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  • 1
    Electronic Resource
    Electronic Resource
    Use of nonlinear convergence accelerators for the efficient evaluation of GTO molecular integrals (1986)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 84 (1986), S. 5617-5623 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Different methods for accelerating convergence of known series expansions for the auxiliary functions Fm(t) encountered in molecular integrals over Gaussian-type orbitals have been tested and compared. We have investigated the numerical properties of the Shanks transformation and the Levin transformations. It turns out that Levin's t transformation is the best accelerator, whereas the Shanks transformation, which nearly yields the same results as the continued-fraction method, is less effective.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.449922
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  • 2
    Electronic Resource
    Electronic Resource
    Addition theorems for B functions and other exponentially declining functions (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 774-784 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this paper addition theorems are derived for a special class of exponentially declining functions, the so-called B functions Bmn,l (α,r) [E. Filter and E. O. Steinborn, Phys. Rev. A 18, 1 (1978)]. Although these B functions have a relatively complicated analytical structure they nevertheless have some mathematical properties that are particularly advantageous in connection with multicenter problems. Also, all the commonly occurring exponentially declining functions like, for instance, bound-state hydrogen eigenfunctions and Slater-type functions, can be expressed by simple finite sums of B functions. Consequently, addition theorems for these functions can also be written down immediately. The various addition theorems for B functions are derived by applying suitable generating differential operators to the well-known addition theorem of the special B function Bm−l,l, which is that solution of the modified Helmholtz equation that is irregular at the origin and regular at infinity [E. J. Weniger and E. O. Steinborn, J. Math. Phys. 26, 664 (1985)]. All differentiations, which have to be done in this approach, can be expressed in closed form leading to comparatively compact addition theorems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528396
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  • 3
    Electronic Resource
    Electronic Resource
    A simple derivation of the addition theorems of the irregular solid harmonics, the Helmholtz harmonics, and the modified Helmholtz harmonics (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 664-670 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this article a simple derivation of the addition theorems of the irregular solid harmonics, the Helmholtz harmonics, and the modified Helmholtz harmonics is presented. Our derivation is based upon properties of the differential operator Yml(∇), which is obtained from the regular solid harmonic Yml(r) by replacing the Cartesian components of r by the Cartesian components of ∇. With the help of this differential operator Yml(∇), which is an irreducible spherical tensor of rank l, the addition theorems of the anisotropic functions are obtained by differentiating the addition theorems of the isotropic functions. The performance of the necessary differentiations is greatly facilitated by a systematic exploitation of the tensorial nature of the differential operator Yml(∇).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.526604
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  • 4
    Electronic Resource
    Electronic Resource
    Comment on "Molecular overlap integrals with exponential-type orbitals'' (1987)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 87 (1987), S. 3709-3711 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A recent article by A. K. Bhattacharya and S. C. Dhabal [J. Chem. Phys. 84, 1598 (1986)], dealing with the fast and accurate evaluation of molecular overlap integrals with exponentially declining atomic orbitals, is analyzed critically. It is shown that some of their analytical results can be improved, and that some others are already known in the literature. Concerning their numerical investigations, it is shown that Gauss–Jacobi and not Gauss–Legendre quadrature should be used for the evaluation of the one-dimensional integral representation for overlap integrals with different scaling parameters.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.452975
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  • 5
    Electronic Resource
    Electronic Resource
    Analytische Lösung des Molekülintegrals vom Drei-Zentren-Ein-Elektron-Typ über Slater-Atomorbitale und eine reguläre räumliche Kugelfunktion (1974)
    Springer
    Theoretical chemistry accounts 34 (1974), S. 251-262 
    Details
    ISSN: 1432-2234
    Keywords: Molecular integrals ; Slater-type atomic orbitals ; Solid spherical harmonics ; Elliptic coordinates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Abstract In calculations of molecular properties matrix elements of solid spherical harmonics are often needed. For the molecular three-center one-electron integral over two Slater-type orbitals of different atoms and a regular solid spherical harmonic, which is defined with respect to a third center, a completely analytical solution is given that contains only finite sums. Due to the strict observation of the concept of separate computation of the charge density distribution, and since elliptic coordinates are used, expressions can be utilized which occur in calculations of diatomic molecules. This is advantageous for practical applications.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00553751
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  • 6
    Electronic Resource
    Electronic Resource
    Symmetrie und analytische Struktur der Additionstheoreme räumlicher Funktionen und der Mehrzentren-Molekülintegrale über beliebige Atomfunktionen (1979)
    Springer
    Theoretical chemistry accounts 52 (1979), S. 189-208 
    Details
    ISSN: 1432-2234
    Keywords: Addition theorems of spatial functions ; One-center expansions in terms of spherical harmonics ; Slater-type orbitals ; Multi-center integrals, series expansions of ∼
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Abstract Three-dimensional functions f(r) = g(r) · Y m l (θ, ø), which transform like an irreducible tensor, are transformed simultaneously under rotations and translations. The relationships governing the transformation reveal some general properties. If the addition theorem of a function f(r) can be represented by a one-center expansion in terms of surface spherical harmonics Y m l , each expansion coefficient is given by a Clebsch-Gordan coefficient and a radial function. Because of these properties, addition theorems are especially helpful for the simplification and evaluation of quantum-mechanical matrix elements and multi-center energy integrals in molecular LCAO calculations. The application of addition theorems has two major advantages: First, because addition theorems are equivalent to translation formulas, the number of centers of an integral can be reduced by translation of orbitals and operators. Second, due to the typical analytical structure of the series expansion representing the addition theorem, the dimensionality of a molecular integral can be reduced, because the integration over the angular variables can be executed. Then, a molecular multi-center integral is represented by a series of one-center integrals over functions of the radial variable only.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00547678
