Condon-Shortley parameters

Occasionally ihe Slalcr-Condon-Shortley parameters F2 and F4 are used instead. Their relation to the Racah parameters is B = F2 - 5Ft and C = 35F4. 

As with ligand-field absorption spectra, the Nephelauxetic effect [41, 42] will also impact L-edge XAS data. For L-edge spectra, the effect of a reduction in interelectron repulsion is also to reduce the ffee-ion state splitting and make the spectra somewhat more orbital like. This is demonstrated by the sequence of simulated low-spin Fe(III) L-edge X-ray absorption spectra shown in Fig. 13, where the e - e repulsion is reduced systematically from [i = 100% to 60%. This starts from 80% of the Hartree-Fock calculated values of the Slater Condon Shortley parameters for e - e repulsion. 

Slater-Condon-Shortley parameters of interelectronic repulsion photon energy ionization energy... 

Interelectron repulsion integrals are parametrized via central-field Condon-Shortley parameters F2 and F4 (or the Racah equivalents B and C) for the (7-block. For the /-block, the related Ei, E2, and F3 parameters are used together with an additional nine parameters to account for... 

In further refinement, one could consider additional pressure effects on C for some ions. (One might expect that the Slater-Condon-Shortley parameter would be more affected by pressure than F. ) Alternatively one might assume with Stout that the higher energy Cg) levels are more spread out by the crystal field than the 2 levels. One could then calculate the covalency parameter e for these as a function of pressure. 

The energy levels of the free Cr3+-ion (d ) are quite well known (4). A quantitative description with the Slater-Condon-Shortley method (5) using the Racah-parameters B and C ) and the L—S-coupling parameter A does not lead to a satisfactory agreement between theory and experiment this can be achieved only by assuming different C/R-ratios for different doublet-levels (Table 1). The depression of the P-level is especially remarkable. The deviations from the three-parameter theory are less pronounced for Ni + or Co + (2), for example. 

Firstly, these /8 parameters are rather ill-defined but most immediately, the qualitative trends between the values has been shown merely to reflect a greater reduction in Fi than in from the corresponding free-ion values. In short these parameter trends furnish little or no evidence for differential orbital expansion. Further, even if such differentiation is really there - by which we presumably mean within a conventional m.o. sheme - it must manifest itself only within the numerical values of (and ligand field parameters). While working within a (f basis, therefore, there is no inconsistency whatever by recognizing the non-spherical hgand field within Fl.f. of (1-5) at the same time as representing the interelectron repulsions - l/r,y of (1-5) - in the usual free-ion Condon-Shortley theory. 

Ligand-field models serve the purpose of parameterizing experiments ). Their beauty and applicability stem from their derivation from the elementary theory of atomic spectra they are first-order perturbation models based upon a basis set of I functions (assumption 1, p. 71), and hence the interelectronic repulsion within the I shell may be accounted for in terms of Condon and Shortley parameters or Racah parameters. We obtain the expanded radial function model (25, 13,8). 

Once it was realized that there is an almost invariant ratio between the parameters of interelectronic repulsion in the f shell, both in the Slater-Condon-Shortley parametrization 1 0.75 0.5, and in the treatment of Racah... 

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 Akm can be calculated from a model such as the modified point-charge model presented in section 3.2.4, the Rt l) can be calculated as well, and matrix elements of ju, can be computed between crystal-field split sublevels for a particular lanthanide ion in a particular host crystal a priori, without fitting experimental intensity measurements (Esterowitz et al., 1979a Leavitt and Morrison, 1980). However, this method is not prevalent in the literature rather, usually the theory is expressed in terms of a few adjustable parameters and a fit is made to intensity data. To this end, we consider the line strength, defined by (Condon and Shortley, 1959)... 

Respective mean errors of only 158cm and 144cm were obtained, and in the process of the analysis 126 new levels were found. This sophisticated treatment is a far cry from the example of La II 4f given by Condon and Shortley (1935) in their classic monograph. For that spectrum only four parameters were used to fit the seven LS terms of P. Even that represents a considerable advance in content over table 5.4 of White (1934), where the ground configurations of eleven lanthanide atoms (Pr, Nd, Pm, Sm, Eu, Tb, Dy, Ho, Er, Tm and Yb) are given incorrectly. 

The attraction of the second spectrum of lanthanum for atomic spectroscopists has been mentioned in section 2.1. The analyses of the two-electron configurations 4f5d and 4f carried out by Condon and Shortley (1931) achieved great importance by being the only lanthanide spectrum described in detail in their subsequent monograph (Condon and Shortley 1935). The least-squares fits to the experimental data provided by Russell and Meggers (1932) yielded Slater parameters given (in cm ) by... 

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