Simulation of electrostatic systems in periodic boundary conditions. II. Equivalence of boundary conditions

We consider simulations of dipolar systems under periodic boundary conditions in which a large sphere consisting of periodic replications of a central simulation cell is surrounded by a continuum of dielectric constant $\epsilon'$. We develop a perturbation theory expressing correlation functions with $\epsilon''$ in terms of correlation functions with $\epsilon'$ exactly to order N$^{-1}$, N being the number of particles in the sample. In the thermodynamic limit, the correlation functions and internal energy density are independent of $\epsilon'$. The Kirkwood g-factor is strongly dependent on $\epsilon'$ but in such a way as to make the dielectric constant independent of $\epsilon'$. The dependence upon $\epsilon'$ of h$_\Delta$(r) at large r, described in paper I, is explained in terms of the perturbation series.