Ewald approximation

Fig. 2. Contrast between (a) realistie polarization of the surroundings by a fluctuation dipole moment in the central box and (b) that corresponding to the Ewald approximation.

Fig. 3. It is evident from symmetry that the effeetive force between the two particles, under the Ewald approximation, exactly vanishes for this configuration of the two particles within the central box.

In the case of the reciprocal sum, two methods have been implemented, smooth particle mesh Ewald (SPME) [65] and fast Fourier Poisson (FFP) [66], SPME is based on the realization that the complex exponential in the structure factors can be approximated by a well behaved function with continuous derivatives. For example, in the case of Hermite charge distributions, the structure factor can be approximated by... 

The tangent plane approximation is valid the curvature of the Ewald sphere is negligible at small scattering angles. 

Geometrically, electron diffraction patterns of crystals can be approximated as sections of the reciprocal lattice, since the Ewald sphere can be regarded as a plane (i.e. the radius of the Ewald sphere, 1/2, is much larger than the lengths of low-index reciprocal lattice vectors). 

The deviation of the electron beam from the Bragg condition is measured by the distance from the reciprocal lattice vector to the Ewald sphere along the zone axis direction, which approximately is defined by... 

An analogous implementation for the standard Ewald method has been presented [44]. Conversely, direct use of the Ewald sum [45] or approximations to it [46 18], which are pairwise decomposable and hence suitable for MC simulations, have generally proven to be too inefficient for most modern applications [49]. Additionally, it should be pointed out that Ewald sums — independent of implementation — are incompatible with implicit solvent models that model a spatially varying dielectric with anything more than trivial functional dependencies [45]. 

Of course, concerns about periodicity only relate to systems that are not periodic. The discussion above pertains primarily to the simulations of liquids, or solutes in liquid solutions, where PBCs are a useful approximation that helps to model solvation phenomena more realistically than would be the case for a small cluster. If the system truly is periodic, e.g., a zeolite crystal, tlien PBCs are integral to the model. Moreover, imposing PBCs can provide certain advantages in a simulation. For instance, Ewald summation, which accounts for electrostatic interactions to infinite length as discussed in Chapter 2, can only be carried out within the context of PBCs. 

In the first part of this introductory section, we summarize the main collective phenomena acquired by the dipolar exciton from the lattice-symmetry collectivization of molecular properties. The crystal is considered as an assembly of electrically neutral systems, the molecules, physically separated from each other and in electromagnetic interaction. This /V-body problem will be treated quantum-mechanically in the limit of low exciton densities. We redemonstrate the complete equivalence of this treatment with the theories of Lorentz and Ewald, as well as with the semiclassical approximation. In Section I.A, in a more compact but still gradual way, we establish the model of the rigid lattice of dipoles and the general theory of low-exciton-density systems in interaction with the radiation field. Coulombic excitons, photons,... 

The motions of proteins are usually simulated in aqueous solvent. The water molecules can be represented either explicitly or implicitly. To include water molecules explicitly implies more time-consuming calculations, because the interactions of each protein atom with the water atoms and the water molecules with each other are computed at each integration time step. The most expensive part of the energy and force calculations is the nonbonded interactions because these scale as 77 where N is the number of atoms in the system. Therefore, it is common to neglect nonbonded interactions between atoms separated by more than a defined cut-off ( 10 A). This cut-off is questionable for electrostatic interactions because of their 1/r dependence. Therefore, in molecular dynamics simulations, a Particle Mesh Ewald method is usually used to approximate the long-range electrostatic interactions (71, 72). 

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