Ewald summation parameters

TABLE 8. Refinement factor R(%), mean electronegativities e 5 ) (eV) atomic sizes r(M) (pm), Mulliken electronegativities (eV), Ewald summation indexes p, Ewald summation parameters K (A ), Madelung energies M (eV), ionicity indexes / (%), minimum interatomic distances Rmin (pm) and Madelung constants Rmin) of the eleven pyrosilicates displayed in figure 8. ( ) Mg values (KBa) = 198 pm, exo (Ba) = 1.81 eV). 

The interaction parameters for the water molecules were taken from nonempirical configuration interaction calculations for water dimers (41) that have been shown to give good agreement between experimental radial distribution functions and simulations at low sorbate densities. The potential terms for the water-ferrierite interaction consisted of repulsion, dispersion, and electrostatic terms. The first two of these terms are the components of the 6-12 Lennard-Jones function, and the electrostatic term accounts for long-range contributions and is evaluated by an Ewald summation. The... 

Auerbach et al. (101) used a variant of the TST model of diffusion to characterize the motion of benzene in NaY zeolite. The computational efficiency of this method, as already discussed for the diffusion of Xe in NaY zeolite (72), means that long-time-scale motions such as intercage jumps can be investigated. Auerbach et al. used a zeolite-hydrocarbon potential energy surface that they recently developed themselves. A Si/Al ratio of 3.0 was assumed and the potential parameters were fitted to reproduce crystallographic and thermodynamic data for the benzene-NaY zeolite system. The functional form of the potential was similar to all others, including a Lennard-Jones function to describe the short-range interactions and a Coulombic repulsion term calculated by Ewald summation. 

The influence of a cut-off relative to the full treatment of electrostatic interactions by Ewald summation on various water parameters has been investigated by Feller et al. [33], These authors performed simulations of pure water and water-DPPC bilayers and also compared the effect of different truncation methods. In the simulations with Ewald summation, the water polarization profiles were in excellent agreement with experimental values from determinations of the hydration force, while they were significantly higher when a cut-off was employed. In addition, the calculated electrostatic potential profile across the bilayer was in much better agreement with experimental values in case of infinite cut-off. However, the values of surface tension and diffusion coefficient of pure water deviated from experiment in the simulations with Ewald summation, pointing out the necessity to reparameterize the water model for use with Ewald summation. 

Descriptions of the method and its physical background are given by Bertaut, by Jackson and Catlow, and by Tosi. Jackson and Catlow and Karasawa and Goddard have deduced optimal values of the parameters for the summations in direct and reciprocal spaces. Computer code for the Ewald summation is available from several sources. 

The long-range Coulomb interactions were treated using the 3D Ewald summation method with Ewald convergence parameter a = 0.284 A-1 and an Ewald sum precision of 1 10-5 (from the standard in the DL POLY package [35]). 

Here qi is the effective charge of an atom is a dispersion interaction constant and Ay and bij are parameters of the Born - Mayer atom-atom repulsion potential. To calculate the long range Coulomb term in Eq. (1) one generally has to employ the Ewald summation technique. To obviate this inconvenience, the Coulomb term has been multiplied by the screening factor (and the dispersion term has been neglected) ... 

The inverse length a is the splitting parameter of the Ewald summation which should be chosen so as to optimize the performance. The form Eq. 83 given for the surface correction assumes that the set of the periodic replications of the simulation box tends in a spherical way toward an infinite cluster, and that the medium outside this sphere is a uniform dielectric with dielectric constant s [7,29]. The case of a surrounding vacuiun corresponds to e = 1 and the surface term vanishes for the metallic boimdary conditions (s = oo). 

In the case of 5 < 5 x 10", Eq. 105 could be used instead of Eq. 100 to show the dependence of the accuracy on the parameters more clearly. Equations 119 and 120 provide the qualitative function relations of Tc and kc with a as rda) - AVln S/a and kc a) - uVln 6a. Inserting them into Eq. 118 and differentiating it with respect to a yields a cx and thus rc a and kc oc The minimized computation time is then proportional to with the proportionality constant depending on the accuracy. The same results can be found for the Coulomb Ewald method in Refs. [9,35,46]. This can be easily understood by comparing Eqs. 100 and 112 in Sect. 3 of this paper with Eqs. 18 and 32 in Ref. [35], and finding the same exponential dependences of the cutoff errors on a, rc, and kc for the dipolar and Coulomb Ewald summations. 

In the implementation of the Ewald summation according to Eq. 20, the value of the potential energy is controlled by three parameters a, the upper limit of m (ntcut), and the upper limit of n (ncut). At equal truncation error in the two spaces, the summation in the real space is often Umited to interactions involving only the nearest image (m = 0), and consequently a spherical cutoff distance i cut < in the real space can be applied. Moreover, the number of replicas in reciprocal space can be reduced by applying a spherical cutoff of n according to jnj < cut. 

The role of the different parameters controlling the Ewald summation will now be considered in more detail. A single equilibrium configuration of System IV with Nm = 80 macroions will be used, but the outcome does not critically depend on the choice of system or configuration. Values of the potential energy for different truncations of the Ewald summation will be compared with essentially the exact value, the latter obtained from an Ewald siunmation with large values of J cut and cut. 

The electrostatic energy is the dominant term for many inorganic materials, particularly oxides, and therefore it is important to evaluate it accurately. For small- to moderate-sized systems this is most efficiently achieved through the Ewald summation (Ewald 1921) in which the inverse distance is rewritten as its Laplace transform and then split into two rapidly convergent series, one in reciprocal-space and one in real-space. The distribution of the summation between real- and reciprocal-space is controlled by a parameter t]. The resulting expression for the energy is ... 

The conventional Ewald summation method works well for simulations of small periodic systems, but the computation can become prohibitively expensive when large systems are involved, in which the particle number exceeds lO. Several numerical techniques have been used to enhance the performance of the traditional Ewald method with mixed results. For example, look-up tables and polynomial approximations have been suggested. The algorithmic performance can also be optimized through the parameter 1 113,114 determines both the extension of the short-range interaction... 

The Na-AOT reverse micelle is a widely investigated reverse micelle system made up of the sodium salt of a two-tailed anionic surfactant, sodium di(2-ethylhexyl) sulfosuccinate. The interior of the aqueous reverse micelle is modeled as a rigid cavity, with a united atom representation for the sulfonate head group (Faeder and Ladanyi 2000 Pal et al. 2005). The head groups protrude from the cavity boundary and are tethered only in the radial direction by means of a harmonic potential. Interactions between reverse micelles are neglected in the model hence periodic boundary conditions and Ewald summations for the electrostatics are not required. Water is treated using the extended simple point charge, or SPC/E, model and the potential parameters for all the species are listed in Table 6.1. 

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