Lekner summations and Ewald summations for quasi-two-dimensional systems

Abstract

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. 49, 433 (1975); ibid., 54, 195 (1976)], the Hautman-Klein method [J. Hautman and M.L. Klein, Mol. Phys. 75, 379 (1992)] and the Lekner summations method [J. Lekner, Physica A176, 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.

Yukawa potentials in systems with partial periodic boundary conditions. II. Lekner sums for quasi-two-dimensional systems

“…(15). Taking into account that the energy of Coulomb systems computed with Ewald sums or with Lekner sums agree very well [7], then the result obtained in Eq. (23) is fully equivalent to the result obtained with Ewald sums.…”

“…As outlined before for Coulomb potentials [5,6,7], the uses of Lekner sums to obtain interaction energies (or forces in molecular dynamics implementations) have to be done cautiously. For some configurations of particles, the convergence of sums may be very slow [7] and many contributions have to be included to obtain rather accurate results and to avoid bias in the sampling of the phase space [5].…”

“…(8) and (9), one substitutes x to y and L x to L y then the energy E cc (Yukawa; κ) has to remain unchanged (for a detailed discussion and numerical tests about Coulomb potential, see refs. [7,5]) ; in the following, when substitutions…”

“…This last property of Lekner sums introduces some complications in numerical implementations of Lekner sums. There are mainly two methods to improve the convergence rate of these summations : the Lekner-cyclic [4,7] and Lekner-Sperb [8,9,10] methods.…”

“…(8) and (9) have to be truncated to some finite values p c and m c , therefore, in numerical implementations of Lekner sums, for both Coulomb and Yukawa potentials, one should take care of the slow convergence of the sums over the Bessel functions for some particular configurations of the system of particles [7,5,6]. There are mainly two ways to counter the slow convergence behaviour of the Lekner sums, Lekner-Sperb [10] and the Lekner-cyclic methods [7].…”