Spectral parameter fitting processes

Apart from the programs mentioned, most spectrum-processing suites contain modules for simulating DNMR spectra and for fitting spectral parameters (MNova -NMR,40 iNMR,41 SpinWorks,42 NUTS,43 NMRLoop44). 

Dynamic analysis of photochromic systems under continuous irradiation represents a powerful method of investigation of the reaction mechanisms. The characteristic kinetic and spectral parameters such as the quantum yields of the photochemical steps and the molar extinction coefficients of the transient species can be derived using this method. The essence of the method is the inverse treatment based on numerical simulation and fitting of the plots (Abs versus t) obtained under continuous irradiation. This also exploits the information contained in the irradiation kinetics. In order to extract one or more of the relevant parameters of a given process, specially designed experiments need to be carried out in which the effect of the process under consideration is conspicuous. 

The central-limit theorem (Section III.B) suggests that when a measurement is subject to many simultaneous error processes, the composite error is often additive and Gaussian distributed with zero mean. In this case, the least-squares criterion is an appropriate measure of goodness of fit. The least-squares criterion is even appropriate in many cases where the error is not Gaussian distributed (Kendall and Stuart, 1961). We may thus construct an objective function that can be minimized to obtain a best estimate. Suppose that our data i(x) represent the measurements of a spectral segment containing spectral-line components that are specified by the N parameters... 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

Fits to single (one floating parameter) and double (three floating parameters) exponential decay laws are always poorer as judged by the x2 and residual traces. In the case where we assume that there is some type of excited-state process (e.g., solvent relaxation) we find that the spectral relaxation time is > 20 ns. This is much, much greater than any reasonable solvent relaxation process in supercritical CF3H. For example, in liquid water, the solvent relaxation times are near 1 ps (56). 

In Figure 2.10 we show a selection of results, in which experimental and calculated spectra are compared at 292 and 155K. The results are quite satisfactory, especially when considering that no fitted parameters, but only calculated quantities (via QM and hydrodynamic models) have been employed. The overall satisfactory agreement of the spectral line shapes, particularly at low temperatures, is a convincing proof that the simplified dynamic modelling implemented in the SLE through the purely rotational stochastic diffusive operator f, and the hydrodynamic calculation of the rotational diffusion tensor, is sufficient to describe the main slow relaxation processes. 

Figure 1.3. Inversion-recovery experimental results for determining Tj relaxation times of ibuprofen and maleic add. The spectral region from 5.6 to 7.6 ppm is displayed as a function of the variable delay time t. The t values used were (from left to right across the figure) 0.125s, 0.25s, 0.50s, 1.0s, 2.0s, 4.0s, 8.0s, 16s, 32s, and 64s. The acquisition and processing parameters are the same as for Figure 1.1. The inset shows the signal intensities of the maleic acid peak (5.93 ppm) fit to Eq. (1.5).

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