Rigorous foundations for the angular-averaged (Yakub-Ronchi) Ewald potential derivation: the lattice-sum identity $S(k)=\delta_{1,k}$ and the Euler-Maclaurin-to-Poisson function class in three dimensions

The angular-averaged (Yakub-Ronchi) Ewald potential replaces the orientation-dependent Ewald pair interaction in periodic Coulomb simulations by a simple finite-range spherically symmetric potential, and its 2022 derivation by Demyanov and Levashov rests on two steps that the authors themselves flag as non-rigorous. First, the small-$\delta$ analysis of the underlying cubic lattice sums depends on the finite-sum identity $S(k)=\delta_{1,k}$, which the authors verified symbolically only for $2\le k\le 60$ and left unproven in general. Second, the demonstration that the three-dimensional Euler-Maclaurin summation formula reduces to the Poisson summation formula, the two routes used interchangeably in the derivation, exchanges the limit of a conditionally convergent lattice Green's-function series with an integral without justification, and the class of functions for which this reduction is valid is undetermined. The problem is to settle both components: the finite-sum identity $S(k)=\delta_{1,k}$ for all integers $k\ge 1$, an elementary supporting identity of the same derivation, and the function class for which the Euler-Maclaurin-to-Poisson reduction holds in three dimensions, with counterexamples delimiting sharpness. Both are required; the function class is the open research question.

Background

Simulations of charged particles (plasmas, ionic fluids, polyelectrolytes) impose periodic boundary conditions, so the electrostatic energy requires summing the Coulomb interaction $1/r$ over infinitely many periodic images of the simulation cell, a conditionally convergent sum. Ewald summation makes this tractable by attaching to each charge a smooth Gaussian screening charge whose dimensionless width is controlled by a parameter $\delta$; the interaction splits into a short-ranged real-space part and a smooth reciprocal-space part over the dual lattice of integer wave vectors, both converging rapidly. The resulting pair interaction depends not only on the distance between two particles but also on the orientation of their separation relative to the periodic lattice.

For disordered, nearly isotropic systems such as strongly coupled plasmas, Yakub and Ronchi (2003) proposed replacing the full Ewald pair interaction by its average over the angular orientations of the periodic image lattice. In the limit of an infinitely narrow screening Gaussian ($\delta\to\infty$) this yields a remarkably simple finite-range pair potential,

$$\varphi(r)=\frac{1}{r}\left[1+\frac{1}{2}(r/r_m)^3\right]\quad\text{truncated at }r_m=(3/(4\pi))^{1/3}L,$$

where $L$ is the cubic cell edge (the sphere of radius $r_m$ has the cell volume $L^3$): each particle interacts as an electrically neutral sphere carrying a point charge and a uniformly distributed compensating charge. Because it is short-ranged and isotropic, the potential costs $O(N)$ work per simulation step, and it has been used extensively in plasma and soft-matter simulation. Yakub and Ronchi presented it as a power series in $r$ in which all coefficients beyond the first two vanish, asserted on the basis of numerical consistency and without a derivation.

Demyanov and Levashov (2022) supplied the missing derivation and identified its gaps, which are the subject of this problem. The series coefficients reduce to three-dimensional lattice sums $\sum_{\mathbf{n}}f_k(\mathbf{n})$ over integer vectors $\mathbf{n}\in\mathbb{Z}^3$ with $f_k(\mathbf{r})=e^{-\pi^2r^2/\delta^2}r^{2(k-1)}$, evaluated by two routes. The Poisson route applies the Poisson summation formula, valid for Schwartz functions (smooth functions whose derivatives all decay faster than any power, a class the Gaussians $f_k$ belong to), and expresses each sum as a finite series of derivatives of the Jacobi theta function $\vartheta_3(0,e^{-\delta^2})=\sum_{n\in\mathbb{Z}}e^{-\delta^2n^2}$, with coefficients built from the Kummer confluent hypergeometric function $M(1-k,3/2,x)={}_1F_1(1-k;3/2;x)$; for integer $k\ge 1$ this is a polynomial $M(1-k,3/2,x)=\sum_{s=0}^{k-1}a_{s,k}x^s$ with $a_{s,k}=\frac{(1-k)^{(s)}}{(3/2)^{(s)}s!}$, where $x^{(s)}=x(x+1)\cdots(x+s-1)$ is the rising factorial (Pochhammer symbol). The Euler-Maclaurin route uses the three-dimensional Euler-Maclaurin formula over a regular region $D$ with continuously differentiable boundary,

$$\sum_{\mathbf{n}\in D}f(\mathbf{n})=\int_D f\,d^3r+\int_{\partial D}(G\nabla f-f\nabla G)\cdot\mathbf{h}\,ds-\int_D G\,\Delta f\,d^3r,$$

