Radial orbitals

Theoretically, for a particle of a given size that moves in the highly rotating fluid flow in a cyclone, a particular radial orbit position may be found in every horizontal plane of the cyclone where the outward centrifugal force is just balanced by the drag exerted on the particle by the radial inward fluid flow. If Stokes s law (13.16) is assumed, then the position of the equilibrium orbit on each horizontal plane of the cyclone may be obtained and is given by... 

The division of the ( + 1) bonding AO s into two types, a unique radial orbital and n surface orbitals, incidentally is also a feature of a free-electron MO treatment (198) which has been applied to borane polyhedra, as distinct from the linear combination of atomic orbitals (LCAO) MO treatments mentioned above. 

The density associated with the Hartree-Fock-Raffenetti wavefunction is denoted by puVif)- We take this to be the initial density in our local-scaling transformation, i.e., pi r) = puVir)- We take as the final density, that associated with the 650-term Cl wavefunction of Esquivel and Bunge [73], which we call P2ir) = pair). These two densities are practically about the same, as can be seen clearly in Fig. 4, where we have also plotted their difference. The transformed radial orbitals are given by ... 

The spherically averaged atomic core and valence densities are obtained as the sum over products of the radial orbital functions, or, including normalization,... 

The radial momental Uni p) is related to e radial orbital r fnt r) by a Hankel... 

Figure 8.16 Schematic representations of the MOs of. The upper set are those formed by inwardly directed radial orbitals (s, p or sp. hybrids). The lower ones show one each of the tangential sets.

According to the general relationship (5.9), rotations in isospin space transform the electron creation operators by the D-matrix of rank 1/2. If we go over from these operators to the one-electron wave functions they produce, then we shall have the unitary transformation of radial orbitals... 

Calculations carried out with hydrogen radial orbitals indicate that the supermultiplet basis S U4 in many cases is more diagonal than the basis in which an additional classification of states is achieved using the quantum numbers v, t) of the five-dimensional quasispin group. 

To sum up the potentialities of the isospin method are not exhausted by the results stated above. There is a deep connection between orthogonal transformations of radial orbitals and rotations in isospin space (see (18.40) and (18.41)). This shows that the tensorial properties of wave functions and operators in isospin space must be dominant in the Hartree-Fock method. This issue is in need of further consideration. 

It is necessary to underline that the dependence of the matrix elements of the relativistic energy operator on orbital quantum numbers is contained only in phase multipliers and radial integrals. There are only a few types of radial integrals and their coefficients, which considerably simplifies their calculation. Radial integrals have no derivatives of radial orbitals, whereas... 

Let us notice that for n2l2 = nih and the same radial orbitals expression (25.6) equals zero. Therefore the radial orbitals, depending on the term quantum numbers, must be employed when calculating /c-transitions using the second form (4.13). 

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]. 

The Hartree-Fock approach is also called the self-consistent field method. Indeed, the potential of the field in which the electron nl is orbiting is also expressed in terms of the wave functions we are looking for. Therefore the procedure for determination of the radial orbitals must be coordinated with the process of finding the expression for the potential starting with the initial form of the wave function, we find the expression for the potential needed to determine the more accurate wave function. We must continue this process until we reach the desired consistency between these quantities. 

Equations (28.22) are valid for both ground and excited configurations of atoms and ions for the cases when all radial orbitals are varied, or some of them are frozen . 

Usually the solutions of any version of the Hartree-Fock equations are presented in numerical form, producing the most accurate wave function of the approximation considered. Many details of their solution may be found in [45], However, in many cases, especially for light atoms or ions, it is very common to have analytical radial orbitals, leading then to analytical expressions for matrix elements of physical operators. Unfortunately, as a rule they are slightly less accurate than numerical ones. 

Analytical radial orbitals may be found from the variational principle starting with a given analytical form of the initial function. The accuracy of the final radial orbital obtained depends essentially on the type of initial function and on the number of parameters varied. With increase of the number of shells in an atom the number of varied parameters grows rapidly. This, in turn, complicates substantially the optimization procedure. For these reasons analytical radial orbitals are impractical for obtaining the energy spectra of middle and, particularly, heavy atoms. 

The simplest analytical radial orbitals may be found by solving the radial Schrodinger equation for a one-electron hydrogen-like atom with arbitrary Z. They are usually called Coulomb functions and are expressed... 

There exist a number of attempts to generalize hydrogenic radial orbitals to cover the case of a many-electron atom. In [180] the so-called generalized analytical radial orbitals (Kupliauskis orbitals) were proposed ... 

Here af and cf for the cases n = l + 1 are found from the variational principle requiring the minimum of the non-relativistic energy, whereas cf (n > l + 1) - form the orthogonality conditions for wave functions. More complex, but more accurate, are the analytical approximations of numerical Hartree-Fock wave functions, presented as the sums of Slater type radial orbitals (28.31), namely... 

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