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

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, $$\mathcal{U}=\sum _{i<j}^N{q}_i{q}_j\nu ({\mathbf{r}}_{ij})+\sum _{j=1}^N{q}_j{\int }_{V}d{\mathbf{r}}_0\rho ({\mathbf{r}}_0)\nu ({\mathbf{r}}_{0j})+\frac{1}{2}{\int }_{V}d{\mathbf{r}}_0{\int }_{V}d{\mathbf{r}}_1\rho ({\mathbf{r}}_0)\rho ({\mathbf{r}}_1)\nu ({\mathbf{r}}_{01}),$$ 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 $1\mathrm{/}\mid \mathbf{r}\mid$ 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.