Simultaneous fitting

In conventional fitting, as has been widely used within this community in the past, the gradients and properties have been calculated at the experimental crystal structure and the potential parameters have been varied so as to minimize the error in these calculated quantities. This approach takes the experimental gradients to be zero at the observed atomic co-ordinates. 

These above two methods yield slightly different results, if properties other than the crystal structure are included in the fit, since in the first technique the shells are purely minimized with respect to the energy, whereas in the second case the shells are optimized with respect to the sum of the squares of the residuals. 

The shell model can allow the simulation of a significantly covalent material using an ionic model because of the similarity between polarization and covalency - both are just shifts in the electron density distribution, but with different partitioning. In this case, the dipolar shell model can subsume covalent effects because of the low symmetry at oxygen. 

Modeling of silicates within the ionic model employing formal charges is now well established (Catlow and Cormack 1987), however, earlier attempts to extend the scope of such calculations to their aluminophosphate analogues had proved unsuccessful because the cores 

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Pullerits T, Visscher K J, Hess S, Sundstrom V, Freiberg A, Timpmann K and Van Grondelle R 1994 Energy-transfer in the inhomogeneously broadened core antenna of purple bacteria-a simultaneous fit of low-intensity picosecond absorption and fluorescence kinetics Biophys. J. 66 236-48... 

Fig. 6. Pilot-scale kiln results for a fill fraction of 0.08% at 0.5 rpm and an initial toluene loading, on a dry, calcined, montmorillonite clay adsorbent, of 0.25 wt %, at A, 790°C B, 330°C and C, 190°C. The soHd lines are model fits using equation 24. The model simultaneously fits to all of the data (24).

Recent papers [4-6] of the NRL group have concentrated on a tight-binding methodology that simultaneously fits the energy bands and the total ener — of the fee and bcc structures as a function of volume, and correctly predicts the ground state for those metals that crystallize in the hep or even the Of-Mn structure. 

Fig. 17. High-spin low-spin relaxation of [Fe(ptz)j](BF4)2 a normalized relaxation curves at temperatures between 57.5 and 66.5 K b activation energy AE as a function of n. The solid line represents a simultaneous fit of the five experimental curves displayed under a. The broken line is the result of a linear regression determined by AE° = — 164 cm" = — 0.47 kcal mol" as slope and AE = 797 cm" = 2.28 kcal mol" as intercept. Notation in the figure uses yi, A instead of Hl,...

Great progress has been made, however, in our later study using in situ SXS [Stamenkovic et ak, 2007a], where, by simultaneously fitting the intensity ratio between two different sets of crystal truncation rod (CTR) data that constrain the fit to the full CTR data [Robinson, 1986 Warren, 1990], it was possible to reveal the elemental concentration profile at the surface (Fig. 8.13c). Based on the in situ SXS results depicted in Fig. 8.13a, the topmost surface layer is confirmed to be 100 at%... 

The spectrum recorded at 230 K was discarded in the fit procedure because above 200 K the effective thickness decreases drastically because of a significant softening of protein-specific modes [16]. From the simultaneous fit of the spectra in the temperature range 3.2-200 K, the Debye temperature was determined as do = 215 K. AEq proved to be a temperature-dependent quantity, which is discussed later (see Sect. 9.4.2). 

The temperature-dependent fluctuation rates a>i2 of the EFG, which result from this simultaneous fit, were interpreted in terms of low-temperature tunneling plus a temperature-dependent activation of the terminal O-atom from position 1 to 2 and vice versa, described by the expression... 

Q2 behavior takes place at decreasing Q. The position of the crossover point Q (c) is a direct measure of the dynamic correlation length (c) = 1/Q (c). The plateau value at low Q determines the collective diffusion coefficient Dc. A simultaneous fit of all low Q spectra where a simple exponential decay was found led to the concentration dependence and the numerical values of Dc... 

The experimental data also show that the crossover to the many-chain regime does not appear in the Q-window accessible to the method. Owing to this lack, the direct experimental evidence that the upturn has to be assigned to a single chain mode and does not result from a collective mode, is still missing. Nevertheless, assuming Dc c, and expressing Dg by DC/(Q/), from a simultaneous fit (c)/A = (6.0 0.5)c-1 and //A = 7.4 + 0.6 are derived. 

Fig. 65. Concentration dependence of the hydrodynamic screening length (c). The solid line represents the result of the simultaneous fit, the dashed line in correlation length (c) related to the transition from single to many chain behavior. (Reprinted with permission from [40]. Copyright 1984 American Chemical Society, Washington)...

Ishikawa etal. proposed an approach for the determination of the ligand-field (LF) parameters of a set of isostructural lanthanide complexes. This method consists of a simultaneous fit of the temperature dependence of magnetic susceptibilities and NMR spectra for the whole isostructural series [18]. In order to avoid over-parametrization a key restriction is imposed each parameter is expressed as a linear function of the number of f electrons, n ... 

