of Hougen (1970, p. 31), Townes and Schawlow (1955, p. 96), and Table 2.1 assume the basis set definition derived from Eq. (2.3.40) and the phase choice a = 7t/2 ... 

The three most us ul equations for determining the density at the gas-liquid surface are the tirst YBG equation (4.34), the integral equation over the direct correlation function (4.S2), and the potential distribution theorem (4.69). At a planar surface with K- 0 these can be re-written in simpler forms. If the direction normal to the surface (the heig ) is that of the z-axis, then the only non-zero gradient is d/dz and we have cylindrical symmetry about this axis. The volume element can then be more usefully expressed in cylindrical or spherical polar coordinates. 

What information do the A, g, and D tensors contain The six independent elements of each tensor (D, being traceless, has only five) may be transformed to yield three sets of values consisting of one, two, amd three members. These sets contain different types of information and may be determined under different conditions. The set of one member (which D lacks) is the isotropic or average value of the interaction amd may be determined from samples in amy phase. The set with two members describes the magnitude of the interaction s amisotropy and contains symmetry information. Its determination usually requires a rigid sample. The set of three members describes the orientation of the anisotropy, giving for example the spherical polar coordinates of a molecular symmetry axis and the phase of rotation about it. The orientation can only be determined with an oriented sample, usually a single crystal. 

---