Ewald parameter

Regarding the use of Eq. (6.26) in practice we note that the same comments made earlier apply here as well [see discussion after Eq. (6.17)]. A detailed discussion of optimal choices for the Ewald parameters a and for dipolar systems can bo found in Rc fs. 243 and 244. Finally, readers who are interested in performing MD simulations of dipolar fluids are referred to Appendix F.2.2 where we present explicit expressions for forces and torques associated with the three-dimensional Ewald sum [see Eq. (6.26)]. Moreover, explicit expressions for various components of the stress tensor can be found in Appendix F.2.3. 

Calculations were performed with the Ewald parameters nfi = 7.0 and m = 80. 

But the methods have not really changed. The Verlet algorithm to solve Newton s equations, introduced by Verlet in 1967 [7], and it s variants are still the most popular algorithms today, possibly because they are time-reversible and symplectic, but surely because they are simple. The force field description was then, and still is, a combination of Lennard-Jones and Coulombic terms, with (mostly) harmonic bonds and periodic dihedrals. Modern extensions have added many more parameters but only modestly more reliability. The now almost universal use of constraints for bonds (and sometimes bond angles) was already introduced in 1977 [8]. That polarisability would be necessary was realized then [9], but it is still not routinely implemented today. Long-range interactions are still troublesome, but the methods that now become popular date back to Ewald in 1921 [10] and Hockney and Eastwood in 1981 [11]. 

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

The virial coefficients at 190°K have been calculated from Ewald s results22 and may be combined with the measurements at room temperature of Michels and Boerboom,47 of Cottrell and his colleagues,11 12 and of Harper and Miller,32 to give the parameters for helium + carbon dioxide... 

Data for which no reference is given are from the Slrukturbericht of P. P. Ewald and C. Hermann. 6 R. W. G. Wyckoff, Z. Krisl., 75,529 (1930). W. H. Zachariasen, ibid., 71, 501, 517 (1929). d The very small paramagnetic susceptibility of pyrite requires the presence of electron-pair bonds, eliminating an ionic structure Fe++S2. Angles are calculated for FeS2, for which the parameters have been most accurately determined. The parameter value (correct value = 0.371) and interatomic distances for molybdenite are incorrectly given in the Slrukturbericht. 

Figure 2. Ewald construction for X-ray (soUd sphere) and electron (dotted sphere). ( kO, k wave-vectors, X - wave-length, a, b - parameters of reciprocal unit cell).

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. 

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