Simultaneous fits of a series of related complexes, as mentioned above. In this case, a handful of parameters can be used to reproduce the experimental behaviour of a large quantity of compounds. This reduces the possibilities of numerical artefacts, and also the propagation of experimental errors into the final parameters. 

Figure 2.8 is a partial snapshot of the recent progress in the construction of this general library, taken from Ref. [23]. Only the homoleptic complexes which can be fitted by the REC model are included. In the case of the halides F-, Cl-, Br-, energy levels were used [21]. For oxygen, two series of a total of 10 POM complexes were used, but in this case only xT data were available. For nitrogen, a combination of simultaneous fits, xT data and spectroscopic data, was used [22, 24]. In this comparison we chose to depict the position... 

If a collisional quencher of the fluorophore is also incorporated into the membrane, the lifetime will be shortened. The time resolution of the fluorescence anisotropy decay is then increased,(63) providing the collisional quenching itself does not alter the anisotropy decay. If the latter condition does not hold, this will be indicated by an inability to simultaneously fit the data measured at several different quencher concentrations to a single anisotropy decay process. This method has so far been applied to the case of tryptophans in proteins(63) but could potentially be extended to lipid-bound fluorophores in membranes. If the quencher distribution in the membrane differed from that of the fluorophore, it would also be possible to extract information on selected populations of fluorophores possibly locating in different membrane environments. 

The activity coefficients can be calculated using any of the existing models if the binary parameters for all combinations of binary pairs are known. These parameters are obtained by fitting to experimental data. For ternary systems, one can either simultaneously fit all six parameters or first determine the parameters using binary data for those binary systems that have a phase separation and the rest of the parameters from ternary data. 

The smaller contribution to solvent proton relaxation due to the slow exchanging regime also allows detection of second and outer sphere contributions (62). In fact outer-sphere and/or second sphere protons contribute less than 5% of proton relaxivity for the highest temperature profile, and to about 30% for the lowest temperature profile. The fact that they affect differently the profiles acquired at different temperature influences the best-fit values of all parameters with respect to the values obtained without including outer and second sphere contributions, and not only the value of the first sphere proton-metal ion distance (as it usually happens for the other metal aqua ions). A simultaneous fit of longitudinal and transverse relaxation rates provides the values of the distance of the 12 water protons from the metal ion (2.71 A), of the transient ZFS (0.11 cm ), of the correlation time for electron relaxation (about 2 x 10 s at room temperature), of the reorienta-tional time (about 70 x 10 s at room temperature), of the lifetime (about 7 x 10 s at room temperature), of the constant of contact interaction (2.1 MHz). A second coordination sphere was considered with 26 fast exchanging water protons at 4.5 A from the metal ion (99), and the distance of closest approach was fixed in the range between 5.5 and 6.5 A. 

The algorithms can be Interpreted in this case as trying to obtain a "best" simultaneous fit of the data to two straight lines (in the sense of minimizing Equation 5). If r had been chosen as r=0 in the above problem, the distances would have reduced to just... 

Refinement is thus the optimization of the model parameters to simultaneously fit both the experimental diffraction observations and a set of a priori known stereochemical information. It should become clear from the formula above, that solvent content crucially affects the observation-to-parameter ratio. Since crystals that diffract to lower resolution typically have higher solvent content, refinement can often be carried out decently well even at nominally low resolution. 

The implemented model is based on the Global Fit approach, i.e. all the limbscanning spectra are simultaneously fitted, and on the analysis in narrow spectral intervals (microwindows). 

Max m is the maximum molality at which the coefficients will give reliable values. a is the standard deviation over this molality range. The letter a indicates that data of several types were simultaneously fitted, leading to higher precision for those coefficients than for the ones designated by b and c where only limited data were available. 

FIGURE 7.8 X-ray (square) and neutron (circle) scattering data for DMDBTDMA in n-dodecane contacted with water. Lines correspond to the simultaneous fit to the experimental X-ray and neutron data with the Baxter sticky hard-sphere approach. [DMDBTDMA] = 0.5 M, [Monomers] = 0.28 M, aggregation number = 4.4, and U/kBT = -1.7. 

Fig. 7.19. The time dependence of the mean ortho-positronium kinetic energy measured in silica aerogel under vacuum (o) and with 0.92 amagat of helium gas added ( ). The two curves were simultaneously fitted to the data by Nagashima et al. (1998), to obtain estimates of the Ps-He momentum transfer cross section. Reprinted from Journal of Physics B31, Nagashima et al, Momentum transfer cross section for slow positronium-He scattering, 329-339, copyright 1998, with permission from IOP Publishing.

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