Ewald term

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]. 

In order to And rjopt, we minimise an expression of the total CPU time, T, required for the Ewald sums. We assume that T is proportional to the number of vectors used in both the reciprocal and direct space sums, G /(2j ) and R /a . and to t, and t<, the CPU times required for the evaluation of a single term in each series. In formulae. 

In this section we describe the methods to extend Ewald sum methodologies to accelerate the calculation of the intermolecular interactions using PBC. For simplicity, we begin with a generalization of Ewald sums to interacting spherical Hermite Gaussians (e.g. GEM-0 [14]). This is followed by the extension to arbitrary angular momentum. Finally, we describe the implementation of methods to speed up both the direct an reciprocal terms in the Ewald sum [62],... 

The fourth term on the right-hand side of eq. (11.3) is the electrostatic interaction (Coulomb s law) between pairs of charged atoms i and j, separated by distance r j. Since electrostatic interactions fall off slowly with r (only as r-1) they are referred to as long-range and, for an infinite system such as a periodic solid, special techniques, such as the Ewald method, are required to sum up all the electrostatic interactions (cf. Section 7.1) (see e.g. Leach, Jensen (Further reading)). The... 

These two systems are examples from non-linear physics, where the equations can be solved in terms of elliptic functions and elliptic integrals. The reader who is not familiar with these functions, which do not arise in the same way as the previously mentioned special functions, is referred to the excellent book by Whittaker and Watson [6]. In that book, the reader will see that there are two flavours of elliptic functions, the Weierstrass and Jacobi representations, three kinds of elliptic integrals, and six kinds of pseudo-periodic functions, the Weierstrass zeta and sigma functions and the four kinds of Jacobi theta functions. Of historical interest for theoretical chemists is the fact that Jacobi s imaginary transformation of the theta functions is the same as the Ewald transformation of crystal physics [7]. 

Now, in the TF-HK equation, all the potential terms can be set up by conventional plane-wave-basis teehniques with essentially linear scaling. However, for very large systems with more than 5000 nuclei, the computational cost associated with the nuelear-nuelear Coulomb repulsion energy becomes the major bottleneck." In this case, linear-scaling Ewald sirmmation techniques should be utilized. 

For coupling purposes the QM particles are separated into two sets — those which are near the QM center and those located close to the QM/MM border. For QM atoms close to the center the intermolecular distances to MM particles are typically larger than the non-Coulombic cutoff distances and, therefore, these atoms only require a Coulombic term to account for the coupling between the QM and the MM particles. A correction term compensating for the Coulombic cutoff such as Ewald summation or a reaction field is typically applied. Atoms close to the interface region have small intermolecular distances and consequently, non-Coulombic interactions have to be included in addition to the Coulombic forces to achieve a proper coupling. 

As already noted in Chapter 2, for electrostatic interactions, Ewald sums are generally to be preferred over cut-offs because of tlie long-range nature of the interactions. For van der Waals type terms, cut-offs do not introduce significant artifacts provided they are reasonably large (typically 8-12 A). 

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. 

Uf all the different types of atomic aggregates, ionic crystals have been found to be most suited to simple theoretical treatment. The theory of the structure of ionic crystals described briefly in the following sections was developed about 40 years ago by Born, Haber, Land6, Madelung, Ewald, Fajans, and other investigators. The simplicity of the theory is due in part to the importance in the interionic interactions of the well-understood Coulomb terms and in part to the spherical symmetry of the electron distributions of the ions with noble-gas configurations. 

Following Fraser et al. (4), we choose to represent the scattered intensity in terms of a cylindrically symmetric "specimen intensity transform" I (D), where D is a position vector in reciprocal space. Figure 10 shows the Ewald sphere construction, the wavelength of the radiation being represented by X. The angles p and X define the direction of the diffracted beam and are related to the reciprocal-space coordinates (R, Z) and the pattern coordinates (u,v) as follows ... 

The first section recalls the Frenkel-Davydov model in terms of a set of electromagnetically coupled point dipoles. A compact version of Tyablikov s quantum-mechanical solution is displayed and found equivalent to the usual semiclassical theory. The general solution is then applied to a 3D lattice. Ewald summation and nonanalyticity at the zone center are discussed.14 Separating short and long-range terms in the equations allows us to introduce Coulomb (dipolar) excitons and polaritons.15,16 Lastly, the finite extent of actual molecules is considered, and consequent modifications of the above theory qualitatively discussed.14-22... 

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