Generalized spherical functions

The orientation is not strictly identical for all structural units and is rather spread over a certain statistical distribution. The distribution of orientation can be fully described by a mathematical function, N(6, q>, >//), the so-called ODF. Based on the theory of orthogonal polynomials, Roe and Krigbaum [1,2] have shown that N(6, generalized spherical harmonics that form a complete set of orthogonal functions, so that... 

One vexed question concerning polarization sets is the number of functions in a given shell, as discussed elsewhere. The early quantum chemistry codes employed Cartesian Gaussian functions, so that a d set actually comprises the five spherical harmonic d functions and a 3s function generally termed a contaminant. The reason for this emotive terminology is that with multiple polarization sets the contaminants,... 

As was mentioned in Chapter 2, there exists another method of constructing the theory of many-electron systems in jj coupling, alternative to the one discussed above. It is based on the exploitation of non-relativistic or relativistic wave functions, expressed in terms of generalized spherical functions [28] (see Eqs. (2.15) and (2.18)). Spin-angular parts of all operators may also be expressed in terms of these functions (2.19). The dependence of the spin-angular part of the wave function (2.18) on orbital quantum number is contained only in the form of a phase multiplier, therefore this method allows us to obtain directly optimal expressions for the matrix elements of any operator. The coefficients of their radial integrals will not depend, except phase multipliers, on these quantum numbers. This is the case for both relativistic and non-relativistic approaches in jj coupling. 

Let us also mention that using a number of functional relations between the products of 3n/-coefficients and submatrix elements (/ C(k) / ), the spin-angular parts of matrix elements (26.1) and (26.2) are transformed to a form, whose dependence on orbital quantum numbers (as was also in the case of matrix elements of the energy operator, see Chapters 19 and 20) is contained only in the phase multiplier. In some cases this mathematical procedure is rather complicated. Therefore, the use of the relativistic radial orbitals, expressed in terms of the generalized spherical functions (2.18), is much more efficient. In such a representation this final form of submatrix element of relativistic Ek-radiation operators follows straightforwardly [28]. 

It can be straightforwardly obtained while using generalized spherical functions (see Chapter 2). Selection rules for Mk-radiation, described by non-relativistic formulas as usual, follow from the non-zero conditions for the quantities in this expression. They were discussed in Chapter 24 (see formulas (24.22) and (24.24)). 

Another option is the exploitation of the so-called generalized spherical functions D m, instead of the usual spherical functions (harmonics) Y K In this approach one can express all operators in terms of these D-functions. It turned out [28] that it is possible to present a new form of relativistic atomic wave function, the angular part of which is of the form ... 

These equations are a part of the more general Eq. (40). They describe the dispersion of the quadratic term of the power series of the scattering function. For spherical structures like the ferritin molecule p is zero, as there is no change of the centre of mass at different solvent densities. Furthermore, the contributions to q and q" are small compared to q. The radius of gyration exhibits the dispersion of f ... 

The general spherical harmonics are familiar, in low order, as the mutually orthonormal angular components of valence atomic orbitals. Now, the sufficient number of these functions to provide basis functions for the regular representations of the molecular point groups, in... 

Figure 3.7 The general spherical harmonics in the range 0 < I < 4 displayed as elliptical projections on the unit sphere. The shadings in the diagrams reflect areas on the unit sphere of positive function amplitude. 

The crystallite orientation distribution w(a, / , y) can similarly be expanded in a series. Since the distribution is a function of three variables, the series expansion requires orthonormal functions more general than the spherical harmonics. The generalized spherical harmonics to be used are now of the form45... 

What we are essentially looking for is the asymptotic behavior of the solution of Eq. (142) with F(r) = F(r). In this event tpj is a function only of 6 and r, as the scattering is symmetric about the polar axis z. But we delay consideration of the general spherically symmetric potential while we observe the effect of the simpler potential, F(r) = 0, since the solution of the equation... 

The functions are the generalized spherical harmonic functions which... 

Our notation for these functions refers implicitly to generalized spherical harmonic functions which have been adapted for the crystal symmetry according to the previously discussed definition of an orientation (Sect. 3.1). This means that the functions, and consequently the ODF, f g), are invariant for... 

We start by defining the singlet and pair distribution functions for a system of rigid, not necessarily spherical particles. This is a rather simple generalization of the corresponding functions for spherical particles which are discussed in current texts on the theory of liquids. 

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