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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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