Simulation of two dimensional systems with two dimensional electrostatics

Some of the problems associated with Monte Carlo and molecular dynamics simulations of three dimensional ionic and dipolar systems are discussed, with emphasis on the use of periodic boundary conditions (PBC). It is shown that analogous problems may arise in two dimensional systems provided that the interactions are two dimensional electrostatic interactions, that is, interactions derived from the two dimensional Laplace equation. The PBC hamiltonian is evaluated by considering the appropriate two dimensional lattice sums, and a computable form for the effective pair interactions in PBC developed. The idea of an external dielectric constant is introduced and its effects included in the PBC hamiltonian. Formulae for evaluating the dielectric constant from a simulation with any external dielectric constant are given. Perturbation formulae showing the effects on the structure and mean square dipole moment of a dipolar system which are caused by a change in external dielectric constant are derived. A formula for the shift in mean square dipole moment of an ionic system is also developed. The problem of interpreting results from such simulations is discussed briefly.