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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 ε.' changes this positive term, the coefficient being zero as ε' ->∞ . We report on the simulation of a dense fluid of hard spheres with embedded point dipoles; simulations are made for different values of showing how the Kirkwood gr-factor and the long-range part of hA (r) depend on ε' in a finite simulation. We show how this dependence on ε' nonetheless leads to a dielectric constant for the system that is independent of ε . In particular, the Clausius-Mosotti and Kirkwood formulae for the dielectric constant e of the system give consistent ε values.

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