--- title: Angular-Averaged Ewald Potential url: https://www.emergentmind.com/topics/angular-averaged-ewald-potential type: topic --- # Angular-Averaged Ewald Potential 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 $r=|\mathbf r|$ [2203.15025]. 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 [2203.15025], [2509.02390], [2504.00737]. ## 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 [2203.15025]. 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 \[ \phi_2(r_{ij}) = \frac{1}{L}\left[ \frac{\mathrm{erfc}(\delta r_{ij}/L)}{r_{ij}/L} + \frac{1}{\pi} \sum_{\mathbf n\neq\mathbf 0} e^{-\frac{\pi^2}{\delta^2}n^2}n^{-2} \cos\bigl(2\pi\,\mathbf n\cdot \mathbf r_{ij}/L\bigr) \right]. \] The anisotropy is explicit in the factor $\cos(2\pi\,\mathbf n\cdot \mathbf r_{ij}/L)$, so $\phi_2$ depends not only on $r_{ij}$ but on the orientation of $\mathbf r_{ij}$ relative to the lattice [2203.15025]. The angular-averaged pair potential is defined by averaging $\phi_2(\mathbf r)$ over the solid angle at fixed $r$, \[ \phi_2^{a}(r) = \frac{1}{4\pi}\int d\Omega\,\phi_2(\mathbf r), \] or, in the OCP notation, \[ v_\mathrm{AAEP}(r) \equiv \frac{1}{4\pi} \int \mathrm{d}\Omega_{\hat{\mathbf{r}}}\, v_\mathrm{Ewald}(\mathbf{r}) . \] This averaging removes the directional dependence induced by the periodic lattice and leaves a central potential suitable for isotropic simulations [2203.15025], [2509.02390]. ## 2. Derivation from the standard Ewald representation The angular averaging acts trivially on the real-space error-function term, since $\mathrm{erfc}(\delta r/L)/(r/L)$ is already isotropic. The nontrivial step concerns the reciprocal-lattice cosine factor. Using $\mathbf n\cdot\mathbf r = nr\cos\theta$, one obtains \[ \frac{1}{4\pi}\int_{-1}^{1}d(\cos\theta)\int_0^{2\pi}d\psi\, \cos\left(2\pi n r\cos\theta/L\right) = \frac{L/r}{2\pi n}\,\sin\left(2\pi n r/L\right), \] which yields \[ \phi_2^{a}(r) = \frac{1}{r}\left[ \mathrm{erfc}(\delta r/L) + \frac{1}{2\pi^2}\sum_{n\neq 0} e^{-\pi^2 n^2/\delta^2}\,n^{-3} \sin\left(2\pi n r/L\right) \right]. \] Thus the angular-averaged reciprocal-space contribution becomes radial and contains spherical-Bessel-type $\sin(kr)/(kr)$ behavior; the OCP formulation states the same point as \[ \frac{1}{4\pi}\int \mathrm{d}\Omega_{\hat{\mathbf{r}}}\,\cos(\mathbf{k}\cdot \mathbf{r}) = \frac{\sin(kr)}{kr}. \] Consequently, the reciprocal-space piece \[ \frac{4\pi}{V}\sum_{\mathbf{k}\ne 0} \frac{e^{-k^2/(4\alpha^2)}}{k^2}\cos(\mathbf{k}\cdot \mathbf{r}) \] is replaced by \[ \frac{4\pi}{V} \sum_{\mathbf{k}\ne 0} \frac{e^{-k^2/(4\alpha^2)}}{k^2} \frac{\sin(kr)}{kr} \] after angular averaging [2203.15025], [2509.02390]. 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 [2203.15025]. In their construction, the coefficients are represented as finite series containing derivatives of Jacobi theta functions [2203.15025]. ## 3. Closed form and series structure A central result of the point-charge limit is the simple radial expression \[ \varphi(r) = \frac{1}{r}\left[\,1 + \frac{1}{2}\left(\frac{r}{r_m}\right)^3\right], \quad r_m = \left(\frac{3}{4\pi}\right)^{1/3}L, \] with $\varphi(r)=0$ for $r>r_m$ in practical implementations [2203.15025]. In the warm-dense electron-gas literature, the same spherically averaged periodic pair potential is written in a finite-range polynomial form, \[ \phi_\text{YR}(r_{ab}) = \begin{cases} \displaystyle \frac{1}{r_{ab}}\left\{ 1 + \frac{1}{2}\Big(\frac{r_{ab}}{r_\text{cut}}\Big) \Big[\Big(\frac{r_{ab}}{r_\text{cut}}\Big)^2 - 3\Big] \right\}, & r_{ab} < r_\text{cut},\\[1.2ex] 0, & r_{ab} \ge r_\text{cut}, \end{cases} \] with \[ r_\text{cut} = L\left(\frac{3}{4\pi}\right)^{1/3}. \] These two formulas are the same finite-range radial construction expressed in different notations [2203.15025], [2504.00737]. The derivation also admits a formal power-series representation, \[ \phi_2^{a}(r) = \frac{1}{r\Bigl(1 + \displaystyle\sum_{k=1}^{\infty} C_k\,r^{2k+1}\Bigr)}, \] with coefficients \[ C_k = \frac{2(-1)^k}{(2k+1)L^{2k+1}} \left[ \frac{(2\pi)^{2k-1}}{(2k)!