Three-dimensional Ewald summation

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. 

Equation (6.40) is our final expression for the energy of a charged system between insulating walls within the slab-adapted three-dimensional Ewald summation method. On the right side, the contribution f/ le =oo is defined by Eqs. (6.15), (6.16), and (6.17b). The reader should also realize that, for the current system, the volume V appearing in the Fourier pjurt of the energy [see Eq. (6.16b)] includes the vacuum space that is, V = = 7S -... 

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 ( ).

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. 

Having understood the concepts of Ewald summation techniques for three-dimensional bulk systems, we now turn to systems that are finite in at least one spatial dimension. We focus on a slab-like geometry where the fluid is confined by two plane parallel and structureless solid surfaces separated by a distance s along the z-axis of the coordinate system and of infinite extent in the x-y plane (see also Section 1.3.2). Hence, for the time being, we shall be... 

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