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.
Categories: physics.comp-ph physics.chem-ph physics.plasm-ph