标题： 无限边界项与成对相互作用：周期性库仑系统的统一框架 显示英文标题

标题： Infinite Boundary Terms and Pairwise Interactions: A Unified Framework for Periodic Coulomb Systems

摘要： 无限边界项和成对相互作用的引入[J. Chem. Theory Comput., 10, 5254, (2014)]使得在周期性边界条件（PBC）下，对于中性和非中性系统，可以采用一种物理直观的方法来推导静电能和压力。 对于由$N$个点电荷组成的周期性系统（电荷为$q_j$，位于${\mathbf r}_j$处，其中$j=1,2,\cdots N$），以及在一个体积为$V$的原胞内密度为$\rho({\mathbf r})$的电荷分布，推导出的静电能可以表示为，\[ {\mathcal U} = \sum_{i<j}^N q_iq_j\nu({\mathbf r}_{ij} ) + \sum_{j=1}^N q_j \int_V d{\mathbf r}_0\,\rho({\mathbf r}_0) \nu({\mathbf r}_{0j} ) + \frac{1}{2}\int_V d{\mathbf r}_0 \int_V d{\mathbf r}_1\,\rho({\mathbf r}_0)\rho({\mathbf r}_1) \nu({\mathbf r}_{01}), \]，其中${\mathbf r}_{ij}={\mathbf r}_i - {\mathbf r}_j$是相对向量，$\nu({\mathbf r})$表示在PBC下的有效成对相互作用。 电荷密度$\rho({\mathbf r})$在整个体积内没有类似狄拉克δ函数的发散，但可能会出现不连续性。这种统一的公式直接来源于孤立系统的公式，通过将库仑相互作用$1/\lvert {\mathbf r} \rvert$或其他修改后的库仑相互作用替换为$\nu({\mathbf r})$而得到。对于具有均匀中和背景的一组分等离子体系统，各种成对公式的实现明确了背景的贡献，并随后揭示了设计保持能量与压力之间简单关系的体积依赖势能的标准。 显示英文摘要

摘要： The introduction of the infinite boundary terms and the pairwise interactions [J. Chem. Theory Comput., 10, 5254, (2014)] enables a physically intuitive approach for deriving electrostatic energy and pressure for both neutral and non-neutral systems under the periodic boundary condition (PBC). For a periodic system consisting of $N$ point charges (with charge $q_j$ located at ${\mathbf r}_j$ where $j=1,2,\cdots N$) and one charge distribution of density $\rho({\mathbf r})$ within a primary cell of volume $V$, the derived electrostatic energy can be expressed as, \[ {\mathcal U} = \sum_{i<j}^N q_iq_j\nu({\mathbf r}_{ij} ) + \sum_{j=1}^N q_j \int_V d{\mathbf r}_0\,\rho({\mathbf r}_0) \nu({\mathbf r}_{0j} ) + \frac{1}{2}\int_V d{\mathbf r}_0 \int_V d{\mathbf r}_1\,\rho({\mathbf r}_0)\rho({\mathbf r}_1) \nu({\mathbf r}_{01}), \] where ${\mathbf r}_{ij}={\mathbf r}_i - {\mathbf r}_j$ is the relative vector and $\nu({\mathbf r})$ represents the effective pairwise interaction under PBC. The charge density $\rho({\mathbf r})$ is free of Delta-function-like divergence throughout the volume but may exhibit discontinuity. This unified formulation directly follows that of the isolated system by replacing the Coulomb interaction $1/\lvert {\mathbf r} \rvert$ or other modified Coulomb interactions with $\nu({\mathbf r})$. For a particular system of one-component plasma with a uniform neutralizing background, the implementation of various pairwise formulations clarifies the contribution of the background and subsequently reveals criteria for designing volume-dependent potentials that preserve the simple relation between energy and pressure.