Ewald summation rigorous

Thus, for both the ionic and the dipolar systems, the actual use of the rigorously derived Ewald summation for slab systems loads to a substantial increase in computer time. One way of dealing with this problem would be to employ precalculated tables [252] for potential energies (and forces) on a three-dimensional spatial grid amended by a suitable interpolation scheme. Another strategy is to employ approximate methods such as the one presented in the subsequent Section 6.3.2. 

Similar expressions are obteiined for case 1. Data plotted in Figs. 6.2 have been obtained by truncating the sums over Z and ly at 5000, which yields convergent results as long as n, < 4. Comparing these results with those from the two Ewald methods, we conclude that not only the rigorous Ewald summation, but also the slab-adapted three-dimensional version provide quasi-exact results for th( dipolar energy. 

A more rigorous approach is the Ewald summation. This method, first presented by P. P. Ewald in 1921, exploits the periodicity of the system by calculating part of the summations in reciprocal space. When this method is used, the energy converges much faster, and a more accurate result can be obtained. 

Coulomb interactions show critically poor convergence properties as a function of distance (i.e., 1/r interactions). Interaction cutoffs have shown prone to artifacts and motivated the development of long-range electrostatic methods, such as Ewald summation (see, e.g.. Reference [25] and references therein). A number of Ewald summation methods have been extended to MTPs (e.g., [43, 54, 120]), providing a rigorous treatment of electrostatics in MD simulations. 

Our goal in this chapter is to present a simple and physically meaningful derivation of various Ewald summation techniques. For a mathematically more rigorous presentation, we refer the reader to the original papers by de Leeuw et al. [239-241],... 

Figure 6.2 Dimensionless energy per particle for dipolar crystalline (fee) slabs as a function of the number of lattice layers, assuming perfect order along the ar-axis (a) and along the z-axis (b). Included are results from direct summation (O), the rigorous Ewald sum for slab systems (A) (sec Appendix F.3.1.2], and the slab-adapted three-dimensional Ewald sum (x) [see Eq. (6.44)j. Part (b) additionally includes results from the latter method when the correction term [see Eq. (6.43)) is neglected ( ).

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