where $\mathbf{h}$ is the outward unit normal, $\Delta$ is the Laplacian, and $G(\mathbf{r})=\frac{1}{4\pi^2}\sum_{\mathbf{q}\neq\mathbf{0}}\frac{e^{2\pi i\mathbf{q}\cdot\mathbf{r}}}{q^2}$ is the periodic Green's function of the Laplacian on the cubic lattice, given by a series that converges only conditionally. In the limit $\delta\to\infty$ the derivation gives $C_k=\frac{2\pi}{3L^3}\delta_{1,k}$ (with $\delta_{1,k}$ the Kronecker delta), proving the vanishing of all higher series coefficients in that limit, and the paper also demonstrates the formal equivalence of the two summation routes.

Two steps are, by the authors' own statement, not rigorous. First, analyzing the $\delta\to 0$ limit by substituting the theta-function asymptotic $\vartheta_3(0,e^{-\delta^2})\sim\sqrt{\pi}/\delta$ into the theta-derivative series leads to the finite combinatorial sum

$$S(k)=\sum_{s=0}^{k-1}a_{s,k}\,\frac{s!}{2^{2s}}\,A(s),\qquad A(s)=\sum_{\alpha_1+\alpha_2+\alpha_3=s}\frac{(2\alpha_1)!(2\alpha_2)!(2\alpha_3)!}{(\alpha_1!\alpha_2!\alpha_3!)^2},$$

where the triple sum $A(s)$ arises from products of three theta-derivative asymptotics (each summand is a product of central binomial coefficients $\binom{2\alpha_i}{\alpha_i}$). The authors verified $S(1)=1$ and, by exact symbolic computation in Mathematica, $S(k)=0$ for $2\le k\le 60$, and conjectured $S(k)=\delta_{1,k}$ for all integers $k\ge 1$, without proof. Second, their reduction of the Euler-Maclaurin formula to the Poisson formula over the expanding ball $D=B_R$, $R\to\infty$, uses the conditionally convergent Green's-function series and, in their words, exchanges the limit of the partial sums of the series with the integral without any justification; they state that the class of functions $f(\mathbf{r})$ for which this limit is valid remains uncertain and express the hope that it will be defined in the future. Multidimensional Euler and Poisson summation formulas have a classical literature, notably Muller and Freeden (1980) and the lattice-function theory of Freeden's school, but the specific class of validity for this ball-limit reduction is not stated there.

Problem Statement

This problem asks for rigorous resolutions of the two gaps that Demyanov and Levashov (2022) identified in their derivation of the angular-averaged Ewald potential. The two components close the rigor deficits of one derivation chain, so a complete answer resolves both: component (a) completes the proof of the small-$\delta$ asymptotics of the lattice sums that enter the series representation of the potential, and component (b) establishes for which functions the two summation routes used in the derivation are genuinely equivalent.

(a) Prove or refute: for every integer $k\ge 1$,

$$S(k)=\delta_{1,k},$$

where $S(k)=\sum_{s=0}^{k-1}a_{s,k}\frac{s!}{2^{2s}}A(s)$ with $a_{s,k}=\frac{(1-k)^{(s)}}{(3/2)^{(s)}s!}$ (the coefficients of the Kummer polynomial $M(1-k,3/2,x)$, with $(x)^{(s)}$ the rising factorial) and $A(s)=\sum_{\alpha_1+\alpha_2+\alpha_3=s}\frac{(2\alpha_1)!(2\alpha_2)!(2\alpha_3)!}{(\alpha_1!\alpha_2!\alpha_3!)^2}$. This elementary supporting identity, which the source verified symbolically for $2\le k\le 60$ but left unproven there, is what makes the theta-function route reproduce the small-$\delta$ limit $\sum_{\mathbf{n}}f_k(\mathbf{n})\to\delta_{1,k}$ for $f_k(\mathbf{r})=e^{-\pi^2r^2/\delta^2}r^{2(k-1)}$.

(b) Determine the class of functions $f:\mathbb{R}^3\to\mathbb{R}$ for which the three-dimensional Euler-Maclaurin formula over the expanding ball $D=B_R$ (with the lattice Green's function $G(\mathbf{r})=\frac{1}{4\pi^2}\sum_{\mathbf{q}\neq\mathbf{0}}\frac{e^{2\pi i\mathbf{q}\cdot\mathbf{r}}}{q^2}$, a conditionally convergent series) rigorously reduces, in the limit $R\to\infty$, to the Poisson summation formula

$$\sum_{\mathbf{n}\in\mathbb{Z}^3}f(\mathbf{n})=\sum_{\mathbf{q}\in\mathbb{Z}^3}\int_{\mathbb{R}^3}f(\mathbf{r})\,e^{-2\pi i\mathbf{q}\cdot\mathbf{r}}\,d^3r,$$

that is, characterize the functions for which the formal transformations of the source's Appendix B, in particular the interchange of the limit of the partial sums of the Green's-function series with the integral over $D$ and the passage $D\to\mathbb{R}^3$, are justified. The characterization must be sharp and accompanied by explicit counterexamples: a complete answer states which functions satisfy the reduction and exhibits functions for which it fails.

