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

Abstract

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 $\ddot{I}(\mathbf{r})$ within a primary cell of volume $V$, the derived electrostatic energy can be expressed as, \[ {\mathcal U} = \sum_{i<j}^N q_iq_jν({\mathbf r}_{ij} ) + \sum_{j=1}^N q_j \int_V d{\mathbf r}_0\,ρ({\mathbf r}_0) ν({\mathbf r}_{0j} ) + \frac{1}{2}\int_V d{\mathbf r}_0 \int_V d{\mathbf r}_1\,ρ({\mathbf r}_0)ρ({\mathbf r}_1) ν({\mathbf r}_{01}), \] where ${\mathbf{r}}_{ij}={\mathbf{r}}_i-{\mathbf{r}}_j$ is the relative vector and $\hat{I}½(\mathbf{r})$ represents the effective pairwise interaction under PBC. The charge density $\ddot{I}(\mathbf{r})$ 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 $1/|\mathbf{r}|$ or other modified Coulomb interactions with $\hat{I}½(\mathbf{r})$. 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.

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