Polar angles, spherical harmonics

Here, Yx m( j) denotes a spherical harmonic, coj represents the spherical polar angles made by the symmetry axis of molecule i in a frame containing the intermolecular vector as the z axis. The choice of the x and y axes is arbitrary because the product of the functions being averaged depends on the difference of the azimuthal angles for the two molecules which are separated by distance r. At the second rank level the independent correlation coefficients are... 

The applicability of Eq. (21) rests on the validity of the assumption that the averages over internal and external variables are uncorrelated and thus can be calculated separately. Furthermore, theexpression of Eq. (21) emphasizes the close similarity of the irreducible Cartesian representation to the expression of the problem in terms of polar angles and the normalized 2nd rank spherical harmonics Y (see Eq. (7)). The corresponding polar angles ( (1), (t)) and (C(t), (t)), shown in Fig. 2B, describe the orientation of the internuclear vector and the magnetic field relative to the arbitrary reference frame, respectively. The different representations are related according to the following relationships.37... 

Equations (4.329) for a solid assembly and (4.332) for a magnetic suspension are solved by expanding W with respect to the appropriate sets of functions. Convenient as such are the spherical harmonics defined by Eq. (4.318). In this context, the internal spherical harmonics used for solving Eq. (4.329) are written Xf (e, n). In the case of a magnetic fluid on this basis, a set of external harmonics is added, which are built on the angles of e with h as the polar axis. Application of a field couples [see the kinetic equation (4.332)] the internal and external degrees of freedom of the particle so that the dynamic variables become inseparable. With regard to this fact, the solution of equation (4.332) is constructed in the functional space that is a direct product of the internal and external harmonics ... 

Here 0 and are the spherical polar angles (only two angles are required to define the orientation of the vector r in space). Since these operators are the same as the infinitesimal rotation operators, all the results of the previous sections apply. The eigenfunctions of L2 and Lz are known as the spherical harmonics,... 

To relate Dff to the anisotropic motion of a molecule in a liquid crystalline solvent, we employ the function P(d, < ), defined as the probability per unit solid angle of a molecular orientation specified by the angles 6 and <3>, the polar coordinates of the applied magnetic field direction relative to a molecule-fixed Cartesian coordinate system. We expand P(0, ) in real spherical harmonics ... 

It is a mathematical property of spherical harmonic functions Yem(6, ) that they obey an addition rule which is known as the biaxial theorem if ri and r2 are two vectors, with directions described by the polar angles (6i, 4> ) and (62,4>2), and if a is the included angle between the two vectors, then... 

By definition the components of the second-rank Cartesian tensor ax transform under rotation just like the product of coordinates xy (e.q., see Jeffreys, 1961) The motivation for what ensues springs from the observation that the spherical harmonics Ym (0, ft) (where 6, ft) are the polar and azimuthal angles of the unit vector (r/1 r )) can be written in terms of the coordinates (x, y, z) of the vector r, for example,... 

The set of all square integrable functions of the polar angles (9, fi) forms a Hilbert space. This space is spanned by the spherical harmonics Yim(9, 4) and can be decomposed into subspaces such that the Ith subspace is spanned by the (21 + 1) spherical harmonics of index /. 

In eqn. 5.2.14 the coefficients of the series expansions depend on the variable x = /cr, and S v(0, surface spherical harmonics depending on the polar angles 0 and q>. The vector coefficients 1 /Jc) are independent of the position of ion j which may be considered stationary according to eqn. 5.2.6. At infinity y) and/ vanish. Besides, Pitts assumed that for r = a the perturbations may be neglected thus the ionic potentials and distribution functions on the surface of the central ion are not affected by the external field. 

The actual implementation does not involve the explicit calculation of the polar angles, we calculate the spherical harmonics in term of the Cartesian coordinates X, y, z and zo- The first two four-dimensional spherical harmonics are... 

In these expressions, (r") represents the average (i.e., expectation value) of r with respect to the radial part of the d-electron wavefunctions, r is the radius of the electron cloud about the metal center, and L is the metal-ligand bond distance. There are five 3d-orbital wavefunctions, and each one exhibits a different spherical harmonic expression in terms of polar angle 0 and azimuthal angle (p in spherical coordinates. 

The shape of such a surface may be described in a mathematical way which has several important advantages. The shape is analysed into harmonic components functions much as one might analyse a wave into a Fourier series. The spherical harmonics, as these components are called, have the necessary property of orthogonality however, their form is more complicated than the cos(njc) type of component of a Fourier series for a plane wave. The spherical harmonics are functions of the polar coordinate angle, referred to the director axis. The first four components, abbreviated Pq, P2(cosa), P4(cos Of) and PeCcos a) or simply Pq, P2, P4, Pe, are defined below and drawn in Fig. 4. 

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