Simplified derivation of a one‐range addition theorem of the Yukawa potential

Homeier, Herbert H. H.; Weniger, Ernst Joachim; Steinborn, E. Otto

doi:10.1002/qua.560440308

Cited by 22 publications

(19 citation statements)

References 18 publications

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“…[113][114][115][116][117][118][119][120][121][122][123]] , yet we have not found these alternative formulations to be particularly advantageous in the present case compared to the standard Barnett-Coulson scheme, Eq. (8).…”

Section: Theorymentioning

confidence: 81%

Combining Slater-type orbitals and effective core potentials

2017

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We present a general methodology to evaluate matrix elements of the effective core potentials (ECPs) within one-electron basis set of Slater-type orbitals (STOs). The scheme is based on translation of individual STO distributions in the framework of Barnett-Coulson method. We discuss different types of integrals which naturally appear and reduce them to few basic quantities which can be calculated recursively or purely numerically. Additionally, we consider evaluation of the STOs matrix elements involving the core polarisation potentials (CPP) and effective spin-orbit potentials. Construction of the STOs basis sets designed specifically for use with ECPs is discussed and differences in comparison with all-electron basis sets are briefly summarised. We verify the validity of the present approach by calculating excitation energies, static dipole polarisabilities and valence orbital energies for the alkaline earth metals (Ca, Sr, Ba). Finally, we evaluate interaction energies, permanent dipole moments and ionisation energies for barium and strontium hydrides, and compare them with the best available experimental and theoretical data.

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Section: Theorymentioning

confidence: 81%

Combining Slater-type orbitals and effective core potentials

2017

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“…Sturmians also play a major role in books by Avery [6,7], Avery and Avery [8], and Avery et al [9]. We used Sturmians for the construction for an addition theorem of the Yukawa potential [45] with the help of weakly convergent orthogonal and biorthogonal expansions for the plane wave introduced in ref. 5, Section III.…”

Section: Incompleteness Of the Bound-state Hydrogen Eigenfunctionsmentioning

confidence: 99%

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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“…As discussed in more details in [120,122], addition theorems have a two-range form if they are pointwise convergent three-dimensional Taylor expansions and if the function f (r ± r ′ ), which is to be expanded, is not analytic at the origin. The undeniably troublesome two-range form of an addition theorem can be avoided if f : R 3 → C belongs to the Hilbert space L 2 (R 3 ) of square integrable functions or to other, closely related function spaces as for example Sobolev spaces that are proper subspaces of L 2 (R 3 ) (compare for instance [31,46,118,132] and references therein). For the sake of simplicity, let us assume that a discrete function set {Ψ m n,ℓ (r)} n,ℓ,m is complete and orthonormal in the Hilbert space L 2 (R 3 ).…”

Section: Addition Theoremsmentioning

confidence: 99%

The Spherical Tensor Gradient Operator

Weniger

2005

Collect. Czech. Chem. Commun.

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Simplified derivation of a one‐range addition theorem of the Yukawa potential

Cited by 22 publications

References 18 publications

Combining Slater-type orbitals and effective core potentials

Combining Slater-type orbitals and effective core potentials

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

The Spherical Tensor Gradient Operator

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