Ewald s method

It is not feasible to evaluate the series (4.32) directly in fact, it does not tend to a unique value as q 0, but to a value which depends on the relative directions of q and the electrical polarization associated with the mode concerned. Fortunately, there is a very elegant procedure, known as EWALD s method [4.32], by which C(q) can be expressed as the sum of two rapidly convergent series, one involving a summation over a limited region in the reciprocal lattice and the other over a limited region in the direct lattice. Let be the potential energy of a point charge z e at a... 

Whereas dispersion forces are straightforward to model, this is not the case with electrostatic interactions. Coulombic interactions act at significantly longer distances, falling off with r instead of r , and require special treatment, with a method called Ewald summation. A discussion of Ewald s method is beyond the scope of this book, and the interested reader is referred to the text by Allen and Tildesley (Further reading). 

Gil-Villegas A, McGrother S C and Jackson G 1997 Reaction-field and Ewald summation methods in Monte Carlo simulations of dipolar liquid crystals Mol. Phys. 92 723-34... 

Ewald s formalism reduces the infinite lattice sum to a serial complexity of in the number of particles n, which has been reduced to n logn in more recent formulations. A review of variants on Ewald summation methods which includes a more complete derivation of the basic method is in [3]. 

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... 

Figure 3.41 Deviation from Bragg s Law diffraction illustrated by the Ewald sphere method. The deviation from exact Bragg conditions is represented by vector s in reciprocal space, ko is the primary beam vector, k is the scattering beam vector and g is a reciprocal lattice vector.

Weinert M 1981 Solution of Poisson s equation beyond Ewald-type methods J. Math. Phys. 22 2433... 

FIGURE S.17 The method of rotating crystals diffraction the diffraction principle (left-up) and diffraction pattern (left-down) along the Ewald s rationalization on the reflection sphere of diffraction (right) after (Matter Dif action, 2003 X-Rays, 2003 Putz Lacrama, 2005 HyperPhysics, 2010). 

An alternative to the Eyjen method is described by Ewald and formulated by Tosi. This method involves a summation over the reciprocal lattice and, unlike Eyjen s method, is valid over any point in the crystal, not just the lattice sites, and so is important for ealculations involving lattice defects or interstitial ions. Van Gool and Piken have developed a program for automatically calculating lattice energies and Madelung constants based on this method and have used this on compounds such as P-alumina. The Madelung term remains constant for a particular stmcture type. Some values are listed in Table 3.4. ... 

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]. 

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... 

---