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Ernst Joachim Weniger

Publications and source records attributed to Ernst Joachim Weniger.

At least 19 recordsLinked to original sources
arXiv2019

Comment on "Fourier transform of hydrogen-type atomic orbitals'', Can. J. Phys. Vol.\ 96, 724 - 726 (2018) by N. Yükçü and S. A. Yükçü

Podolsky and Pauling (Phys. Rev. \textbf{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.

math-ph↗
arXiv2019

Construction of New Generalizations of Wynn's Epsilon and Rho Algorithm by Solving Finite Difference Equations in the Transformation Order

We construct new sequence transformations based on Wynn's epsilon and rho algorithms. The recursions of the new algorithms include the recursions of Wynn's epsilon and rho algorithm and of Osada's generalized rho algorithm as special cases. We demonstrate the performance of our algorithms numerically by applying them to some linearly and logarithmically convergent sequences as well as some divergent series.

math.NA↗
arXiv2014

Convergence Analysis of the Summation of the Euler Series by Padé Approximants and the Delta Transformation

Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far from satisfactory. The Euler series $\mathcal{E}(z) \sim \sum_{n=0}^{\infty} (-1)^n n! z^n$ is a very important model for the ubiquitous factorially divergent perturbation expansions in physics. In this article, we analyze the summation of the Euler series by Padé approximants and the delta transformation [E. J. Weniger, Comput. Phys. Rep. Vol.10, 189 (1989), Eq. (8.4-4)] which is a powerful nonlinear Levin-type transformation that works very well in the case of strictly alternating convergent or divergent series. Our analysis is based on a new factorial series representation of the truncation error of the Euler series [R. Borghi, Appl. Num. Math. Vol.60, 1242 (2010)]. We derive explicit expressions for the transformation errors of Padé approximants and of the delta transformation. A subsequent asymptotic analysis proves \emph{rigorously} the convergence of both Padé and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Padé. This is in agreement with previous numerical results.

math-ph↗
arXiv2011

On the Mathematical Nature of Guseinov's Rearranged One-Range Addition Theorems for Slater-Type Functions

Starting from one-range addition theorems for Slater-type functions, which are expansion in terms of complete and orthonormal functions based on the generalized Laguerre polynomials, Guseinov constructed addition theorems that are expansions in terms of Slater-type functions with a common scaling parameter and integral principal quantum numbers. This was accomplished by expressing the complete and orthonormal Laguerre-type functions as finite linear combinations of Slater-type functions and by rearranging the order of the nested summations. Essentially, this corresponds to the transformation of a Laguerre expansion, which in general only converges in the mean, to a power series, which converges pointwise. Such a transformation is not necessarily legitimate, and this contribution discusses in detail the difference between truncated expansions and the infinite series that result in the absence of truncation

math-ph↗
arXiv2010

Summation of Divergent Power Series by Means of Factorial Series

Factorial series played a major role in Stirling's classic book "Methodus Differentialis" (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are useful numerical tools for the summation of divergent (inverse) power series. This is documented by summing the divergent asymptotic expansion for the exponential integral $E_{1} (z)$ and the factorially divergent Rayleigh-Schrödinger perturbation expansion for the quartic anharmonic oscillator. Stirling numbers play a key role since they occur as coefficients in expansions of an inverse power in terms of inverse Pochhammer symbols and vice versa. It is shown that the relationships involving Stirling numbers are special cases of more general orthogonal and triangular transformations.

math.NA↗
arXiv2008

The strange history of B functions or how theoretical chemists and mathematicians do (not) interact

$B$ functions are a class of relatively complicated exponentially decaying basis functions. Since the molecular multicenter integrals of the much simpler Slater-type functions are notoriously difficult, it is not at all obvious why $B$ functions should offer any advantages. However, $B$ functions have Fourier transforms of exceptional simplicity, which greatly simplifies many of their molecular multicenter integrals. This article discusses the historical development of $B$ functions from the perspective of the interaction between mathematics and theoretical chemistry, which traditionally has not been very good. Nevertheless, future progress in theoretical chemistry depends very much on a fertile interaction with neighboring disciplines.

