In molecular dynamics (MD) calculations of the free energies of ions and ionic molecules, we often encounter net charged molecular systems where the electrical neutrality condition is broken. This charge causes a problem in the evaluation of long-range Coulombic interactions under periodic boundary conditions. A standard remedy for this problem is to consider a hypothetical homogeneous background charge density to neutralize the total system. Here, we present a new expression for the evaluation of Coulombic interactions for such systems including background charge using the fast multipole method (FMM). Furthermore, an efficient scheme is developed to evaluate solute–solvent interaction energies using the FMM, reducing the computational burden for the far-field part. We calculate the hydration free energies of Mg²⁺, Na⁺, and Cl⁻ ions dissolved in a neutral solvent using the new expression. The calculated free energies show good agreement with the results obtained using the well-established particle mesh Ewald method. This demonstrates the validity of the proposed expression. This work should make a contribution to highly parallelized MD calculations for large-scale charged systems (particularly, those with over million particles).

中文翻译：

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利用快速多极子方法精确计算被均匀分布背景电荷中和的净荷系统的远距离库仑能，并将其应用于有效的自由能计算。

在离子和离子分子的自由能的分子动力学（MD）计算中，我们经常会遇到电中性条件被破坏的净电荷分子系统。这种电荷会在周期性边界条件下评估长距离库仑相互作用时引起问题。解决此问题的标准方法是考虑假设的均匀背景电荷密度，以中和整个系统。在这里，我们提出了一种新的表达式，用于评估此类系统的库仑相互作用，包括使用快速多极方法（FMM）的背景电荷。此外，开发了一种有效的方案来使用FMM评估溶质与溶剂的相互作用能，从而减少了远场部分的计算负担。我们计算Mg ²⁺的水合自由能，钠⁺和Cl ^-溶解在使用新表达中性溶剂的离子。计算出的自由能与使用完善的粒子网格Ewald方法获得的结果显示出良好的一致性。这证明了所提出表达的有效性。这项工作应为大规模带电系统（特别是具有百万个粒子的系统）的高度并行的MD计算做出贡献。