Simulation of electrostatic systems in periodic boundary conditions. I. Lattice sums and dielectric constants

The effective interactions of ions, dipoles and higher-order multipoles under periodic boundary conditions are calculated where the array of periodic replications forms an infinite sphere surrounded by a vacuum. Discrepancies between the results of different methods of calculation are resolved and some shape-dependent effects are discussed briefly. In a simulation under these periodic boundary conditions, the net Hamiltonian contains a positive term proportional to the square of the net dipole moment of the configuration. Surrounding the infinite sphere by a continuum of dielectric constant $\epsilon'$ changes this positive term, the coefficient being zero as $\epsilon'\rightarrow\infty$. We report on the simulation of a dense fluid of hard spheres with embedded point dipoles; simulations are made for different values of $\epsilon'$, showing how the Kirkwood g-factor and the long-range part of h$_\Delta$(r) depend on $\epsilon'$ in a finite simulation. We show how this dependence on $\epsilon'$ nonetheless leads to a dielectric constant for the system that is independent of $\epsilon'$. In particular, the Clausius-Mosotti and Kirkwood formulae for the dielectric constant $\epsilon$ of the system give consistent values.