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Internet Electronic Journal of Molecular Design - IEJMD, ISSN 1538-6414, CODEN IEJMAT
ABSTRACT - Internet Electron. J. Mol. Des. April 2006, Volume 5, Number 4, 201-212

An Efficient Method for Computing NMR Spectral Densities Involving Kohlrausch/Williams-Watts Decay Function
Ahmed Bouferguene and Hassan Safouhi
Internet Electron. J. Mol. Des. 2006, 5, 201-212

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Abstract:
The relaxation behavior of dielectrics in time dependent external electric and magnetic fields plays an important role in the determination and understanding of chemical structures. For example, NMR often requires the evaluation of spectral densities to determine relaxation parameters. Traditionally, spectral densities are represented by semi-infinite integrals with oscillatory integrands. Special algorithms are needed to compute these accurately, and this affects evaluation of all related properties. A new procedure is presented in this paper for the fast and accurate calculation of the spectral densities that involve the Kohlrausch/Williams-Watts decay function. Comparisons with previously published benchmarks show that our procedure is numerically stable and can be used safely for a wide range of parameters. NMR spectral densities obtained using the Kohlrausch/Williams-Watts decay function is usually represented by a semi-infinite integral containing an oscillating integrand. For certain values of the parameters, these oscillations for certain parameters are very strong and pose challenging difficulties from a practical point of view since classical integration techniques are usually unable to interpolate accurately the integrand. In this investigation, we propose to evaluate the spectral densities of interest by means of an infinite series which is obtained from the initial integral representation. However, based on two theoretical results, our series are shown to converge logarithmically which makes direct summation techniques extremely costly. To circumvent this difficulty we apply Wynn's epsilon algorithm to accelerate the convergence of the infinite series hence allowing an efficient numerical procedure to be obtained. Thorough comparisons with previously published data by Dishon et al. (J. Res. Natl. Bur. Stand. 1985, 90, 27-39) are carried. Selected numerical data are presented in several tables showing the agreement of our calculations with those in the literature. In addition, the result of a large scale comparison is provided as a gray level image in which the axes represent typical values of some parameters (used in practice) while the gray level provides the number of exact digits. It is shown that the procedure developed in this paper allows spectral densities to be accurately computed accurately. Thorough comparisons with previous work show the stability of the numerical approach. In addition, the proposed algorithm is very general and is therefore useable for a wide range of parameters that are needed in practice.

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