Effective Slater integrals

This can be seen by expressing Ull2l>l3 in terms of complex spherical harmonics and effective Slater integrals Fk as in Eq. [63]70... 

Ma et al. [147,162,163] argued that pressure-induced covalency effects can be understood in terms of the radial expansion of the valence electron wavefunc-tions as the nearest neighbor bond length decreases with pressure. They considered an isotropic compression model based on the scaHng of the Slater integrals... 

As fits to the crystal spectra became more detailed, a lack of balance in the theory appeared. The Coulomb interactions within the 4f shell and the effects of configuration interaction to second order can be taken into account by means of the four Slater integrals F (4f, 4f), the three Trees parameters a, J5, y, and the six three-electron parameters T . In contrast to these 13 electrostatic parameters, the spin-orbit interaction, until 1968, was represented by the single parameter This scheme overlooks the terms that arise from the Breit interaction, which was developed on relativistic grounds to account for the fine structures of the multiplets of Hel Isnp (see Bethe and Salpeter 1957). In the non-relativisitc limit parts of the Breit interaction, such as the retardation of the Coulomb interaction and the magnetic interactions that exist between the electrons in virtue of their orbital motions, can be represented by adjustments to the electrostatic parameters. Two terms cannot be absorbed in that way the spin-spin interaction and the spin-other-orbit interaction Marvin (1947) showed that, for the configurations I", ... 

The theory presented here is not fully ab initio since it involves the adjustment of the chemical potential as well as the (or Hubbard U) parameter. The U parameter only enters in cases where three or more occupation numbers n off configurations are involved, as is the case in the examples presented here. The size of multiplet splittings within a given configuration is solely determined by the F > 0) parameters, which are calculated from the self consistent f partial wave, as the examples show with quite accurate results. Hence, screening effects do not enter as severely into the higher Slater integrals as they do for the Coulomb U parameter. 

Pressure affects the energetic structure of the Ln ion and the energies of the 4f -4f transitions in two ways (1) the nephelauxetic [115] reduction of the Slater integrals and (2) the spin-orbit coupling constant. This effect was broadly discussed by Shen and Holzapfel [116] in the framework of the covalence model. In this framework, the radial-wave functions of the Ln ion, when it is embedded in the lattice, expand compared with those of a free ion due to the penetration of the ligand electrons into the Ln ion space. Penetration increases when pressure decreases the Ln ion-ligand distances. The second effect is related to the pressure-induced increase of the crystal-field strength. 

Usually, it is assumed that pressure effects under the hydrostatic limit are isotropic, which causes a proportional decrease of all distances in the lattice without a change in the local symmetry of the Ln ion. As the result, a standard assumption is that angular factors C (0, (j>) in the crystal-field Hamiltonian are pressure independent and that the only pressure-sensitive parameters are the Slater integrals, spin-orbit coupling, and radial crystal-field parameters (R)- Then, in most cases, pressure causes small linear shifts of the sharp-lines luminescence related to f-f transitions in the Ln " and Ln ions. 

As to spherical parameters, at least the Slater integrals and the spin-orbit coupling constant, it should be possible to evaluate them with precision from the radial part of the f-electron wave function obtained by solving the Hartree or Hartree-Fock equations. Unfortunately, the F calculated this way are 30-50% greater than the fitted ones. It seems that the lanthanides are almost alone (even if there are fourteen elements ) in the Periodic Table in displaying such pronounced misbehaviour. Elements to help understand this may be found in a paper by Rajnak and Wyboume (1963) who showed that the effects of configuration interactions in configurations may be represented by... 

The practical problems associated with the evaluation of exchange integrals led Slater (1974) many years ago to introduce a localized approximation to the exchange. This has been widely utilized and more recently has been modified to include additional correlation effects. Potentials so obtained are called local-density potentials. ... 

Explicit formulas and numerical tables for the overlap integral S between AOs (atomic orbitals) of two overlapping atoms a and b are given. These cover all the most important combinations of AO pairs involving ns, npSlater type, each containing two parameters i [equal to Z/( - 5)], and n — S, where n — S is an effective principal quantum number. The S formulas are given as functions of two parameters p and t, where p = p- + p,s)R/ao, R being the interatomic distance, and t = — Mb)/(ma + Mb)- Master tables of... 

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