Atomic spectroscopy parameter fitting

The number of experimentally determined and assigned levels often exceeds the number of free-ion and crystal-field parameters by several fold. Even so, local minima are a significant problem in the least-squares adjustment procedure. The practice is to allow only a fraction of the parameters given in eq. (1) to vary freely. Note that fitting of model free-ion parameters to experimental data recorded in condensed phases results in parameter values that differ from the free-ion values found in atomic spectroscopy studies of lanthanide ions (Goldschmidt 1978) and actinide ions (Freed and Blasse 1986) in the vapor phase. For this reason, the term free ion is used to refer to parameters or energy levels determined from studies on ions in condensed phases. 

The parameters l0, Kb, 0o, K , K,p, n, 8, ay, by, qit qy and r belong to the fit parameters, which can be determined by fitting of Equation 1.1 to a sufficient set of data calculated by QM and/or determined experimentally (e.g., X-ray scattering, IR spectroscopy, heats of formation). From a numeric point of view the pair interaction terms (van der Waals and Coulomb) are most demanding. In this connection the typical size of polymer packing models is limited to typically 3000-10000 atoms (leading to lateral sizes of bulk models of a few nm), although in other connections now also models with up to 100000 atoms have been used. 

Preliminary models of the surface topography, for example, can be determined by atomic-probe methods, ion-scattering, electron diffraction, or Auger spectroscopy. The chemical bonds of adsorbates can be estimated from infrared spectroscopy. The surface electronic structure is accessible by photoelectron emission techniques. In case the surface structure is known, its electronic structure has to be computed with sophisticated methods, where existing codes more and more rely on first principles density functional theory (DFT) [16-18], or, in case of tight-binding models [19], they obtain their parameters from a fit to DFT data [20]. The fit is not without ambiguities, since it is unknown whether the density of states used for the fit is really unique. 

In any case, because ion scattering is strongly affected by the thermal vibrations of surface atoms, experimental data must be compared to Monte-Carlo simulations for model surfaces to achieve quantitative results. The available data base for structure-fitting is rather small compared to electron spectroscopies, so the sensitivity to structural parameters is sometimes limited. But when the surface structure is close to the bulk structure, ion channeling data can be strongly sensitive to small variations in structural parameters. 

The basic construction of the mathematical model using simplified metabolic networks to describe the reactions of the citric acid cycle and associated transamination reactions between pyruvate and alanine, oxalacetate and aspartate and a-ketoglutarate and glutamate, and the use of the FACSIMILE program (Chance et al., 1977) to solve the rather large number of simultaneous differential equations generated by the model was the same as previously described (Chance etal., 1983). For the present experiments the model was expanded to include an input flux at the level of succinate to represent propionate metabolism to succinyl-CoA, and a dilution of the aspartate pool to represent net proteolysis. These input fluxes required an output flux of carbon from the citric acid cycle in order to maintain a steady state carbon balance, for which the conversion of malate to pyruvate via malic enzyme was chosen. The model calculates the unknown flux parameters to provide a minimum least squares fit of the C fractional enrichments of specific carbon atoms of metabolic intermediates as measured by C NMR spectroscopy. 

In addition to TEM, XRD, and XPS characterizations, a detailed X-ray absorption fine structure spectroscopy (XAFS) analysis of the atomic-scale coordination structures was carried out, revealing increased heteroatomic coordination with improved alloying structures for the catalyst treated at the elevated temperatures. By comparing the results for the same PtNiFe/C catalyst treated at 400 and 800 C from fitting using either PtNi or PtFe model, we found that the fitted parameters for Pt neighbors including coordination number (CN), bond... 

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