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  • 1
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The mathematical foundation of the methods using addition theorems to evaluate multicenter integrals over Slater-type orbitals is actually well understood. However, many numerical aspects of such approaches still require further investigations. In the framework of these methods, multicenter integrals are generally represented by infinite series which under certain circumstances are very slowly convergent. Accordingly, the determination of the convergence type of such series is of great importance since it allows one to choose adequately the convergence accelerator to be used in the summation procedure. In this work, the convergence of the two-range addition theorem proposed by Barnett and Coulson [Philos. Trans. R. Soc. London, Ser. A 243, 221 (1951)] is analyzed. The results obtained from this study are then applied to study the convergence of three-center nuclear integrals, and most importantly, to discuss the choice of the convergence accelerator to be used in the summation procedure. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Journal of Computational Chemistry 12 (1991), S. 1217-1222 
    ISSN: 0192-8651
    Keywords: Computational Chemistry and Molecular Modeling ; Biochemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology , Computer Science
    Notes: Using the Löwdin alpha-function method in which displaced orbitals are expanded in spherical harmonics, two-center, two-electron repulsion integrals of the Coulomb, hybrid, and exchange type are done analytically using Slater-type orbitals. Computer algebra and integer arithmetic are used to obtain analytic results and avoid cancellation errors by the generation of rational matrix elements for C, E, and F matrices that are used to express the α-function. The formula for the exchange integral is kept simple by reversing the order of integration over each part of a split quadrant. Only two basic integrals are used that are first efficiently evaluated by using look-up tables and then used repeatedly.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Exact formulas for 147 overlap integrals between Slater-type orbitals with equal screening constants are presented in the most simplified form. This represents all combinations of orbitals with quantum numbers: 1 ≤ N ≤ 5, 0 ≤ L ≤ 3, and M ≤ L. The formulas are automatically generated by computer using the “C-matrix” single-center expansion method. There are no limitations to the applicability of this method to orbitals of higher quantum numbers.
    Additional Material: 1 Tab.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    New York, NY : Wiley-Blackwell
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: A triple infinite sum of formulas is generated by a computer algebra implementation of the Löwdin α function method (supplemented by a C matrix with integer elements that characterize an orbital) for a fourcenter two-electron repulsion integral over 1s Slater-type orbitals, with screening constants of 1. In this example, one orbital is placed at the origin and the others at equal distances along the x, y, and z axes. Surprisingly few terms are needed for five figure accuracy. There should be no problem in generalizing these methods to all cases of four-center integrals.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    New York, NY : Wiley-Blackwell
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: A commercial computer algebra program, Mathematica, is used to generate the C matrix that characterizes our implementation of the Löwdin α-function method as applied to Slater-type orbitals. An example of a two-center overlap integral is done to show how the arbitrary precision capability of Mathematica can overcome severe cancellation errors encountered with programming in FORTRAN. This strategy is capable of being generalized to other multicenter molecular integrals. Mathematica programs are included.
    Additional Material: 3 Tab.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    New York, NY : Wiley-Blackwell
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: The Löwdin α-function method, in which displaced orbitals are expanded in an infinite series of spherical harmonics, is implemented for Slater-type orbitals using a commercial computer algebra program, Mathematica, The program, which is included, generates a C matrix with integer elements that characterizes our approach to multicenter molecular integrals. The general two-center, two-electron Coulomb repulsion integral is produced analytically with a finite number of terms. Each Coulomb formula may be evaluated to arbitrary precision, since Mathematica works with integer arithmetic. Hence, cancellation errors can be overcome. © 1993 John Wiley & Sons, Inc.
    Additional Material: 2 Tab.
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  • 7
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: The strategy of using expansions in spherical harmonics of displaced Slater-type orbitals for the evaluation of multicenter molecular integrals is reviewed and projected. Our Löwdin α-function method is augmented by computer algebra and C, E, and F matrices with rational elements. This permits exact evaluations of each term in an infinite sum using a commercial computer algebra program to investigate the three-center nuclear attraction integral. Each term (harmonic potential) is important for proposed use in numerical evaluations of multicenter integrals. © 1994 John Wiley & Sons, Inc.
    Additional Material: 1 Tab.
    Type of Medium: Electronic Resource
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  • 8
    ISSN: 0020-7608
    Keywords: Chemistry ; Theoretical, Physical and Computational Chemistry
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: In this work, a general method of calculating multicenter integrals over Slater-type functions is presented and numerical calculations on diatomic molecules, H2 and N2, are carried out. Our results are then compared with those obtained with Gaussian-type functions. The method, which is essentially an addition theorem for Slater-type functions, is outlined and some of its mathematical and numerical properties are presented. As regards our numerical results, although such very small diatomics are of limited interest in chemistry, they are to be considered as the necessary step that will allow us to tune up the computational machinery. Furthermore, the conclusions drawn throughout this work will be used for more general algorithms which will allow us to treat efficiently more complicated systems, namely, three, four, and many more atomic systems.   © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 89-93, 1998
    Additional Material: 2 Tab.
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  • 9
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Exact computer-generated formulas can be produced for each term of an infinite series that gives the value of three-center Coulomb integrals over Slater-type orbitals of the s-type with equal screening constants. As a specific example, the Coulomb energy of an equilateral triangular arrangement of all 1s orbitals is calculated using seven terms of an infinite expansion to obtain an answer comparable to earlier work in elliptic coordinates. Generalization to all three-center cases of this implementation of the Löwdin α-function method is straightforward.
    Type of Medium: Electronic Resource
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  • 10
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: By use of computer algebra and expansions of orbitals about a single center in spherical harmonics with a matrix representation for the α-function coefficients it is possible to produce formulas for each term of an infinite series that represents the value of a three-center exchange integral. Only seven terms are required to get six decimal digits for triatomic hydrogen. This method is readily generalized to solve all three-center exchange integrals over Slater-type orbitals.
    Type of Medium: Electronic Resource
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