Generalized spherical functions definition

General properties and definitions of polarizabilities can be introduced without invoking the complete DFT formalism by considering first an elementary model the dipole of an isolated, spherical atom induced by a uniform electric field. The variation of the electronic density is represented by a simple scalar the induced atomic dipole moment. This coarse-grained (CG) model of the electronic density permits to derive a useful explicit energy functional where the functional derivatives are formulated in terms of polarizabilities and dipole hardnesses. 

Clebsch-Gordan coefficients have already occurred several times in our considerations in the Introduction (formula (2)) while generalizing the quasispin concept for complex electronic configurations, while defining a relativistic wave function (formulas (2.15) and (2.16)), in the addition theorem of spherical functions (5.5) and in the definition of tensorial product of two tensors (5.12). Let us discuss briefly their definition and properties. There are a number of algebraic expressions for the Clebsch-Gordan coefficients [9, 11], but here we shall present only one ... 

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... 

The vector spherical harmonics YjtM form an orthogonal system. The state of the photon with definite values of j and M is described by a wave function which in general is a linear combination of three vector spherical harmonics... 

Now we are ready to define the spherical harmonic functions. In Section 1,6 we gave examples for t = 0, 1, 2 here is the general definition. 

In this section, we illustrate the general features of the radial distribution function (RDF), g(R), for a system of simple spherical particles. From the definitions (2.31) and (2.39) (applied to spherical particles), we get... 

The spherically averaged electron density is an important quantity [481] as it can be defined even for an atom in a molecule. It follows from the general definition of the electron ground-state density p(r), which is to be calculated from the total electronic ground state wave function To, Eq. (4.9). For a representation in spherical coordinates r = (r, qj, j ) we define a radial electron density D(r) such that... 

In turn, the coefficients of the expanded series JV, and Q can be obtained from equations (8) and (9) respectively. Clearly, from equations (8) and (9), the coefficients and Qj are the averages of the distribution functions with respect to the Imnth and Imth orders of spherical harmonics respectively. In other words, the coefficients are the Imnth and /mth averages of the orientation distribution functions, and are a sort of general definition of the orientation factors as will be discussed later. 

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