On the Evaluation Overlap Integrals with the Same and Different Screening Parameters Over Slater Type Orbitals via the Fourier-Transform Method

Yavuz, Metin; Yükçü, Niyazi; Öztekin, Emin; Yılmaz, Hava; Döndür, S.

doi:10.1088/0253-6102/43/1/029

Cited by 14 publications

(9 citation statements)

References 19 publications

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“…Guseinov's original notation (68) does not allow non-integral values of ␣. To rectify this obvious deficiency, I introduced [23,85] the alternative definition (70), which uses the modern mathematical notation for the generalized Laguerre polynomials. Later, Guseinov [86,87] was forced to change to my notation because he wanted to consider non-integral self-frictional quantum numbers, ␣.…”

Section: Discussionmentioning

confidence: 99%

“…With the help of (60) and (61), it is again a trivial matter to construct the Fourier transform (74) of a Guseinov function in the notation of (70). Equation (74) contains the Fourier transforms of Sturmians and Lambda functions as special cases, but is more complicated.…”

Section: Discussionmentioning

confidence: 99%

“…To disguise the obvious, Guseinov used in these formulas instead of a generalized Laguerre polynomial a terminating confluent hypergeometric series 1 F 1 . Because of (2), Guseinov's formulas are equivalent to my definition (70…”

Section: Expansion In Terms Of Reduced Bessel Functionsmentioning

confidence: 99%

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Comment on “Fourier transform of hydrogen-type atomic orbitals”

Weniger

2019

Can. J. Phys.

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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. Yükçü and Yükçü (Can. ), 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. Résumé: Podolsky et Pauling (Phys. Rev. 34, 109 (1929) doi:10.1103/PhysRev.34.109) ont été les premiers à développer une expression explicite pour la transformée de Fourier de la fonction propre d'un état lié de l'hydrogène. Yükçü et Yükçü (Can. ) ont procédé différemment, apparemment sans connaître le travail de Podolsky et Pauling, ni les nombreuses références antérieures sur la transformée de Fourier. Ils ont exprimé un polynôme généralisé de Laguerre comme une somme finie de puissances ou, de façon équivalente, ils ont exprimé la fonction propre d'un état lié de l'hydrogène comme une somme finie de fonctions de type Slater. Cette approche semble très simple, mais elle mène à des expressions comparativement compliquées qui ne peuvent pas égaler la simplicité du résultat classique de Podolsky et Pauling. Il reste néanmoins possible de reproduire, non seulement la formule de Podolsky et Pauling, mais d'obtenir des résultats de qualité similaires pour les transformées de Fourier d'autres fonctions étroitement reliées, comme les Sturmiens, les fonctions Lambda ou de Guseinov, en faisant l'expansion des polynômes généralisés de Laguerre en termes de fonctions appelées fonctions réduites (reduced) de Bessel. [Traduit par la Rédaction]Mots-clés : fonctions propres d'états liés de l'hydrogène, transformation de Fourier, polynômes généralisés de Laguerre, série hypergéométrique, fonctions de type Slater, fonctions réduites de Bessel.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Comment on “Fourier transform of hydrogen-type atomic orbitals”

Weniger

2019

Can. J. Phys.

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“…No attempt has been made in the present study to analyze the numerical properties of the various atomic orbital parameters for the overlap integrals. These aspects, which are very important for practical application, will be discussed in the following paper by Yavuz et al 25.…”

Section: Summary and Discussionmentioning

confidence: 99%

Overlap integrals with respect to quantum numbers over Slater‐type orbitals via the Fourier‐transform method

Öztekin

2004

Int J of Quantum Chemistry

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ABSTRACT:A unified analytical treatment of overlap integrals over Slater-type orbitals, using the Fourier transform method, is presented. Our approach leads to considerable simplification of the derivation of analytical representations for the overlap integrals over Slater type orbitals. In this approach, the representations of overlap integrals with both the same and different screening parameters have been obtained with the help of the Rayleigh expansion of a plane wave in terms of spherical Bessel functions and spherical harmonics. Analytical expressions obtained by this method have been expressed in terms of Gaunt coefficients, which are the product of spherical harmonics, Gegenbauer, and binomial coefficients. These overlap integrals have also been calculated with respect to the differences in n 1 Ϫ l 1 and n 2 Ϫ l 2 numbers that are even or odd.

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“…Because of Eq. ( 2), Guseinov's formulas are equivalent to my definition (70). Characteristically, Guseinov did not acknowledge my contributions [85,Eq.…”

Section: Expansion In Terms Of Reduced Bessel Functionsmentioning

confidence: 99%

Fourier transform of hydrogen-type atomic orbitals

Yükçü

2018

Can. J. Phys.

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Podolsky and Pauling (Phys. Rev. 34, 109 -116 (1929)) were the first ones to derive an explicit expression for the Fourier transform of a bound-state hydrogen eigenfunction. Yükçü and Yükçü, 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 the Podolsky and Pauling 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.

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On the Evaluation Overlap Integrals with the Same and Different Screening Parameters Over Slater Type Orbitals via the Fourier-Transform Method

Cited by 14 publications

References 19 publications

Comment on “Fourier transform of hydrogen-type atomic orbitals”

Comment on “Fourier transform of hydrogen-type atomic orbitals”

Overlap integrals with respect to quantum numbers over Slater‐type orbitals via the Fourier‐transform method

Fourier transform of hydrogen-type atomic orbitals

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