Explicit real and reciprocal Ewald potential, energy, and force tail bounds

Fix a collision-free configuration and choose a small closed product of coordinate balls around its source positions, using local torus lifts, that remains collision-free. Finitely many nearby nonself images then stay a positive distance from the singularity. For large real-space shell index $m$, [F1] bounds the kernel by a constant times $e^{-{\alpha }^2{m}^2}$ and its first position derivatives by $(2\alpha +{\alpha }^{-1})e^{-{\alpha }^2{m}^2}\mathrm{/}\sqrt{\pi }$. [F2] bounds the shell count uniformly over every translate; since the source-difference vectors stay in a bounded set, $\mathrm{\mid }\lambda \mathrm{\mid }\to \mathrm{\infty }$ forces the corresponding distance to infinity uniformly. Thus the real energy series and each first-derivative series converge uniformly on this product. Finite partial sums are continuously differentiable and their values converge at one point by [F4]. Under [A1], apply [F5] along each real coordinate line to justify termwise differentiation of the real energy.

For the reciprocal energy, [F3] gives $\mathrm{\mid }{\sigma }_k\mathrm{\mid }\le Q$ and Gaussian decay of $c_k$, while [F6] gives $\mathrm{\mid }{\mathrm{\nabla }}_{x_i}{\sigma }_k\mathrm{\mid }=\mathrm{\mid }{q}_i\mathrm{\mid }\mathrm{\mid }{\kappa }_k\mathrm{\mid }$. The derivative of each summand $\frac{1}{2}{c}_k\mathrm{\mid }{\sigma }_k{\mathrm{\mid }}^2$ is bounded by $c_{k}Q\mathrm{\mid }{q}_i\mathrm{\mid }\mathrm{\mid }{\kappa }_k\mathrm{\mid }$. All nonzero modes in shells $m\ge 1$ therefore have a summable position-uniform majorant $C(m+1{)}^3{e}^{-m^2\mathrm{/}(4{\alpha }^2)}$ by [F2]; the remaining bounded ball contains finitely many modes because the reciprocal lattice is full rank. The energy summands themselves are bounded by $Q^2{c}_k\mathrm{/}2$, whose shell sum also converges by [F2]–[F3]. Thus the reciprocal energy and all its first-derivative series converge uniformly on the same product. Finite partial sums have values converging at one point by [F4], so [F5] gives their termwise derivatives along every coordinate line.

Under Countable Choice [A1], use the Ewald representation [F4]. For $r>R$, the shell index $m$ is at least $M_R\ge 1$. By [F1], each omitted real potential term is at most $e^{-{\alpha }^2{m}^2}\mathrm{/}(\sqrt{\pi }\alpha m^2)$. The shell count [F2] and $m^{-2}\le 1$ bound the total image tail for one source by $C_R(\sqrt{\pi }\alpha {)}^{-1}{T}_3({\alpha }^2,M_R)$. Summing $\mathrm{\mid }{q}_i\mathrm{\mid }$ gives $B_R^U$.

For a reciprocal mode in the omitted shell $m\le \mathrm{\mid }{\kappa }_k\mathrm{\mid }<m+1$, with $m\ge M_K\ge 1$, [F3] gives the potential summand bound $(4\pi Q\mathrm{/}V)e^{-m^2\mathrm{/}(4{\alpha }^2)}\mathrm{/}{m}^2$. Multiply by [F2], use $m^{-2}\le 1$, and sum to obtain $B_K^U$. The energy summand is one half the coefficient times $\mathrm{\mid }{\sigma }_k{\mathrm{\mid }}^2\le Q^2$, so the same count yields $B_K^E$.

Each coordinate derivative obtained in steps 1.1–1.2 is a locally uniform limit of continuous finite-sum derivatives, hence is continuous by [F7]. The open collision-free configuration set is a finite-dimensional Euclidean open set in local lifts. Applying the continuous-partials clause of [F7] shows the Ewald energy is totally differentiable there, with the termwise derivatives computed below as its gradient.