References

1. G. S. Demyanov and P. R. Levashov, 'Systematic derivation of angular-averaged Ewald potential', Journal of Physics A: Mathematical and Theoretical 55, 385202 (2022). DOI: 10.1088/1751-8121/ac870b; arXiv:2203.15025. URL: https://arxiv.org/abs/2203.15025
2. Eugene Yakub and Claudio Ronchi, 'An efficient method for computation of long-ranged Coulomb forces in computer simulation of ionic fluids', The Journal of Chemical Physics 119, 11556-11560 (2003). DOI: 10.1063/1.1624364. URL: https://doi.org/10.1063/1.1624364
3. Claus Müller and Willi Freeden, 'Multidimensional Euler and Poisson summation formulas', Results in Mathematics 3, 33-63 (1980). DOI: 10.1007/BF03323347. URL: https://doi.org/10.1007/BF03323347
4. Willi Freeden, 'Metaharmonic Lattice Point Theory', Chapman & Hall/CRC, Boca Raton (2011). DOI: 10.1201/b10876. URL: https://doi.org/10.1201/b10876
5. Willi Freeden and Martin Gutting, 'Special Functions of Mathematical (Geo-)Physics', Birkhäuser/Springer, Basel (2013). DOI: 10.1007/978-3-0348-0563-6. URL: https://doi.org/10.1007/978-3-0348-0563-6
6. G. S. Demyanov and P. R. Levashov, 'One-component plasma of a million particles via angular-averaged Ewald potential: A Monte Carlo study', Physical Review E 106, 015204 (2022). DOI: 10.1103/PhysRevE.106.015204; arXiv:2206.14894. URL: https://arxiv.org/abs/2206.14894
7. G. S. Demyanov, A. S. Onegin, and P. R. Levashov, 'N-convergence in one-component plasma: Comparison of Coulomb, Ewald, and angular-averaged Ewald potentials', Contributions to Plasma Physics 64, e202300164 (2024). DOI: 10.1002/ctpp.202300164. URL: https://doi.org/10.1002/ctpp.202300164
8. Tobias Dornheim, Thomas M. Chuna, Hannah M. Bellenbaum, Zhandos A. Moldabekov, Panagiotis Tolias, and Jan Vorberger, 'Application of a spherically averaged pair potential in ab initio path integral Monte Carlo simulations of a warm dense electron gas', Physical Review E 112, 035203 (2025). DOI: 10.1103/lj9c-bh48; arXiv:2504.00737. URL: https://arxiv.org/abs/2504.00737
9. Ikuo Fukuda and Haruki Nakamura, 'Non-Ewald methods for evaluating the electrostatic interactions of charge systems: similarity and difference', Biophysical Reviews 14, 1315-1340 (2022). DOI: 10.1007/s12551-022-01029-2. URL: https://doi.org/10.1007/s12551-022-01029-2
10. G. S. Demyanov and P. R. Levashov, 'Molecular dynamics of nondegenerate hydrogen plasma using improved Kelbg pseudopotential with electron thermal de Broglie wavelength correction', Physics of Plasmas 32, 123901 (2025). DOI: 10.1063/5.0298952; arXiv:2508.19820. URL: https://arxiv.org/abs/2508.19820
11. A. S. Onegin, G. S. Demyanov, and P. R. Levashov, 'Pressure of Coulomb systems with volume-dependent long-range potentials', Journal of Physics A: Mathematical and Theoretical 57, 205002 (2024). DOI: 10.1088/1751-8121/ad40e5. URL: https://doi.org/10.1088/1751-8121/ad40e5
12. G. S. Demyanov, A. S. Onegin, and P. R. Levashov, 'Reply to Comment on Pressure of Coulomb systems with volume-dependent long-range potentials', Journal of Physics A: Mathematical and Theoretical 58, 088002 (2025). DOI: 10.1088/1751-8121/adb400. URL: https://doi.org/10.1088/1751-8121/adb400
13. Jean-Christophe Feauveau, 'A unified approach for summation formulae', arXiv:1604.05578 (2016). URL: https://arxiv.org/abs/1604.05578