Ewald sum methods

There are problems with using both of these methods in the simulation of inhomogeneous systems. Because the periodicity of the system is lost in the direction normal to the interface (unless one uses image charges with the flat wall model, which effectively results in a 3D periodic system implementation of the ES method is not straightforward for certain type of systems. Hautman and Klein have presented a modified Ewald sum method for the simulation of systems that are periodic in two... 

A more refined approach uses the two-centers potential expansion, which considers the interaction between point multipoles located at the centers of the two charge distributions. Since the exchange integral interactions vary as 1/r and the numbers of interactions increase as r, one has to choose spherical shapes for the convergence of the sum to be guaranteed (Ewald sums method)... 

The motion equations have been solved by the Verlet Leap-frog algorithm subject to periodic boundary conditions in a cubic simulation cell and a time step of 2 fs. The simulations have been performed in the NVT ensemble with the Nose-Hoover thermostat [62]. The SHAKE constraints scheme [65] was used. The spherical cutoff radius comprises 1.2 nm. The Ewald sum method was used to treat long-range electrostatic interactions. 

We note, however, in the present context that as discussed in Chapter 4, there are two alternative techniques to the Ewald sum method for evaluating the long range Coulomb interactions. One is the Particle-Particle/Particle-Mesh method (PPPM) (Eastwood et al., 1980) and the other is the Cell Multipole Method (CMM) (Greengard and Rokhlin, 1987). The computational cost for both PPPM and CMM scale as N, the number of particles, while for the Ewald sum the cost scales as Ni>2 (Fincham, 1994). Of the two alternative techniques, the CMM is gaining more popularity mainly because it is applicable to non-periodic and inhomogeneous systems as well and it is more amenable to parallelization. CMM is slower than the Ewald sum for small systems but it is faster for very large systems. However, it is not certain yet at which value of N the crossover occurs. Values between 300 and 30000 have been quoted (Fincham, 1994). 

Recently we proposed a new method called MMM2D [24,58], which has an 0(N / ) complexity and full error control that is based on a convergence factor approach similar to MMM [12]. In two dimensions the convergence factor based methods and the Ewald sum methods yield exactly the same results. However, this will still only allow simulations including up to a few thousand charges due to the power law scaling. 

When carried out properly, the results of the reaction field method and the Ewald sum are consistent [67]. Recently, the reaction field method has been reconnnended on grounds of elTiciency and ease of progrannning [68, 69]. The... 

Darden T, York D and Pedersen L 1993 Particle mesh Ewald—an N.log(N) method for Ewald sums in large systems J. Chem. Phys. 98 10089-92... 

U. Essmann, L. Perera, M. L. Berkowitz, T. Darden, H. Lee, and L. G. Pedersen. The smooth particle mesh ewald method. J. Chem. Phys., 103 8577, 1995. Brock A. Luty, Ilario G. Tironi, and Wilfried F. van Gunsteren. Lattice-sum methods for calculating electrostatic interactions in molecular simulations. J. Chem. Phys., 103 3014-3021, 1995. 

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... 

Eor instance, the contribution of water beyond 12 A from a singly charged ion is 13.7 kcal/mol to the solvation free energy or 27.3 kcal/mol to the solvation energy of that ion. The optimal treatment is to use Ewald sums, and the development of fast methods for biological systems is a valuable addition (see Chapter 4). However, proper account must be made for the finite size of the system in free energy calculations [48]. 

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],... 

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]. 

Conventional methods for performing the Ewald sum scale as 0(N / ) or 0(N ), and formulations specifically designed to include dipole-dipole interactions are in fairly wide use. Easter scaling methods, such as the fast multipole and particle-mesh algorithms, have also been extended to the treatment of point dipoles. 

Ewald An Nlog(N) Method for Ewald Sums in Large Systems. 

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