Lekner summations and Ewald summations for quasi-two dimensional systems

Using the specific model of a bilayer of classical charged particles (bilayer Wigner crystal), we compare the predictions for energies and pair distribution functions obtained by Monte Carlo simulations using three different methods available to treat the long range Coulomb interactions in systems periodic in two directions but bound in the third one. The three methods compared are: the Ewald method for quasi-two dimensional systems [D.E. Parry, Surf. Sci. $\bm{49}$, 433 (1975); \it{ibid.}, $\bm{54}$, 195 (1976)], the Hautman-Klein method [J. Hautman and M.L. Klein, Mol. Phys. $\bm{75}$, 379 (1992)] and the Lekner summations method [J. Lekner, Physica A$\bm{176}$, 485 (1991)]. All of the three method studied in this paper may be applied to any quasi-two dimensional systems, including those having not the specific symmetry of slab systems. For the particular system used in this work, the Ewald method for quasi-two dimensional systems is exact and may be implemented with efficiency; results obtained with the other two methods are systematically compared to results found with the Ewald method. General recommendations to implement with accuracy, but not always with efficiency, the Lekner summations technique in Monte Carlo algorithms are given.