Spectral transformations

The equilibrium between the complexes formed according to Equation (80) depends both on the concentration of fluorine ions and on the potential of interionic interactions, namely the nature of the outer-sphere cations [358]. The influence of the concentration of fluorine ions and of the nature of the outer-sphere cations on the equilibrium in Equation (80) can be demonstrated by the spectral transformations observed at 850°C for M2TaF7 - MF systems, where M = alkali metal [358]. 

IR spectra of the molten system K2TaF7 - KF are more sensitive to the concentration of KF (Fig. 76, b). Molten K2TaF7 is characterized by a strong band at 605 cm"1 and a shoulder at 540 cm 1, as shown before (see Fig. 74, curve 3). The addition of KF to molten K2TaF7 leads to a decrease in the intensity of the above band and to the appearance of a band at 540 cm 1. When the KF concentration equals or exceeds 0.9 mol fraction, a band is observed at 540 cm 1 only. This spectral transformation indicates that the equilibrium in Equation (80) shifts to the left with the increase in KF concentration, and that at KF concentrations above 0.9 mol fraction, only TaF72 complexes are present, while TaF6 ions are not observed by IR emission spectral methods. 

The conception of the formation of hetero-ligand complexes and the nature of anion-anion interactions can be clarified using IR spectra of K2TaF7 - KX mixtures, where X = Cl, Br or I [356, 360]. Fig. 78 shows the spectral transformation due to the dilution of molten K2TaF7 with potassium halide. 

Increasing the concentration of potassium chloride, KCI, in the molten system K2TaF7 - KCI leads to the spectral transformation (Fig. 78, a), which is similar to that observed in the case of K2TaF7 - KF. The intensity of the band at 600 cm 1 decreases while the intensity of the band at 540 cm 1 increases. When KCI concentration is equal to, or greater than 0.8 mol fractions, the band at 600 cm 1 disappears and the only band observed is at 540 cm 1. The above behavior of the system can be explained by the following interaction between hexafluorotantalate ions, TaF6 and chloride ions, CF ... 

An examination of the IR spectral transformations versus concentration of the components in K2TaF7 - KX systems indicates that an increase in the KX concentration leads to a shift to the right in the following equilibrium ... 

As discussed in [91], the shape of a static spectrum determines significantly the spectral transformation as frequency exchange increases. In particular, spectral narrowing will take place only if the second moment of the spectrum is finite. In our case... 

The intensity at the periphery of the line ( Ageneral rule (2.62) [20, 104]. However, the most valuable advantage of general formula (3.34) is its ability to describe continuously the spectral transformation from a static contour to that narrowed by motion (Fig. 3.1). In the process of the spectrum s transformation its maximum is gradually shifted, the asymmetry disappears and it takes the form established by perturbation theory. 

Although from a mathematical point of view formulae (3.34) and (3.40) have little in common, the spectral transformation described by them proceeds in a similar way (Fig. 3.2). Just as with strong collisions, the contour is gradually symmetrized and its centre is shifted to the average frequency coq with an increase in the density. When the spectrum is narrowed (at T 1), its central part ( Aco] < 1/tj) takes the following form ... 

The impact theory defines uniquely the spectral transformation in the limit of weak collisions. Expanding in a series over J — J the integrand of Eq. (6.4) one can obtain at T = /coq%j 1... 

One possibility for this was demonstrated in Chapter 3. If impact theory is still valid in a moderately dense fluid where non-model stochastic perturbation theory has been already found applicable, then evidently the continuation of the theory to liquid densities is justified. This simplest opportunity of unified description of nitrogen isotropic Q-branch from rarefied gas to liquid is validated due to the small enough frequency scale of rotation-vibration interaction. The frequency scales corresponding to IR and anisotropic Raman spectra are much larger. So the common applicability region for perturbation and impact theories hardly exists. The analysis of numerous experimental data proves that in simple (non-associated) systems there are three different scenarios of linear rotator spectral transformation. The IR spectrum in rarefied gas is a P-R doublet with either resolved or unresolved rotational structure. In the process of condensation the following may happen. 

However, there is a price to pay in a spectral transform Lanczos algorithm At each recursion step, the action of the filter operator onto the Lanczos vectors has to be evaluated. In the original version, Ericsson and Ruhe update the Lanczos vectors by solving the following linear equation ... 

Interestingly, the spectral transform Lanczos algorithm can be made more efficient if the filtering is not executed to the fullest extent. This can be achieved by truncating the Chebyshev expansion of the filter,76,81 or by terminating the recursive linear equation solver prematurely.82 In doing so, the number of vector-matrix multiplications can be reduced substantially. 

PIST distinguishes itself from other spectral transform Lanczos methods by using two important innovations. First, the linear equation Eq. [38] is solved by QMR but not to a high degree of accuracy. In practice, the QMR recursion is terminated once a prespecified (and relatively large) tolerance is reached. Consequently, the resulting Lanczos vectors are only approximately filtered. This inexact spectral transform is efficient because many less matrix-vector multiplications are needed, and its deficiencies can subsequently... 

Quasiminimal Residuals Method to Accelerate an Inexact Spectral Transform Calculation of Energy Levels and Wave Functions. 

Scattering Study of the Cl + CH4 — HC1 + CH3 Reaction via Spectral Transform Iteration. 

Spectral Transform Method for Calculating Resonance Energies and Widths, as Applied to HCO. 

Calogero, F., and Degasperis, A. (1982), Spectral Transform and Solitons, North-Holland, Amsterdam. 

Fig. 8. Spectral transformations in the reaction between Craq002 + (9 x 10 M) and L1(H20)RhH2+ (2 x 10 1 M) in 02-saturated aqueous 0.1 M HC104.

F. Calogero and A. Degasperis, Spectral Transformations and Solitons. Methods for Solution and Study of Evolutionary Equations, Mir, Moscow, 1985 (in Russian). 

Due to the nature of FFT, or spectral transforms in general, parallelization of the solver is more suited to a shared memory architecture than to the Message Passing Interface approach. Further, since the most time-consuming portion of the code deals with calculating the RHS of the equation, we choose to apply a single threaded library for ODE solvers (gsl in our case) and only parallelize the calculation of derivatives needed in the ODE solver. 

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