} \sum_{\mathbf n\neq 0} f_k(\mathbf n) -\frac{\delta^{2k+1}}{\sqrt{\pi}\,k!} \right], \quad k\ge 1, \] and \[ f_k(\mathbf n) = \exp\left(-\frac{\pi^2}{\delta^2} n^2\right) n^{2(k-1)}. \] The unary coefficient is \[ C_0 = \frac{2}{L}\left[ \frac{1}{2\pi}\sum_{\mathbf n\neq0} e^{-\pi^2n^2/\delta^2}\,n^{-2} -\frac{\delta}{\sqrt{\pi}} \right] = 2\phi_1 . \] Demyanov and Levashov show that, for point charges, only the first two nontrivial coefficients survive, recovering the simple cubic correction embodied in $\varphi(r)$ [2203.15025]. ## 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 \[ v_\mathrm{Ewald}(\mathbf{r}) = Z^2 e^2 \left[ \sum_{\mathbf{n}} \frac{\mathrm{erfc}\big(\alpha|\mathbf{r}+\mathbf{n}L|\big)}{|\mathbf{r}+\mathbf{n}L|} + \frac{4\pi}{V} \sum_{\mathbf{k}\ne 0} \frac{e^{-k^2/(4\alpha^2)}}{k^2} \cos(\mathbf{k}\cdot \mathbf{r}) \right] + \text{ (background and self terms)}. \] In neutral periodic Coulomb systems, this decomposition comprises real-space, reciprocal-space, self, and, depending on formulation, background and surface terms [2509.02390], [1404.3534]. 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 \[ \phi^{i}(r)= \phi_{0}^i(r) + \phi_{self}^i(r) + \phi_{surf}^i(r), \] after applying the Ewald splitting to the potential first and deriving fields, energies, and forces by differentiation [1805.10363]. 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 [2509.02390]. 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 [1709.07753]. ## 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 $r_{ij}$ [2203.15025]. They also show its effectiveness by calculating the Madelung constant for a number of crystal lattices [2203.15025]. In the classical OCP, the EOS work employing AAEP studies point ions of charge $Ze$ in a uniform neutralizing background, with coupling parameter \[ \Gamma \equiv \frac{Z^2 e^2}{a k_B T}, \qquad a = \left(\frac{3}{4\pi n}\right)^{1/3}. \] The paper presents analytic fits of internal energy over \[ 0.01 \le \Gamma \le 170 \] using Monte Carlo data in the thermodynamic limit, extending a prior dataset with additional points at strong coupling $\Gamma=120,\ 150,\ 170$ obtained using the angular-averaged Ewald potential [2509.02390]. The excess internal energy per particle is written as \[ \frac{U}{N} = 2\pi n \int_0^\infty r^2\, g(r)\, v_\mathrm{AAEP}(r)\,\mathrm{d}r, \] and finite-size extrapolation is performed with \[ \frac{U}{N}\,(1/N) = \left(\frac{U}{N}\right)_\infty + b\left(\frac{1}{N}\right)^{\gamma}. \] For large $\Gamma\ge 120$, the authors fix $\gamma=1$; for weaker coupling, $\gamma$ is fitted [2509.02390]. The same work reports standard Metropolis Monte Carlo with AAEP for system sizes \[ N = 10^2, 10^3, 10^4, 5\times 10^4, 10^5, 10^6 \] at weak coupling $(\Gamma=0.01,0.05,0.1,1)$, and up to $10^5$ at strong coupling $(\Gamma=10\text{--}170)$, using $10^7$ Monte Carlo steps for $\Gamma \le 1$ and $10^8$ steps for $10\le\Gamma\le 170$ [2509.02390]. The paper attributes the simple power-law finite-size behavior in $1/N$ to the isotropy and good convergence properties of AAEP and notes agreement with Debye–Hückel and HNC results at weak coupling [2509.02390]. ## 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, \[ U = \sum_{i