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• 1
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
Description / Table of Contents: L'état fondamental des systèmes atomiques à deux électrons est décrit par des fonctions variationelles “échellonnées” de type \documentclass{article}\pagestyle{empty}\begin{document}$$\phi _{\alpha k} = e^{ - (Z - 1 + e^{ - \alpha kr})kr}$$\end{document} et \documentclass{article}\pagestyle{empty}\begin{document}$$\phi _{\beta \alpha k} = e^{ - (\beta + e^{ - \alpha kr})kr}$$\end{document} Le “scale factor” k a été introduit pour satisfaire au théorème du viriel. Sauf l'énergie totale, on a calculé les valeurs moyennes de plusieurs opérateurs à un électron. Ces résultats ont été comparés aux ceux obtenus avec d'autres fonctions variationelles.La fonction φαk déjà représente une bonne approximation de la solution Hartree-Fock, tandis qu'avec la fonction φβαk les résultats Hartree-Fock sont pratiquement reproduits.
Abstract: Variationsfunktionen mit einem Koordinatenstreckungsfaktor k, von der Form \documentclass{article}\pagestyle{empty}\begin{document}$$\phi _{\alpha k} = e^{ - (Z - 1 + e^{ - \alpha kr})kr}$$\end{document} und \documentclass{article}\pagestyle{empty}\begin{document}$$\phi _{\beta \alpha k} = e^{ - (\beta + e^{ - \alpha kr})kr}$$\end{document} werden auf den Grundzustand atomarer Zweielektronensysteme angewendet. Der Faktor k wird eingeführt um dem Virialsatz zu befriedigen.Ausser der Energie werden Mittelwerte von mehreren Einelektronoperatoren berechnet und mit entsprechenden Resultaten für andere Variationsfunktionen verglichen.Die Funktion φαk schon representiert eine gute Annäherung zu der Hartree-Fock-Lösung, während mit der Funktion φβαk die Hartree-Fock-Resultate praktisch reproduziert werden.
Notes: Scaled variational functions of the form \documentclass{article}\pagestyle{empty}\begin{document}$\phi _{\alpha k} = e^{ - (Z - 1 + e^{ - \alpha kr})kr}$\end{document}, where Z is the nuclear charge and α is a parameter, and \documentclass{article}\pagestyle{empty}\begin{document}$\phi _{\beta \alpha k} = e^{ - (\beta + e^{ - \alpha kr})kr}$\end{document}, where both α and β are parameters, are used for the description of the ground state of atomic two-electron systems in the independent particle model. The scale factor k has been introduced in order to satisfy the virial theorem.Apart from the energy, a number of one-electron expectation values have been calculated and the results are compared with those obtained from other variational functions.The function φαk yields already a good approximation to the Hartree-Fock solution whereas with the function φβαk the Hartree-Fock results are practically reproduced.
Type of Medium: Electronic Resource
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• 2
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 Gaussian lobe representation of 2p and 3dxy functions of a one-electron atom has been reconsidered. Contrary to earlier results, it has been found that the minimum energy associated with the simplest case of these functions is reached when the expansion centers coincide with the nucleus. This also seems to be true for representations with larger expansion lengths. Under similar conditions, Whitten's octahedral representation of the 2p function gives a better energy value than Poshusta's tetrahedral one.
Type of Medium: Electronic Resource
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• 3
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: It is advocated to carry out an optimization procedure, which is based upon the variational method, in such a way that the optimum values of the variational parameters are expressed as functions of physical constants, such as the atomic number, Z. The three stages involved in this treatment are illustrated by the optimization of nine correlated wave functions, which describe the ground states of atomic two-electron systems. An analysis of the Z-expansions of the total energies associated with these functions leads to the concept of a class of variational functions. The performances of functions belonging to the same class differ only marginally, especially at larger values of Z. Consequently, the concept of class may be used to bring some order in the plethora of variational functions. © 1994 John Wiley & Sons, Inc.
Type of Medium: Electronic Resource
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• 4
Electronic Resource
New York, NY : Wiley-Blackwell
International Journal of Quantum Chemistry 17 (1980), S. 1099-1109
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
Description / Table of Contents: Des développments gaussiens complètement optimisés des fonctions hydrogénoïdes de type 1s, 2pz, 3dxy, et 4fxyz ont été pratiquement reproduits par une construction spéciale des paramètres variationnels. L'utilité des fonctions génératrices utilisées est discutée brièvement.
Abstract: Vollständig optimisierte Gaussentwicklungen wasserstoffähnlicher Funktionen von Typ 1s, 2pz, 3dxy, und 4fxyz sind mittels eines speziellen Verfahrens für die bestimmung der Variationsparameter praktisch reproduziert worden. Der Nutzen der angewandten erzeugenden Funktionen wird kurz diskutiert.
Notes: Fully optimized lengthy Gaussian expansions of hydrogenic 1s and 2pz functions have been practically reproduced by generating the variational parameters by means of suitably chosen functions involving only few parameters. Hydrogenic 3dxy and 4fxyz functions have been approximated in a similar way. The usefulness of such generating functions is briefly discussed.
Type of Medium: Electronic Resource
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• 5
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: For the Hartree-Fock ground state of atomic two-electron systems, the variational function of Wilson and Silverstone, φ(r) = (a + kr)-1 exp(-kr) / (4π)1/2, can be optimized in two complementary ways. For small values of the atomic number Z, all intergrals have been calculated numerically and optimization can be performed accurately. However, as Z increases, loss of significant figures is increasingly detrimental to the optimization process. For sufficiently large values of Z, the integrals may be replaced by asymptotic expansions in terms of (2a)-. As a result of optimization, the parameters and expectation values can be given as expansions in terms of (32Z)-1/2. Both methods yield good results for Z ≈ 25, so that the whole range of Z can be treated accurately. The results have been compared with those derived from other analytical two-parameter functions. It is found that φ(r) is indeed the outstanding two-parameter function, at least for small and intermediate values of Z.