Crystal Slater integrals

Model Hartree-Fock calculations which include only the electrostatic interaction in terms of the Slater integrals F0, F2, F and F6, and the spin-orbit interaction , result in differences between calculated and experimentally observed levels596 which can be more than 500 cm-1 even for the f2 ion Pr3. However, inclusion of configuration interaction terms, either two-particle or three-particle, considerably improves the correlations.597,598 In this way, an ion such as Nd3+ can be described in terms of 18 parameters (including crystal field... 

Here E (fd) and G7(fd) stand for the direct and exchange Slater integrals for the Coulomb interaction between the 4f and 5d electrons (Cowan, 1981). The f(dd) parameter is related to the spin-orbit interaction for the 5d electron. The interaction between 5d electron and crystal field is described by the following term ... 

These differences are the well-known Slater integrals, which determine how the orbital moments orient themselves for a free ion. hi a real crystal, there will always exist a competition between the crystal field parameters (described by the am s) and integrals hke those appearing in Eq. 52 which will condition the ionic state. 

The most straightforward modifications of simple- crystal field-theory-that make allowance for orbital overlap involve using all parameters of interelectronic interactions as variables rather than taking them equal to the values found for the free ions. Of these parameters, three are of decisive importance, namely, the spin-orbit coupling constant, A, and the interelectronic repulsion parameters, which may be the Slater integrals, F , or certain usually more convenient linear combinations of these called the Racah parameters, B and C. 

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", ... 

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. 

In this section we show how the general form of Renner-Teller interaction matrices can be obtained at any order in the phonon variables and with electron orbital functions of different symmetry (p-like, < like, /-like, etc.). For this purpose, we use an intuitive approach [18] based on the Slater-Koster [19] technique and its generalization [20] to express crystal field or two-center integrals in terms of independent parameters in the tight-binding band theory [21] then we apply standard series developments in terms of normal coordinates. 

The HF CO method is especially efficient if the Bloch orbitals are calculated in the form of a linear combination of atomic orbitals (LCAO)1 2 since in this case the large amount of experience collected in the field of molecular quantum mechanics can be used in crystal HF studies. The atomic basis orbitals applied for the above mentioned expansion are usually optimized in atoms and molecules. They can be Slater-type exponential functions if the integrals are evaluated in momentum space3 or Gaussian orbitals if one prefers to work in configuration space. The specific computational problems arising from the infinite periodic crystal potential will be discussed later. 

This basic parameterization scheme, used at the time of the last A.C.S. symposium on lanthanide and actinide chemistry ( 5), has been discussed in detail by Wybourne 06). In applying the scheme, the free-ion Hamiltonian was first diagonalized and then the crystal-field interaction was treated as a perturbation. This procedure yielded free-ion energy levels that frequently deviated by several hundred cm from the observed energy levels.a In addition, the derived parameters such as the Slater radial integral, f(2), and the spin-orbit radial integral did not follow an expected systematic pattern across the lanthanide or actinide series ( 7). ... 

Figure 12.8 The Bethe Slater curve for the magnitude of the exchange integral as a function of D/d, where D is the separation of the atoms in a crystal, and d is the diameter of the 3d orbital...

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