Angular-Averaged Ewald Potential

• Angular-averaged Ewald potential is a spherically averaged representation of the periodic Coulomb interaction that removes directional bias by averaging over all orientations.
• Its derivation reformulates both real- and reciprocal-space contributions, resulting in a simplified radial expression that aids efficient Monte Carlo simulations and Madelung constant evaluations.
• The method delivers significant computational speedups and accuracy for integrated properties, though it may underperform in capturing small-q anisotropic observables.

Searching arXiv for recent and foundational papers on angular-averaged Ewald potential and related Ewald formulations. Angular-averaged Ewald potential is a spherically averaged form of the periodic Ewald pair interaction for Coulomb systems, obtained by averaging the anisotropic Ewald potential over all orientations of the separation vector at fixed radius. In a cubic periodic cell, the standard Ewald pair potential depends on the direction of $\mathbf r$ through lattice-image and reciprocal-lattice contributions, whereas the angular-averaged construction produces a radial effective interaction $v_{\mathrm{AAEP}}(r)$ or $\varphi(r)$ that depends only on $r=|\mathbf r|$ (Demyanov et al., 2022). This object is used primarily for disordered and isotropic Coulomb systems, where the cubic anisotropy of the simulation cell is artificial, and it has been employed in classical one-component plasma (OCP) Monte Carlo calculations, Madelung-constant evaluations, and warm-dense electron-gas simulations as a reduced-cost alternative to the full Ewald sum (Demyanov et al., 2022, Demyanov et al., 2 Sep 2025, Dornheim et al., 1 Apr 2025).

1. Definition within periodic Coulomb summation

For a 3D periodic cubic cell of size $L$ containing point charges $Q_i$ at positions $\mathbf r_i$, the Coulomb energy is a conditionally convergent lattice sum,

$$E = \frac{1}{2}\sum_{\mathbf n}'\sum_{i,j=1}^N \frac{Q_iQ_j}{|\mathbf r_i-\mathbf r_j+L\mathbf n|},$$

and Ewald’s method rewrites it as rapidly convergent real-space and reciprocal-space contributions (Demyanov et al., 2022). In the point-charge limit, the energy can be written as

$E = \phi_1 \sum_{i=1}^N Q_i^2 + \frac{1}{2}\sum_{i=1}^N\sum_{\substack{j=1\i\neq j}}^N Q_i Q_j\,\phi_2(r_{ij}),$

with

$$\phi_1 = \frac{1}{L}\left[ \frac{1}{2\pi}\sum_{\mathbf n\neq\mathbf 0} e^{-\frac{\pi^2}{\delta^2}n^2}\,n^{-2} -\frac{\delta}{\sqrt{\pi}} \right],$$

and

$v_{\mathrm{AAEP}}(r)$0

The anisotropy is explicit in the factor $v_{\mathrm{AAEP}}(r)$1, so $v_{\mathrm{AAEP}}(r)$2 depends not only on $v_{\mathrm{AAEP}}(r)$3 but on the orientation of $v_{\mathrm{AAEP}}(r)$4 relative to the lattice (Demyanov et al., 2022).

The angular-averaged pair potential is defined by averaging $v_{\mathrm{AAEP}}(r)$5 over the solid angle at fixed $v_{\mathrm{AAEP}}(r)$6,

$v_{\mathrm{AAEP}}(r)$7

or, in the OCP notation,

$v_{\mathrm{AAEP}}(r)$8

This averaging removes the directional dependence induced by the periodic lattice and leaves a central potential suitable for isotropic simulations (Demyanov et al., 2022, Demyanov et al., 2 Sep 2025).

2. Derivation from the standard Ewald representation

The angular averaging acts trivially on the real-space error-function term, since $v_{\mathrm{AAEP}}(r)$9 is already isotropic. The nontrivial step concerns the reciprocal-lattice cosine factor. Using $\varphi(r)$0, one obtains

$\varphi(r)$1

which yields

$\varphi(r)$2

Thus the angular-averaged reciprocal-space contribution becomes radial and contains spherical-Bessel-type $\varphi(r)$3 behavior; the OCP formulation states the same point as

$\varphi(r)$4

Consequently, the reciprocal-space piece

$\varphi(r)$5

is replaced by

$\varphi(r)$6

after angular averaging (Demyanov et al., 2022, Demyanov et al., 2 Sep 2025).

