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Comment on "Fourier transform of hydrogen-type atomic orbital"

CANADIAN JOURNAL OF PHYSICS 2019年第12期97卷 1349-1360页

作者： Weniger, Ernst Joachim Univ Regensburg Inst Phys & Theoret Chem D-93040 Regensburg Germany

Podolsky and Pauling (Phys. Rev. 34, 109 (1929) doi:10.1103/PhysRev.34.109) were the first ones to derive an explicit expression for the Fourier transform of a bound-state hydrogen eigenfunction. Yukcu and Yukcu (Can. J. Phys. 96, 724 (2018) doi:10.1139/cjp-2017-0728), who were apparently unaware of the work of Podolsky and Pauling or of the numerous other earlier references on this Fourier transform, proceeded differently. They expressed a generalized Laguerre polynomial as a finite sum of powers, or equivalently, they expressed a bound-state hydrogen eigenfunction as a finite sum of slater-type functions. This approach looks very simple, but it leads to comparatively complicated expressions that cannot match the simplicity of the classic result obtained by Podolsky and Pauling. It is, however, possible to reproduce not only Podolsky and Pauling's formula for the bound-state hydrogen eigenfunction, but to obtain results of similar quality also for the Fourier transforms of other, closely related, functions, such as Sturmians, Lambda functions, or Guseinov's functions, by expanding generalized Laguerre polynomials in terms of so-called reduced Bessel functions.

关键词： bound-state hydrogen eigenfunctions Fourier transform generalized Laguerre polynomials hypergeometric series slater-type functions reduced Bessel functions

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Highly accurate numerical results for three-center nuclear attraction and two-electron Coulomb and exchange integrals over slater-type functions

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2004年第2期100卷 172-183页

作者： Safouhi, H Univ Alberta Fac St Jean Edmonton AB T6C 4G9 Canada

The present work focuses on the recent progress in nonlinear transformation methods for improving the convergence of highly oscillatory integrals and in their applications for an efficient and rapid numerical evaluation of molecular electronic integrals over slater-type functions (STFs). The nonlinear D transformation, which is probably the most effective general approach for increasing the rate of convergence of semi-infinite oscillatory integrals, is presented. Molecular integrals over STFs are expressed as finite linear combinations of integrals over B functions. The basis set of B functions is suitable to apply the Fourier transform method, which led to analytic expressions, involving highly oscillatory functions, for molecular integrals. Efficient algorithms based on the D transformation are now developed for a numerical evaluation of molecular integrals over STFs. Numerical results that we obtained with linear and nonlinear molecules, and which are in agreement with those obtained using existing codes (Alchemy, STOP, and ADGGSTNGINT), show that the algorithms described in this work are relevant. (C) 2004 Wiley Periodicals, Inc.

关键词： nonlinear transformations convergence accelerators molecular multicenter integrals slater-type functions B functions

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The W algorithm and the (D)over-bar transformation for the numerical evaluation of three-center nuclear attraction integrals

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2007年第5期107卷 1060-1066页

作者： Duret, Stefan Safouhi, Hassan Univ Alberta Edmonton AB T6C 4G9 Canada

Three-center nuclear attraction integrals over exponential-type functions are required for ab initio molecular structure calculations and density functional theory (DFT). These integrals occur in many millions of terms, even for small molecules, and they require rapid and accurate numerical evaluation. The use of a basis set of B functions to represent atomic orbitals, combined with the Fourier transform method, led to the development of analytic expressions for these molecular integrals. Unfortunately, the numerical evaluation of the analytic expressions obtained turned out to be extremely difficult due to the presence of two-dimensional integral representations, involving spherical Bessel integral functions. The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of these spherical Bessel integrals. This algorithm, which is based on the nonlinear (D) over bar transformation and the W algorithm of Sidi, can be computed recursively, allowing the control of the degree of accuracy. Numerical analysis tests were performed to further improve the efficiency of our algorithm. The numerical results section demonstrates the efficiency of this new algorithm for the numerical evaluation of three-center nuclear attraction integrals. (c) 2006 Wiley Periodicals, Inc.

关键词： nonlinear transformations molecular integrals convergence accelerators numerical integration extrapolation methods slater-type functions B functions

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Numerical treatment of two-center overlap integrals

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JOURNAL OF MOLECULAR MODELING 2006年第2期12卷 213-220页

作者： Safouhi, H Univ Alberta Edmonton AB T6C 4G9 Canada

Among the two-center integrals occurring in the molecular context, the two-center overlap integrals are numerous and difficult to evaluate to a level of high accuracy. The analytical and numerical difficulties arise mainly from the presence of the spherical Bessel integrals in the analytic expressions of these molecular integrals. Different approaches have been used to develop efficient algorithms for the numerical evaluation of the molecular integrals under consideration. These approaches are based on quadrature rules, Levin's u transform, or the epsilon-algorithm of Wynn. In the present work, we use the nonlinear (D) over bar transformation of Sidi. This transformation is shown to be highly efficient in improving the convergence of highly oscillatory integrals, and it has been applied to molecular multicenter integrals, namely three-center attraction, hybrid, two-, three-, and four-center two-electron Coulomb and exchange integrals over B functions and over slater-type functions. It is also been shown that when evaluating these molecular multicenter integrals the (D) over bar> transformation is more efficient compared with the methods cited above. It is now proven that the integrand occurring in the analytic expression of the two-center overlap integrals satisfies all the conditions required to apply the (D) over bar transformation. A highly accurate algorithm based on this transformation is now developed. Special cases are presented and discussed for a better optimization of the algorithm. The numerical results section illustrates clearly the high efficiency of our algorithm.