In a finite real-space partial sum truncated by the symmetric condition $\mathrm{\mid }\lambda \mathrm{\mid }\le L$, differentiating terms with $x_i$ in the first position gives one half of $q_i{\sum }_{j,\lambda }^{\mathrm{\prime }}{q}_j\mathrm{\nabla }{\varphi }_s(\mathrm{\mid }{x}_i-x_j+\lambda \mathrm{\mid })$. Terms with $x_i$ in the second position give $-\frac{1}{2}{q}_i{\sum }_{j,\lambda }^{\mathrm{\prime }}{q}_j\mathrm{\nabla }{\varphi }_s(\mathrm{\mid }{x}_j-x_i+\lambda \mathrm{\mid })$. By [F6] and the chain rule, $\mathrm{\nabla }{\varphi }_s(\mathrm{\mid }y\mathrm{\mid })={\varphi }_s^{\mathrm{\prime }}(\mathrm{\mid }y\mathrm{\mid })y\mathrm{/}\mathrm{\mid }y\mathrm{\mid }$ for $y\mathrm{\ne }0$, so this radial vector gradient is odd. Reindexing the second sum by $\lambda \mapsto -\lambda$ preserves the finite truncation and the excluded self term, and makes it equal to the first half. Passing to the limit is justified by step 1.1, so the full real energy gradient is $q_i{\sum }_{j,\lambda }^{\mathrm{\prime }}{q}_j{\varphi }_s^{\mathrm{\prime }}(r_{ij\lambda })(x_i-x_j+\lambda )\mathrm{/}{r}_{ij\lambda }$.

For each reciprocal mode, [F6] gives ${\mathrm{\nabla }}_{x_i}{\sigma }_k=-i{q}_i{\kappa }_k{a}_k(t_i)$. In real and imaginary coordinates, $\mathrm{\nabla }\mathrm{\mid }{\sigma }_k{\mathrm{\mid }}^2=2\mathrm{Re}(\overline{{\sigma }_k}\mathrm{\nabla }{\sigma }_k)$, so differentiating a finite reciprocal partial energy gives $q_i{\sum }_k{c}_k\mathrm{Re}(\overline{{\sigma }_k}{\mathrm{\nabla }}_x{a}_k(t_i))$. Step 1.2 justifies passage to the full sum. The constant self term has zero derivative. Negating the real gradient in step 2.2 and this reciprocal gradient gives the force formula in the Statement; opposite reciprocal modes are conjugate pairs, so their sum is real.

The real energy has the factor $1\mathrm{/}2$ and charge-pair absolute sum at most $Q^2$, so the same shell estimate as in step 1.3 gives $B_R^E$. The radial derivative bound [F1] and $r\ge m\ge 1$ imply $\mathrm{\mid }{\varphi }_s^{\mathrm{\prime }}(r)\mathrm{\mid }\le (2\alpha +{\alpha }^{-1})e^{-{\alpha }^2{m}^2}\mathrm{/}\sqrt{\pi }$. The real force formula proved in step 2.2, the shell count [F2], and $\mathrm{\mid }{q}_i\mathrm{\mid }{\sum }_j\mathrm{\mid }{q}_j\mathrm{\mid }=\mathrm{\mid }{q}_i\mathrm{\mid }Q$ give exactly $B_R^{F,i}$.

By the reciprocal force formula in step 3.1, each omitted reciprocal force mode has magnitude at most $(4\pi \mathrm{\mid }{q}_i\mathrm{\mid }Q\mathrm{/}V)e^{-\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2\mathrm{/}(4{\alpha }^2)}\mathrm{/}\mathrm{\mid }{\kappa }_k\mathrm{\mid }$, using $\mathrm{\mid }{\mathrm{\nabla }}_x{a}_k\mathrm{\mid }=\mathrm{\mid }{\kappa }_k\mathrm{\mid }$ from [F6]. Since $\mathrm{\mid }{\kappa }_k\mathrm{\mid }\ge m\ge 1$, this is at most $(4\pi \mathrm{\mid }{q}_i\mathrm{\mid }Q\mathrm{/}V)e^{-m^2\mathrm{/}(4{\alpha }^2)}$. The shell count [F2] and sum over $m\ge M_K$ give $B_K^{F,i}$.

The shell bounds are uniform in every translate, so no source-position dependence was introduced. Replacing $T_3$ by its finite upper controller preserves each inequality. Adding the real and reciprocal omitted terms by the triangle inequality gives the total Ewald cutoff error for potential, energy, and force.