Demyanov and Levashov derive the coefficients of the resulting series expansion by two independent routes, based on the Euler–Maclaurin and Poisson summation formulas, and show the formal equivalence of these two summation approaches in the three-dimensional case (Demyanov et al., 2022). In their construction, the coefficients are represented as finite series containing derivatives of Jacobi theta functions (Demyanov et al., 2022).

3. Closed form and series structure

A central result of the point-charge limit is the simple radial expression

$\varphi(r)$7

with $\varphi(r)$8 for $\varphi(r)$9 in practical implementations (Demyanov et al., 2022). In the warm-dense electron-gas literature, the same spherically averaged periodic pair potential is written in a finite-range polynomial form,

$r=|\mathbf r|$0

with

$r=|\mathbf r|$1

These two formulas are the same finite-range radial construction expressed in different notations (Demyanov et al., 2022, Dornheim et al., 1 Apr 2025).

The derivation also admits a formal power-series representation,

$r=|\mathbf r|$2

with coefficients

$r=|\mathbf r|$3

and

$r=|\mathbf r|$4

The unary coefficient is

$r=|\mathbf r|$5

Demyanov and Levashov show that, for point charges, only the first two nontrivial coefficients survive, recovering the simple cubic correction embodied in $r=|\mathbf r|$6 (Demyanov et al., 2022).

4. Relation to standard Ewald theory and boundary conditions

Angular-averaged Ewald potential is not a replacement for Ewald theory in general; rather, it is a particular radial reduction of the triply periodic Coulomb Green’s function. In the standard 3D Ewald construction, the pair interaction is

$r=|\mathbf r|$7

In neutral periodic Coulomb systems, this decomposition comprises real-space, reciprocal-space, self, and, depending on formulation, background and surface terms (Demyanov et al., 2 Sep 2025, Tornberg, 2014).

The general Ewald framework separates the potential into absolutely convergent direct and reciprocal parts plus self and surface contributions. For multipoles, a consistent formulation writes

$r=|\mathbf r|$8

after applying the Ewald splitting to the potential first and deriving fields, energies, and forces by differentiation (Stamm et al., 2018). The angular-averaged construction focuses on the pair potential itself and, in the OCP setting summarized in the EOS work, is used together with self, background, and surface terms chosen so that the total neutral-system energy is well defined and consistent with conductor or tin-foil boundary conditions (Demyanov et al., 2 Sep 2025).

This suggests that AAEP should be viewed as an isotropic effective pair potential extracted from the full periodic electrostatic problem rather than as a general-purpose substitute for all boundary-condition-sensitive observables. The literature on pairwise periodic Coulomb potentials also emphasizes that exact Ewald-based pair potentials can encode specific boundary prescriptions and background choices in a translationally invariant form (Yi et al., 2017).

5. Uses in disordered Coulomb systems and plasma thermodynamics

The principal motivation for angular averaging is the treatment of isotropic, disordered Coulomb systems. Demyanov and Levashov state that, for ionic liquids, plasmas, and disordered electrolytes, the anisotropic lattice-induced structure of the Ewald potential is artificial and unnecessary, while the averaged potential provides a spherically symmetric long-range interaction that can be expressed as a simple series in $r=|\mathbf r|$9 (Demyanov et al., 2022). They also show its effectiveness by calculating the Madelung constant for a number of crystal lattices (Demyanov et al., 2022).

In the classical OCP, the EOS work employing AAEP studies point ions of charge $L$0 in a uniform neutralizing background, with coupling parameter

$L$1

The paper presents analytic fits of internal energy over

$L$2

using Monte Carlo data in the thermodynamic limit, extending a prior dataset with additional points at strong coupling $L$3 obtained using the angular-averaged Ewald potential (Demyanov et al., 2 Sep 2025). The excess internal energy per particle is written as

$L$4

and finite-size extrapolation is performed with

$L$5

For large $L$6, the authors fix $L$7; for weaker coupling, $L$8 is fitted (Demyanov et al., 2 Sep 2025).