关键词： nonlinear transformations extrapolation methods numerical integration molecular integrals slater-type functions B functions

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Extrapolation methods for improving convergence of spherical bessel integrals for the two-center Coulomb integrals

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2006年第11期106卷 2318-2323页

作者： Safouhi, Hassan Bouferguene, Ahmed Univ Alberta Edmonton AB T6C 4G9 Canada

Multi-center two-electron Coulomb integrals over slater-type functions are required for any accurate molecular electronic structure calculations. These integrals, which are numerous, are to be evaluated rapidly and accurately. slater-type functions are expressed in terms of the so-called B functions, which are best suited to apply the Fourier transform method. The Fourier transform method allowed analytic expressions for these integrals to be developed. Unfortunately, the analytic expressions obtained turned out to be extremely difficult to evaluate accurately due to the presence of highly oscillatory spherical Bessel integrals. In this work, we used techniques based on nonlinear transformation and extrapolation methods for improving convergence of these oscillatory spherical Bessel integrals. These techniques, which led to highly efficient and rapid algorithms for the numerical evaluation of three- and four-center two-electron Coulomb and exchange integrals, are now shown to be applicable to the two-center two-electron Coulomb integrals. The numerical results obtained for the molecular integrals under consideration illustrate the efficiency of the algorithm described in the present work compared with algorithms using the epsilon (epsilon) algorithm of Wynn and Levin's u transform. (C) 2006 Wiley Periodicals, Inc.

关键词： molecular integrals Coulomb integrals slater-type functions B functions nonlinear transformations extrapolation methods

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Symbolic programming language in molecular multicenter integral problem

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2006年第1期106卷 65-78页

作者： Safouhi, H Bouferguene, A Univ Alberta Edmonton AB T6C 4G9 Canada

It is well known that in any ab initio molecular orbital (MO) calculation, the major task involves the computation of molecular integrals, among which the computation of three-center nuclear attraction and Coulomb integrals is the most frequently encountered. As the molecular system becomes larger, computation of these integrals becomes one of the most laborious and time-consuming steps in molecular systems calculation. Improvement of the computational methods of molecular integrals would be indispensable to further development in computational studies of large molecular systems. To develop fast and accurate algorithms for the numerical evaluation of these integrals over B functions, we used nonlinear transformations for improving convergence of highly oscillatory integrals. These methods form the basis of new methods for solving various problems that were unsolvable otherwise and have many applications as well. To apply these nonlinear transformations, the integrands should satisfy linear differential equations with coefficients having asymptotic power series in the sense of Poincare, which in their turn should satisfy some limit conditions. These differential equations are very difficult to obtain explicitly. In the case of molecular integrals, we used a symbolic programming language (MAPLE) to demonstrate that all the conditions required to apply these nonlinear transformation methods are satisfied. Differential equations are obtained explicitly, allowing us to demonstrate that the limit conditions are also satisfied. (c) 2005 Wiley Periodicals, Inc.

关键词： nonlinear transformations convergence accelerators molecular integrals slater-type functions B functions symbolic programming language MAPLE

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Symbolic programming language in molecular multicenter integral problem

Symbolic programming language in molecular multicenter integ...

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Symposium on Mathematical Methods and Symbolic Calculation in Chemistry and Chemical Biology

作者： Safouhi, H Bouferguene, A Univ Alberta Edmonton AB T6C 4G9 Canada

关键词： nonlinear transformations convergence accelerators molecular integrals slater-type functions B functions symbolic programming language MAPLE

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Extrapolation methods for improving convergence of spherical bessel integrals for the two-center Coulomb integrals

Extrapolation methods for improving convergence of spherical...

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14th Annual Conference on Current Trends in Computational Chemistry

作者： Safouhi, Hassan Bouferguene, Ahmed Univ Alberta Edmonton AB T6C 4G9 Canada

关键词： molecular integrals Coulomb integrals slater-type functions B functions nonlinear transformations extrapolation methods

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Correspondence between GTO and STO molecular basis sets

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JOURNAL OF COMPUTATIONAL CHEMISTRY 2001年第14期22卷 1655-1665页

作者： Rico, JF López, R Ramírez, G Ema, I Univ Autonoma Madrid Fac Ciencias 100 14 Dept Quim Fis Aplicada E-28049 Madrid Spain

We present a method for the characterization of the distance between two spaces: one generated by a Gaussian basis set, and another by a slater basis set. The method is an extension of one previously developed for atoms that has been modified to cover molecular problems. The current version enables us to obtain slater basis sets capable of reproducing the results (multielectronic wave functions and orbitals) obtained with Gaussian basis sets. The interest of this result arises from the fact that we will be able to profit from the effort invested in the optimization of high-quality Gaussian basis sets. (C) 2001 John Wiley & Sons, Inc.

关键词： molecular integrals slater-type functions basis sets

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Analytical Hartree-Fock wave functions for the atoms Cs to Lr

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THEORETICAL CHEMISTRY ACCOUNTS 2000年第5期104卷 411-413页

作者： Koga, T Kanayama, K Watanabe, T Imai, T Thakkar, AJ Muroran Inst Technol Dept Appl Chem Muroran Hokkaido 0508585 Japan Univ New Brunswick Dept Chem Fredericton NB E3B 6E2 Canada

Analytical approximations to Hartree-Fock wave functions are constructed using slater-type functions for the ground states of all 49 neutral atoms from Cs (Z = 55) to Lr (Z = 103), The current compilation is more extensive and more accurate than previous ones. The wave functions are available upon request from the authors or from the Web page http://***/chem/ ajit/*** on the Internet.

关键词： Hartree-Fock wave functions slater-type functions atoms Cs to Lr

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