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Random Batch Sum-of-Gaussians Method for Molecular Dynamics of Born-Mayer-Huggins Systems

Chen Chen, Jiuyang Liang, Zhenli Xu, Qianru Zhang

TL;DR

The paper tackles the high computational cost of molecular dynamics with the Born-Mayer-Huggins potential by introducing IRBSOG, a unified, linear-scaling framework that combines a sum-of-Gaussians (SOG) decomposition of the Coulomb kernel with random-batch approaches for both short- and long-range interactions. It partitions the energy into a short-range component $U_{\mathcal{N}}$ and a long-range component $U_{\mathcal{F}}$, computes the long-range via stochastic Fourier-space sampling and the short-range via a core-shell random-batch neighbor list, yielding unbiased force estimators with variance decreasing as $O(1/P_R+1/P_F)$. The method delivers substantial performance gains over traditional PPPM and RBSOG approaches (up to about $4\sim10\times$ and $\sim2\times$, respectively) on large-scale systems with up to $5\times10^6$ atoms on thousands of CPU cores, while maintaining accuracy in structure and transport properties for molten NaCl and mixed alkali halide systems. These results demonstrate the practicality of IRBSOG for scalable MD of ionic materials and point toward extensions to quasi-two-dimensional interfaces and dielectric-mismatch scenarios.

Abstract

The Born-Mayer-Huggins (BMH) potential, which combines Coulomb interactions with dispersion and short-range exponential repulsion, is widely used for ionic materials such as molten salts. However, large-scale molecular dynamics simulations of BMH systems are often limited by computation, communication, and memory costs. We recently proposed the random batch sum-of-Gaussians (RBSOG) method, which accelerates Coulomb calculations by using a sum-of-Gaussians (SOG) decomposition to split the potential into short- and long-range parts and by applying importance sampling in Fourier space for the long-range part. In this work, we extend the RBSOG to BMH systems and incorporate a random batch list (RBL) scheme to further accelerate the short-range part, yielding a unified framework for efficient simulations with the BMH potential. The combination of the SOG decomposition and the RBL enables an efficient and scalable treatment of both long- and short-range interactions in BMH system, particularly the RBL well handles the medium-range exponential repulsion and dispersion by the random batch neighbor list. Error estimate is provided to show the theoretical convergence of the RBL force. We evaluate the framework on molten NaCl and mixed alkali halide with up to $5\times10^6$ atoms on $2048$ CPU cores. Compared to the Ewald-based particle-particle particle-mesh method and the RBSOG-only method, our method achieves approximately $4\sim10\times$ and $2\times$ speedups while using $1000$ cores, respectively, under the same level of structural and thermodynamic accuracy and with a reduced memory usage. These results demonstrate the attractive performance of our method in accuracy and scalability for MD simulations with long-range interactions.

Random Batch Sum-of-Gaussians Method for Molecular Dynamics of Born-Mayer-Huggins Systems

TL;DR

The paper tackles the high computational cost of molecular dynamics with the Born-Mayer-Huggins potential by introducing IRBSOG, a unified, linear-scaling framework that combines a sum-of-Gaussians (SOG) decomposition of the Coulomb kernel with random-batch approaches for both short- and long-range interactions. It partitions the energy into a short-range component and a long-range component , computes the long-range via stochastic Fourier-space sampling and the short-range via a core-shell random-batch neighbor list, yielding unbiased force estimators with variance decreasing as . The method delivers substantial performance gains over traditional PPPM and RBSOG approaches (up to about and , respectively) on large-scale systems with up to atoms on thousands of CPU cores, while maintaining accuracy in structure and transport properties for molten NaCl and mixed alkali halide systems. These results demonstrate the practicality of IRBSOG for scalable MD of ionic materials and point toward extensions to quasi-two-dimensional interfaces and dielectric-mismatch scenarios.

