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Modified b function basis sets with generalized hyperbolic cosine functions

COMPUTATIONAL AND THEORETICAL CHEMISTRY 2018年 1127卷 37-43页

作者： Erturk, M. Ozturk, E. Onsekiz Mart Univ Fac Arts & Sci Dept Phys TR-17100 Canakkale Turkey

Efficient exponential type basis sets constructed from new hyperbolic cosine type b functions have been used in self consistent field calculations for the ground states of the atoms from Helium to Argon and their ions. Different hyperbolic cosine type functions are incorporated into the b functions to increase the accuracy of atomic orbitals and correctly describe the electronic density of systems. The presented results for the minimal basis sets are shown that the quality of generalized hyperbolic cosine type b functions is one of the most appropriate basis sets of b functions, especially for increasing atomic number. A comparison with the standard b functions, hyperbolic cosine type b functions and the corresponding numerical Hartree-Fock values are given in tables. Our study shows that the modification of b functions can be an efficient way of increasing the accuracy of the atomic and molecular SCF calculations. These basis sets may also be used in the calculation of atomic properties such as cusp condition and density properties of atomic electrons. Some numerical results and comparisons are given to clarify the basis sets quality from the density point of view. An improved description of atomic orbitals based on the use of modified b functions will also play an important role in ion-atom collisions problems, semi-emprical and density functional methods. (C) 2018 Elsevier b.V. All rights reserved.

关键词： Exponential type orbital b functions Hyperbolic cosine functions Hartree-Fock-Roothaan method

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The Strange History of b functions or How Theoretical Chemists and Mathematicians do (not) Interact

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2009年第8期109卷 1706-1716页

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

b functions area class of relatively complicated exponentially decaying basis functions. because the molecular multicenter integrals of the much simpler Slater-type functions are notoriously difficult, it is not at all obvious why b functions should offer any advantages. However, b functions have Fourier transforms of exceptional simplicity, which greatly simplifies many of their molecular multicenter integrals. This article discusses the historical development of b functions from the perspective of the interaction between mathematics and theoretical chemistry, which traditionally has not been very good. Nevertheless, future progress in theoretical chemistry depends very much on a fertile, interaction with neighboring disciplines. (C) 2009 Wiley Periodicals, Inc. Int J Quantum Chem 109: 1706-1716, 2009

关键词： electronic structure theory exponentially decaying basis functions b functions multicenter integrals interdisciplinary collaboration

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Gaussian Approximation of Exponential Type Orbitals based on b functions

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2009年第2期109卷 135-144页

作者： Pinchon, Didier Hoggan, Philip E. Univ Clermont Ferrand CNRS UMR 6602 LASMEA F-63177 Clermont Ferrand France Univ Toulouse 3 CNRS UMR 5219 Toulouse Inst Math F-31062 Toulouse France

This work gives new, highly accurate optimized gaussian series expansions for the b functions used in molecular quantum mechanics. These functions are generally chosen because of their compact Fourier transform, following Shavitt. The inverse Laplace transform in the square root of the variable is used for Gauss quadrature in this work. Two procedures for obtaining accurate gaussian expansions have been compared for the required extended precision arithmetic. The first is based on Gaussian quadratures and the second on direct optimization. both use the Maple computer algebra system. Numerical results are tabulated and compared with previous work. Special cases are found to agree before Pushing the optimization technique further. The optimal gaussian expansions of b functions obtained in this work are available for reference. (C) 2008 Wiley periodicals, Inc. Int J Quantum Chem 109: 135-144, 2009

关键词： b functions Gaussian approximation Gauss quadrature method best mean-square approximation

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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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Nonlinear transformation methods for accelerating the convergence of Coulomb integrals over exponential type functions

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THEORETICAL CHEMISTRY ACCOUNTS 2007年第2期117卷 213-222页

作者： Safouhi, Hassan bouferguene, Ahmed 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 Coulomb integrals are the most frequently encountered. As the molecular system gets 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 a further development in computational studies of large molecular systems. The atomic orbital basis functions chosen in the present work are Slater type functions. These functions can be expressed as finite linear combinations of b functions which are suitable to apply the Fourier transform method. The difficulties of the numerical evaluation of the analytic expressions of the integrals of interest arise mainly from the presence of highly oscillatory semi-infinite integrals. In this work, we present a generalized algorithm based on the nonlinear (D) over bar transformation of Sidi, for a precise and fast numerical evaluation of molecular integrals over Slater type functions and over b functions. Numerical results obtained for the three-center two-electron Coulomb and hybrid integrals over b functions and over Slater type functions. Comparisons with numerical results obtained using alternatives approaches and an existing code are listed.

关键词： nonlinear transformations extrapolation methods numerical integration molecular integrals 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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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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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

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