physics.chem-ph↗
arXiv2008

On the Analyticity of Laguerre Series

The transformation of a Laguerre series $f (z) = \sum_{n=0}^{\infty} λ_{n}^{(α)} L_{n}^{(α)} (z)$ to a power series $f (z) = \sum_{n=0}^{\infty} γ_{n} z^{n}$ is discussed. Many nonanalytic functions can be expanded in this way. Thus, success is not guaranteed. Simple sufficient conditions based on the decay rates and sign patters of the $λ_{n}^{(α)}$ as $n \to \infty$ can be formulated which guarantee that $f (z$ is analytic at $z=0$. Meaningful result are obtained if the $λ_{n}^{(α)}$ either decay exponentially or factorially as $n \to \infty$. The situation is much more complicated if the $λ_{n}^{(α)}$ decay algebraically as $n \to \infty$. If the $λ_{n}^{(α)}$ ultimately have the same sign, the series expansions for the power series coefficients diverge, and the corresponding function is not analytic at $z=0. If the $λ_{n}^{(α)}$ ultimately have strictly alternating signs, the series expansions for the power series coefficients are summable and the power series represents an analytic function. In the case of simple $λ_{n}^{(α)}$, the summation of the divergent series for the power series coefficients can often be accomplished with the help of analytic continuation formulas for hypergeometric series, but if the $λ_{n}^{(α)}$ are more complicated, numerical techniques have to be employed. Certain nonlinear sequence transformations -- in particular the so-called delta transformation [E. J. Weniger, Comput. Phys. Rep. \textbf{10}, 189 -- 371 (1989), Eq. (8.4-4)] -- sum the divergent series occurring in this context effectively.

math.CA↗
arXiv2007

Extended Comment on "One-Range Addition Theorems for Coulomb Interaction Potential and Its Derivatives" by I. I. Guseinov (Chem. Phys. Vol. 309 (2005), pp. 209 - 213)

Addition theorems are principal tools that express a function $f (\bm{r} \pm \bm{r}')$ in terms of products of other functions that only depend on either $\bm{r}$ or $\bm{r}'$. The best known example of such an addition theorem is the Laplace expansion of the Coulomb potential which possesses a characteristic two-range form. Guseinov [Chem. Phys. {\bf 309}, 209 - 213 (2005)] derived one-range addition theorems for the Coulomb potential via the limit $β\to 0$ in previously derived one-range addition theorems for the Yukawa potential $\exp \bigl(-β| \bm{r}-\bm{r}'| \bigr) /| \bm{r}-\bm{r}'|$. At first sight, this looks like a remarkable achievement, but from a mathematical point of view, Guseinov's work is at best questionable and in some cases fundamentally flawed. One-range addition theorems are expansions in terms of functions that are complete and orthonormal in a given Hilbert space, but Guseinov replaced the complete and orthonormal functions by nonorthogonal Slater-type functions and rearranged the resulting expansions. This is a dangerous operation whose validity must be checked. It is shown that the one-center limit $\bm{r}' = \bm{0}$ of Guseinov's rearranged Yukawa addition theorems as well as of several other addition theorems does not exist. Moreover, the Coulomb potential does not belong to any of the Hilbert spaces implicitly used by Guseinov. Accordingly, one-range addition theorems for the Coulomb potential diverge in the mean. Instead, these one-range addition theorems have to interpreted as expansions of generalized functions in the sense of Schwartz that converge weakly in suitable functionals.

math-ph↗
arXiv2007

Reply to `Extended Rejoinder to "Extended Comment on "One-Range Addition Theorems for Coulomb Interaction Potential and Its Derivatives" by I. I. Guseinov (Chem. Phys. Vol. 309 (2005), pp. 209 - 211)", arXiv:0706.0975v2"

In the years from 2001 to 2006, Guseinov and his coworkers published 40 articles on the derivation and application of one-range addition theorems. In E. J. Weniger, Extended Comment on ``One-Range Addition Theorems for Coulomb Interaction Potential and Its Derivatives'' by I. I. Guseinov (Chem. Phys. Vol. 309 (2005), pp. 209 - 213), arXiv:0704.1088v2 [math-ph], it was argued that Guseinov's treatment of one-range addition theorems is at best questionable and in some cases fundamentally flawed. In I. I. Guseinov, Extended Rejoinder to "Extended Comment on "One-Range Addition Theorems for Coulomb Interaction Potential and Its Derivatives'' by I. I. Guseinov (Chem. Phys. and Vol. 309 (2005)'', pp. 209-213), arXiv:0706.0975v2 [physics.chem-ph], these claims were disputed. To clarify the situation, the most serious mathematical flaws in Guseinov's treatment of one-range addition theorems are discussed in more depth.