Abstract

The Born-Mayer-Huggins (BMH) potential, which combines Coulomb interactions with dispersion and short-range exponential repulsion, is widely used for ionic materials such as molten salts. However, large-scale molecular dynamics simulations of BMH systems are often limited by computation, communication, and memory costs. We recently proposed the random batch sum-of-Gaussians (RBSOG) method, which accelerates Coulomb calculations by using a sum-of-Gaussians (SOG) decomposition to split the potential into short- and long-range parts and by applying importance sampling in Fourier space for the long-range part. In this work, we extend the RBSOG to BMH systems and incorporate a random batch list (RBL) scheme to further accelerate the short-range part, yielding a unified framework for efficient simulations with the BMH potential. The combination of the SOG decomposition and the RBL enables an efficient and scalable treatment of both long- and short-range interactions in BMH system, particularly the RBL well handles the medium-range exponential repulsion and dispersion by the random batch neighbor list. Error estimate is provided to show the theoretical convergence of the RBL force. We evaluate the framework on molten NaCl and mixed alkali halide with up to atoms on CPU cores. Compared to the Ewald-based particle-particle particle-mesh method and the RBSOG-only method, our method achieves approximately and speedups while using cores, respectively, under the same level of structural and thermodynamic accuracy and with a reduced memory usage. These results demonstrate the attractive performance of our method in accuracy and scalability for MD simulations with long-range interactions.
Paper Structure (11 sections, 3 theorems, 51 equations, 5 figures, 3 tables)

This paper contains 11 sections, 3 theorems, 51 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Method
  3. SOG decomposition of the BMH potential
  4. The IRBSOG method
  5. Convergence analysis of IRBSOG-based simulations
  6. Implementation details and complexity analysis
  7. Numerical Examples
  8. Accuracy comparison in molten NaCl systems
  9. Time performance in large-scale simulations
  10. Tests on molten alkali binary halide systems
  11. Conclusion

Key Result

Lemma 2.1

Under the Debye-Hückel (DH) assumption, the variance of the long-range force approximation $\bm{F}_{\mathcal{F},i}^{*}$ satisfies where $C$ is the constant in the DH bound, $q_i$ is the charge of particle $i$, $V$ is the system volume, $P_F$ is the Fourier-space batch size, and $w_m$ are the Gaussian weights used in Eq. (7). Consequently, $\operatorname{var}[\bm{F}_{\mathcal{F},i}^{*}] = O(1/P_F)

Figures (5)

  • Figure 1: Schematic of the IRBSOG method and the Born-Mayer-Huggins (BMH) potential decomposition. The total BMH potential (red solid line) is decomposed into components handled by distinct algorithms. The Coulomb potential is split into a short-range part $\mathcal{N}_{b}^{s}(r)$ (orange dashed line) and a long-range part $\mathcal{F}_{b}^{s}(r)$ (green dot-dashed line). The short-range non-Coulombic BMH terms $\mathcal{B}_{\text{BMH}}(r)$ (blue dot-dashed line) is also illustrated. The upper inset shows the core-shell structure of short-range and the lower one shows the importance sampling of long-range.
  • Figure 2: (a) Na MSD and (b) Na-Na RDF of molten NaCl systems governed by the BMH potential. IRBSOG results with shell-region batch size $P_R\in\{20, 30\}$ and Fourier-space batch size $P_F\in\{50, 100\}$ are compared with PPPM. Panels (c) and (d) compare viscosity and thermal conductivity of NaCl at $1100~K$ and $1500~K$: IRBSOG with $P_R=30$, $P_F=100$ (orange) versus PPPM (blue).
  • Figure 3: The CPU time per step of the PPPM, RBSOG with $P_F=100$ and IRBSOG with $P_R=30$, $P_F=100$ against increasing number of particles using 1000 CPU Cores. The dashed lines show the linear fitting of data.
  • Figure 4: CPU time and strong/weak scalability for the IRBSOG and PPPM, using up to $16$ nodes with 128 CPU core per node. (a-b) present the strong scalability results with a fixed total particle number of $221184$. (c-d) present the weak scalability results with an average of $1024$ particles per core.
  • Figure 5: The RDFs of Cl-Cl(a), Li-Li(b), Na-Na(c), Cl-Li(d) and Na-Cl(e), and the visual model of LiCl-NaCl(f) at 1100 K where the purple, red and green balls represent Na, Li and Cl, respectively. We use the IRBSOG with $P_R=30$ and $P_F=100$ compared to the PPPM.

Theorems & Definitions (4)

  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3