The same work reports standard Metropolis Monte Carlo with AAEP for system sizes

$L$9

at weak coupling $Q_i$0, and up to $Q_i$1 at strong coupling $Q_i$2, using $Q_i$3 Monte Carlo steps for $Q_i$4 and $Q_i$5 steps for $Q_i$6 (Demyanov et al., 2 Sep 2025). The paper attributes the simple power-law finite-size behavior in $Q_i$7 to the isotropy and good convergence properties of AAEP and notes agreement with Debye–Hückel and HNC results at weak coupling (Demyanov et al., 2 Sep 2025).

6. Performance, scope, and limitations

The practical attraction of AAEP is computational simplification. In the OCP context, the interaction energy becomes a sum over isotropic pair potentials,

$Q_i$8

so energy evaluation is independent of the orientation of particle pairs relative to the cubic simulation cell (Demyanov et al., 2 Sep 2025). In the warm-dense uniform electron gas, the spherically averaged periodic pair potential is presented as a reduced-cost alternative to the full Ewald sum, with reported speedups by factors of $Q_i$9 in path-integral Monte Carlo, while preserving high accuracy for integrated properties such as kinetic and potential energy (Dornheim et al., 1 Apr 2025).

The same electron-gas study also delineates the limitations. It finds very accurate results relative to Ewald reference data for integrated quantities, but wave-number-resolved observables such as the static structure factor $\mathbf r_i$0, the static linear density response $\mathbf r_i$1, and the static quadratic density response $\mathbf r_i$2 fluctuate for small $\mathbf r_i$3 (Dornheim et al., 1 Apr 2025). The authors conclude that the YR potential is suitable for equation-of-state properties or $\mathbf r_i$4-resolved observables in the non-collective regime, whereas a full Ewald treatment is mandatory for effects manifesting at smaller $\mathbf r_i$5, including compressibility sum rules, small-angle x-ray scattering, and the estimation of optical and transport properties (Dornheim et al., 1 Apr 2025).

A common misconception is that angular averaging merely smooths numerical noise in the Ewald sum. The derivations instead show that it is a defined projection of the anisotropic periodic Green’s function onto its radial component at fixed $\mathbf r_i$6 (Demyanov et al., 2022, Demyanov et al., 2 Sep 2025). A second misconception is that the averaged potential is universally equivalent to full Ewald electrostatics. The warm-dense electron-gas comparison shows that this equivalence holds only for certain observables; long-wavelength response remains sensitive to the full periodic Coulomb kernel (Dornheim et al., 1 Apr 2025).

7. Connections to broader Ewald generalizations

AAEP belongs to a broader family of Ewald reformulations aimed at controlling anisotropy, convergence, and computational cost. Standard triply periodic Ewald formulas arise from Gaussian screening and Fourier analysis, with real-space terms of the form

$\mathbf r_i$7

and reciprocal-space sums over lattice wave vectors (Tornberg, 2014). Generalizations exist for singly, doubly, and triply periodic systems, where the Fourier kernels exhibit different directional structures; these formulations make explicit where angular dependence enters and therefore where angular averaging would act (Tornberg, 2014).

For arbitrary multipole order, the Ewald splitting can be applied to the scalar potential first, after which electric fields, energies, forces, and self terms are derived consistently. In that setting, angular dependence is generated by Cartesian derivatives of the Coulomb or screened Coulomb kernel, contracted with multipole tensors (Stamm et al., 2018). In interpolated Ewald strategies for distributed multipoles and induced dipoles, the underlying radial kernel

$\mathbf r_i$8

remains central, while angular dependence is carried by differential operators acting on it (Chollet et al., 2022). This suggests that any extension of AAEP beyond point charges would have to average not only the scalar periodic Green’s function but also the tensorial structures generated by multipolar derivatives; the cited multipolar Ewald literature provides the formal machinery for such a program, but does not introduce an angular-averaged multipolar potential as a standard named object (Stamm et al., 2018, Chollet et al., 2022).

In this broader perspective, angular-averaged Ewald potential can be understood as the monopole, spherically symmetric projection of a periodic Coulomb interaction whose full form is intrinsically lattice- and boundary-dependent. Its usefulness is greatest when the target physics is isotropic and thermodynamic, and its limitations become evident when anisotropic or long-wavelength information is itself part of the observable.