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  • 7
    Electronic Resource
    Electronic Resource
    Nuclear attraction and electron interaction integrals of exponentially decaying functions and the Poisson equation (1992)
    Springer
    Theoretical chemistry accounts 83 (1992), S. 105-121 
    Details
    ISSN: 1432-2234
    Keywords: Nuclear attraction ; Electron interaction integrals ; Exponentially decaying functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Summary The technique proposed by O-Ohata and Ruedenberg (J Math Phys 7:547 (1966)) and by Silver and Ruedenberg (J Chem Phys 49:4306 (1968)) of computing nuclear attraction and electron interaction integrals by solving an inhomogeneous Laplace equation can also be applied ifB functions (Filter E, Steinborn EO (1978) Phys Rev A 18:1) are used as basis functions in atomic and molecular calculations. It is shown that because of the remarkable mathematical properties ofB functions the derivation of compact explicit expressions for the multicenter integrals mentioned above is particularly simple. These results are also of interest in the context of other exponentially decaying functions, since all the other commonly occurring exponentially decaying functions as, for instance, Slater functions or bound state hydrogen eigenfunctions can be expressed as simple linear combinations ofB functions. Consequently, their multicenter integrals can also be expressed in terms of multicenter integrals ofB functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01113245
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  • 8
    Electronic Resource
    Electronic Resource
    Analytische Lösung des Molekülintegrals vom Drei-Zentren-Ein-Elektron-Typ über Slater-Atomorbitale und eine reguläre räumliche Kugelfunktion (1974)
    Springer
    Theoretical chemistry accounts 34 (1974), S. 251-262 
    Details
    ISSN: 1432-2234
    Keywords: Molecular integrals ; Slater-type atomic orbitals ; Solid spherical harmonics ; Elliptic coordinates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Abstract In calculations of molecular properties matrix elements of solid spherical harmonics are often needed. For the molecular three-center one-electron integral over two Slater-type orbitals of different atoms and a regular solid spherical harmonic, which is defined with respect to a third center, a completely analytical solution is given that contains only finite sums. Due to the strict observation of the concept of separate computation of the charge density distribution, and since elliptic coordinates are used, expressions can be utilized which occur in calculations of diatomic molecules. This is advantageous for practical applications.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02399013
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  • 9
    Electronic Resource
    Electronic Resource
    Zur Deutung der Ultraviolett-Absorptionsspektren der Alkalihalogenid-Kristalle (1966)
    Springer
    Theoretical chemistry accounts 4 (1966), S. 185-202 
    Details
    ISSN: 1432-2234
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Description / Table of Contents: Abstract A model is developed in which the long wave length ultraviolet absorption bands of alkali halide crystals are assigned to one electron excitations in the anions. These exciton bands are attributed to transitions to the crystal field stabilized splitting terms arising from anion configurations of typep 5d. Consideration of the interionic Coulomb interaction alone leads already to a more satisfactory explanation of the long wave length portion of the spectra than the qualitative predictions of the electron-transfer model ofOverhauser.
    Abstract: Résumé Un modèle est développé où les bandes d'absorption des halogénures alcalins dans l'ultraviolet proche sont attribuées à des excitations monoélectroniques dans les anions. Ces transitions finissent aux termes stabilisés par le champ cristallin et dérivant de la configurationp 5d. de l'anion, dédoublée. N'introduisant que les interactions ioniques de Coulomb, on obtient des résultats qui expliquent plus satisfaisamment la partie de grande longueur d'onde des spectres que ne le puissent les prédictions (qualitatives) du modèle de transfert d'électrons d'Overhauser.
    Notes: Zusammenfassung Es wird ein Modell entwickelt, nach dem die Banden des langwelligen Teils der Ultraviolett-Absorptionsspektren der Alkalihalogenid-Kristalle Einelektronenanregungen in den Anionen zuzuschreiben sind. Diese Exzitonenbanden entstehen danach durch Übergänge in die durch das Kristallfeld stabilisierten Spaltterme aus Anionenkonfigurationenp 5d. Bereits die Berücksichtigung der interionischen Coulomb-Wechselwirkungen führt so auf Resultate, die den langwelligen Teil der Spektren befriedigender erklären, als es die (qualitativen) Aussagen des Elektronen-Transfer-Modells vonOverhauser vermögen.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01007294
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  • 10
    Electronic Resource
    Electronic Resource
    Overlap integrals of B functions (1988)
    Springer
    Theoretical chemistry accounts 73 (1988), S. 323-336 
    Details
    ISSN: 1432-2234
    Keywords: Overlap integrals ; Exponentially declining atomic orbitals ; Numerical quadrature ; Convergence acceleration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Abstract Two different methods for the evaluation of overlap integrals of B functions with different scaling parameters are analyzed critically. The first method consists of an infinite series expansion in terms of overlap integrals with equal scaling parameters [14]. The second method consists of an integral representation for the overlap integral which has to be evaluated numerically. Bhattacharya and Dhabal [13] recommend the use of Gauss-Legendre quadrature for this purpose. However, we show that Gauss-Jacobi quadrature gives better results, in particular for larger quantum number. We also show that the convergence of the infinite series can be improved if suitable convergence accelerators are applied. Since an internal error analysis can be done quite easily in the case of an infinite series even if it is accelerated, whereas it is very costly in the case of Gauss quadratures, the infinite series is probably more efficient than the integral representation. Overlap integrals of all commonly occurring exponentially declining basis functions such as Slater-type functions, can be expressed by finite sums of overlap integrals of B functions, because these basis functions can be represented by linear combinations of B functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00527739
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