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

The introduction of the infinite boundary terms and the pairwise interactions [Hu, Z. J. Chem. Theory Comput. 2014, 10, 5254-5264] 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 r_j where j = 1, 2, ···, N) and one charge distribution of density ρ(r) within a primary cell of volume V, the derived electrostatic energy can be expressed as, Σ_i<j^Nq_iq_j ν(r_ij) + Σ_j=1^Nq_j∫_Vdr₀ρ(r₀)ν(r_0j) + 1/2∫_Vdr₀∫_Vdr₁ρ(r₀)ρ(r₁)ν(r₀₁), where r_ij = r_i - r_j is the relative vector and ν(r) represents the effective pairwise interaction under PBC. The charge density ρ(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/|r| or other modified Coulomb interactions with ν(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.