Slater-Condon factors

R generalized Slater-Condon factors, calculated ab initio... 

There are several theoretical studies on LaO and related lanthanide oxides. We have already mentioned the ligand-field theory model calculations of Field (1982) as well as Carette and Hocquet (1988). More recently, Kotzian et al. (1991a,b) have applied the INDO technique (Pople et al. (1967) extended to include spin-orbit coupling [see also Kotzian et al. (1989a, b)] to lanthanide oxides (LaO, CeO, GdO and LuO). The authors call it INDO/S-CI method. The INDO parameters were derived from atomic spectra, model Dirac Fock calculations on lanthanide atoms and ions to derive ionization potentials, Slater-Condon factors and basis sets. The spin-orbit parameter is derived from atomic spectra in this method. 

Electron-electron interaction is significant in studies where spin-orbital degeneracy, or near degeneracy, is present. The atomic d-orbitals have a common radial factor and their 15 distinct products give rise to 120 density-density integrals. These are expressed in terms of three basic ones, F0, F2, and F4 in the Slater-Condon formulation. The 100 spin-orbital densities are linear combination of the orbital ones and the 100 by 100 interaction integral matrix has a rank of 15 and is expressed by the Slater-Condon parameters. 

Several of the numbers in eq. (15) may readily be uncertain by a factor of two, but for our purpose, the most important feature is Xz close to a half. This qualitative result pervades ligand field calculations, and it is almost surprising that Hartree-Fock radial 4f functions only overestimate the parameters of interelectronic repulsion by 30-40%, and it is close to unbelievable that the relative distances of terms in Ln(III) are better described by Slater-Condon-Shortley-Racah parameters than d and p in monatomic entities. 

The energy levels of 3d 4p have been calculated In the sequence Y XIII —Ag XXI by the Slater-Condon method and Z-expansion formulas are given for the Slater and spin-orbit fitted parameters [8], as well as scaling factors between Hartree-Fock integrals and fitted parameters. 

In conclusion, we note that the phase factors (1.2.3) were necessary to reproduce the Slater-Condon rules for matrix elements between Slater determinants. 

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