math-ph↗
arXiv2005

Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions

Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{ν=0}^{\infty} a_ν$ of infinite series $\sum_{ν=0}^{\infty} a_ν$ for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation $Δr_n = a_{n+1}$. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered -- the Gaussian hypergeometric series ${}_2 F_1 (a, b; c; z)$ and the divergent asymptotic inverse power series for the exponential integral $E_1 (z)$ -- the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples.

math.CA↗
arXiv2005

The Spherical Tensor Gradient Operator

The spherical tensor gradient operator ${\mathcal{Y}}_{\ell}^{m} (\nabla)$, which is obtained by replacing the Cartesian components of $\bm{r}$ by the Cartesian components of $\nabla$ in the regular solid harmonic ${\mathcal{Y}}_{\ell}^{m} (\bm{r})$, is an irreducible spherical tensor of rank $\ell$. Accordingly, its application to a scalar function produces an irreducible spherical tensor of rank $\ell$. Thus, it is in principle sufficient to consider only multicenter integrals of scalar functions: Higher angular momentum states can be generated by differentiation with respect to the nuclear coordinates. Many of the properties of ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ can be understood easily with the help of an old theorem on differentiation by Hobson [Proc. London Math. Soc. {\bf 24}, 54 - 67 (1892)]. It follows from Hobson's theorem that some scalar functions of considerable relevance as for example the Coulomb potential, Gaussian functions, or reduced Bessel functions produce particularly compact results if ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ is applied to them. Fourier transformation is very helpful to understand the properties of ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ since it produces ${\mathcal{Y}}_{\ell}^{m} (-\mathrm{i} \bm{p})$. It is also possible to apply ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ to generalized functions, yielding for instance the spherical delta function $δ_{\ell}^{m} (\bm{r})$. The differential operator ${\mathcal{Y}}_{\ell}^{m} (\nabla)$ can also be used for the derivation of pointwise convergent addition theorems. The feasibility of this approach is demonstrated by deriving the addition theorem of $r^ν {\mathcal{Y}_{\ell}^{m}} (\bm{r})$ with $ν\in \mathbb{R}$.

math-ph↗
arXiv2003

Mathematical Properties of a New Levin-Type Sequence Transformation Introduced by Č\'ıžek, Zamastil, and Skála. I. Algebraic Theory

Č\'ıžek, Zamastil, and Skála [J. Math. Phys. \textbf{44}, 962 - 968 (2003)] introduced in connection with the summation of the divergent perturbation expansion of the hydrogen atom in an external magnetic field a new sequence transformation which uses as input data not only the elements of a sequence $\{s_n \}_{n=0}^{\infty}$ of partial sums, but also explicit estimates $\{ω_n \}_{n=0}^{\infty}$ for the truncation errors. The explicit incorporation of the information contained in the truncation error estimates makes this and related transformations potentially much more powerful than for instance Padé approximants. Special cases of the new transformation are sequence transformations introduced by Levin [Int. J. Comput. Math. B \textbf{3}, 371 - 388 (1973)] and Weniger [Comput. Phys. Rep. \textbf{10}, 189 - 371 (1989), Sections 7 -9; Numer. Algor. \textbf{3}, 477 - 486 (1992)] and also a variant of Richardson extrapolation [Phil. Trans. Roy. Soc. London A \textbf{226}, 299 - 349 (1927)]. The algebraic theory of these transformations - explicit expressions, recurrence formulas, explicit expressions in the case of special remainder estimates, and asymptotic order estimates satisfied by rational approximants to power series - is formulated in terms of hitherto unknown mathematical properties of the new transformation introduced by Č\'ıžek, Zamastil, and Skála. This leads to a considerable formal simplification and unification.

