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New Techniques in Numerical Integration: The Computation of Molecular Integrals over Exponential-Type Functions Open Access


Other title
Trapezoidal Rule
B functions
Conformal maps
Sinc numerical methods
Molecular Integrals
Type of item
Degree grantor
University of Alberta
Author or creator
Slevinsky, Richard M
Supervisor and department
Lau, Anthony To-Ming (Mathematical & Statistical Sciences)
Safouhi, Hassan (Mathematical & Statistical Sciences)
Examining committee member and department
Li, Michael (Mathematical & Statistical Sciences)
Weniger, Ernst Joachim (External Reader, University of Regensburg)
Brown, Alex (Chemistry)
Dai, Feng (Mathematical & Statistical Sciences)
Department of Mathematical and Statistical Sciences
Applied Mathematics
Date accepted
Graduation date
Doctor of Philosophy
Degree level
The numerical evaluation of challenging integrals is a topic of interest in applied mathematics. We investigate molecular integrals in the B function basis, an exponentially decaying basis with a compact analytical Fourier transform. The Fourier property allows analytical expressions for molecular integrals to be formulated in terms of semi-infinite highly oscillatory integrals with limited exponential decay. The semi-infinite integral representations in terms of nonphysical variables stand as the bottleneck in the calculation. To begin our numerical experiments, we conduct a comparative study of the most popular numerical steepest descent methods, extrapolation methods and sequence transformations for computing semi-infinite integrals. It concludes that having asymptotic series representations for integrals and applying sequence transformations leads to the most efficient algorithms. For three-center nuclear attraction integrals, we find an analytical expression for the semi-infinite integrals. Numerical experiments show the resulting algorithm is approximately 300 times more efficient than the state-of-the-art. For the four-center two-electron Coulomb integrals, we take a different approach. The integrand has singularities in the complex plane that can be near the path of integration, making standard quadrature routines unreliable. The trapezoidal rule with double exponential variable transformations has been shown to have very promising properties as a general-purpose integrator. We investigate the use of conformal maps to maximize the convergence rate, resulting in a nonlinear program for the optimized variable transformation.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
Citation for previous publication
R. M. Slevinsky and H. Safouhi, "A comparative study of numerical steepest descent, extrapolation, and sequence transformation methods in computing semi-infinite integrals," Numerical Algorithms, 60:315--337, 2012.

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