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Error estimate of the u-series method for molecular dynamics simulations

Jiuyang Liang, Zhenli Xu, Qi Zhou

TL;DR

The paper addresses the challenge of rigorously bounding errors in the u-series decomposition of the Coulomb kernel for molecular dynamics. By developing a Fourier-space analysis with a kernel split into near- and far-field components, the authors prove that energy errors decay as $O((\log b)^{-3/2} e^{-\pi^{2}/(2\log b)} + b^{-M} + w_{-1} e^{-r_c^{2}/s_{-1}^{2}})$ and force errors as $O((\log b)^{-3/2} e^{-\pi^{2}/(2\log b)} + b^{-3M} + w_{-1}(s_{-1})^{-2} e^{-r_c^{2}/s_{-1}^{2}})$ under reasonable assumptions, with extensions to $C^{1}$-continuous schemes. The work provides closed-form error estimates under an ideal-gas assumption to guide a priori parameter tuning, and validates the theory on a Madelung-constant lattice and an all-atom SPC/E water system, demonstrating reliable energy and force accuracy controlled by the tunable parameters $M$ and $b$. This yields practical guidance for selecting u-series parameters to meet prescribed accuracies while balancing computational cost. The results advance the theoretical understanding of SOG-based electrostatics methods and support their robust application in large-scale MD simulations.

Abstract

This paper provides an error estimate for the u-series method of the Coulomb interaction in molecular dynamics simulations. We show that the number of truncated Gaussians $M$ in the u-series and the base of interpolation nodes $b$ in the bilateral serial approximation are two key parameters for the algorithm accuracy, and that the errors converge as $\mathcal{O}(b^{-M})$ for the energy and $\mathcal{O}(b^{-3M})$ for the force. Error bounds due to numerical quadrature and cutoff in both the electrostatic energy and forces are obtained. Closed-form formulae are also provided, which are useful in the parameter setup for simulations under a given accuracy. The results are verified by analyzing the errors of two practical systems.

Error estimate of the u-series method for molecular dynamics simulations

TL;DR

The paper addresses the challenge of rigorously bounding errors in the u-series decomposition of the Coulomb kernel for molecular dynamics. By developing a Fourier-space analysis with a kernel split into near- and far-field components, the authors prove that energy errors decay as and force errors as under reasonable assumptions, with extensions to -continuous schemes. The work provides closed-form error estimates under an ideal-gas assumption to guide a priori parameter tuning, and validates the theory on a Madelung-constant lattice and an all-atom SPC/E water system, demonstrating reliable energy and force accuracy controlled by the tunable parameters and . This yields practical guidance for selecting u-series parameters to meet prescribed accuracies while balancing computational cost. The results advance the theoretical understanding of SOG-based electrostatics methods and support their robust application in large-scale MD simulations.

Abstract

This paper provides an error estimate for the u-series method of the Coulomb interaction in molecular dynamics simulations. We show that the number of truncated Gaussians in the u-series and the base of interpolation nodes in the bilateral serial approximation are two key parameters for the algorithm accuracy, and that the errors converge as for the energy and for the force. Error bounds due to numerical quadrature and cutoff in both the electrostatic energy and forces are obtained. Closed-form formulae are also provided, which are useful in the parameter setup for simulations under a given accuracy. The results are verified by analyzing the errors of two practical systems.
Paper Structure (22 sections, 15 theorems, 120 equations, 5 figures, 2 tables)

This paper contains 22 sections, 15 theorems, 120 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Error estimate of the u-series decomposition
  3. U-series decomposition for Coulomb kernel
  4. Main results of the convergence rate
  5. Proof of the main results
  6. Fourier spectral expansion and kernel splitting
  7. The estimate of
  8. The estimate of
  9. The estimate of
  10. Extension to -continuity and further discussions
  11. Closed formulae
  12. Numerical results
  13. Madelung constant of crystalline lattice
  14. All-atom water system
  15. Conclusion
  16. ...and 7 more sections

Key Result

Theorem 2

For $C^0$-continuous u-series, $U_{\emph{err}}$ and $\bm{F}_{\emph{err}}(\bm{r}_i)$ hold the following estimates, where $\simeq$ indicates "asymptotically equal" as $b\rightarrow 1$.

Figures (5)

  • Figure 1: Values of (A) the real part and (B) the imaginary part of $\mathcal{J}_{m}(k)$ against the value of $kr_c$ for $m=1,2,3$. The dash-dotted lines in (A-B) indicate the real and imaginary parts of $\mathcal{I}_m$, respectively. The dash lines indicate $x\equiv |\alpha_m|$ for different $m$.
  • Figure 2: Absolute error as a function of the number of truncated terms, $M$, for different $b$, compared to the exact Madelung constant. The symbols indicate the corresponding errors by the u-series method, and the lines display theoretical estimates in this work.
  • Figure 3: Errors in calculations of an SPC/E pure water system using $C^0$-continuous u-series. Panel (A-B) display the relative errors of the energy and the force, respectively, as function of $M$. The dashed lines in both panels indicate $O(b^{-M})$ and $O(b^{-3M})$ scaling, respectively. Panel (C) plots the relative error regarding the energy and the force against the minimal index of Gaussians for $b=1.321$ and 1.218. Panel (D) shows the relative errors as function of $b$. The dash lines in (C-D) show the theoretical estimate using Eqs. \ref{['eq::final3']} and \ref{['eq::Uerr1']} for energy and Eqs. \ref{['eq::force1']} and \ref{['app::eqerr1']} for force.
  • Figure 4: The relative errors of energy and force as a function of the number of truncated terms, $M$, for different $b$ in series of simulations on a SPC/E pure water system. The dashed lines in (A) and (B) plot closed-form theoretical estimates of the energy and force using Eqs. \ref{['eq::deltaUerr']} and \ref{['eq::deltaFerrfinal']}.
  • Figure 5: The relative errors of energy and force in an SPC/E pure water system using $C^1$-continuous u-series. Panel (A) reveals the relative error regarding the energy and the force against the minimal index of Gaussians for $b=1.321$ and $b=1.218$. Panel (B) shows the relative errors as function of $b$. The dash lines in (C-D) show the theoretical estimate using Eqs. \ref{['eq::EgDownC1']} and \ref{['eq::Uerr1']} for energy and Eqs. \ref{['eq::FGDownC1']} and \ref{['app::eqerr1']} for force.

Theorems & Definitions (18)

  • Theorem 2
  • Lemma 3
  • Corollary 4
  • Proposition 5
  • Lemma 6
  • Proposition 7
  • Proposition 8
  • Proposition 9
  • Lemma 10
  • Theorem 11
  • ...and 8 more