math-ph↗
arXiv2003

A Rational Approximant for the Digamma Function

Power series representations for special functions are computationally satisfactory only in the vicinity of the expansion point. Thus, it is an obvious idea to use instead Padé approximants or other rational functions constructed from sequence transformations. However, neither Padé approximants nor sequence transformation utilize the information which is avaliable in the case of a special function -- all power series coefficients as well as the truncation errors are explicitly known -- in an optimal way. Thus, alternative rational approximants, which can profit from additional information of that kind, would be desirable. It is shown that in this way a rational approximant for the digamma function can be constructed which possesses a transformation error given by an explicitly known series expansion.

math.NA↗
arXiv2003

Verallgemeinerte Summationsprozesse als numerische Hilfsmittel für quantenmechanische und quantenchemische Rechnungen

Slowly convergent or divergent sequences and series occur abundantly in quantum physics and quantum chemistry. These convergence problems can be overcome with the help of nonlinear sequence transformations (Wynn's epsilon or rho algorithm, Brezinski's theta algorithm, Levin's transformation, etc.) as for instance described in E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series", Comput. Phys. Rep. Vol. 10, 189 - 371 (1989). A detailed description of the mathematical properties of these transformations is given. The nonlinear sequence transformations are applied for the evaluation of special or auxiliary functions by summing divergent asymptotic expansions via rational approximants, for the evaluation of complicated molecular multicenter integrals, for the summation of the divergent perturbation expansions of the anharmonic oscillators, or for the extrapolation of quantum chemical cluster calculations on stereoregular quasi-one-dimensional polymers to the infinite chain limit.

math-ph↗
arXiv2003

Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

Slowly convergent series and sequences as well as divergent series occur quite frequently in the mathematical treatment of scientific problems. In this report, a large number of mainly nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series are discussed. Some of the sequence transformations of this report as for instance Wynn's $ε$ algorithm or Levin's sequence transformation are well established in the literature on convergence acceleration, but the majority of them is new. Efficient algorithms for the evaluation of these transformations are derived. The theoretical properties of the sequence transformations in convergence acceleration and summation processes are analyzed. Finally, the performance of the sequence transformations of this report are tested by applying them to certain slowly convergent and divergent series, which are hopefully realistic models for a large part of the slowly convergent or divergent series that can occur in scientific problems and in applied mathematics.

math.NA↗
arXiv2001

QED Effective Action Revisited

The derivation of a convergent series representation for the quantum electrodynamic effective action obtained by two of us (S.R.V. and D.R.L.) in [Can. J. Phys. vol. 71, p. 389 (1993)] is reexamined. We present more details of our original derivation. Moreover, we discuss the relation of the electric-magnetic duality to the integral representation for the effective action, and we consider the application of nonlinear convergence acceleration techniques which permit the efficient and reliable numerical evaluation of the quantum correction to the Maxwell Lagrangian.

hep-th↗
arXiv2001

Addition Theorems as Three-Dimensional Taylor Expansions. II. $B$ Functions and Other Exponentially Decaying Functions

Addition theorems can be constructed by doing three-dimensional Taylor expansions according to $f (\mathbf{r} + \mathbf{r}') = \exp (\mathbf{r}' \cdot \mathbf{\nabla}) f (\mathbf{r})$. Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form $\exp (x' \partial /\partial x) \exp (y' \partial /\partial y) \exp (z' \partial /\partial z)$ would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator $\exp (\mathbf{r}' \cdot \mathbf{\nabla})$ involving powers of the Laplacian $\mathbf{\nabla}^2$ and spherical tensor gradient operators $\mathcal{Y}_{\ell}^{m} (\nabla)$, which are irreducible spherical tensors of ranks zero and $\ell$, respectively [F.D.\ Santos, Nucl. Phys. A {\bf 212}, 341 (1973)]. In this way, it is indeed possible to derive addition theorems by doing three-dimensional Taylor expansions [E.J. Weniger, Int. J. Quantum Chem. {\bf 76}, 280 (2000)]. The application of the translation operator in its spherical form is particularly simple in the case of $B$ functions and leads to an addition theorem with a comparatively compact structure. Since other exponentially decaying functions like Slater-type functions, bound-state hydrogenic eigenfunctions, and other functions based on generalized Laguerre polynomials can be expressed by simple finite sums of $B$ functions, the addition theorems for these functions can be written down immediately.

math-ph↗