Ewald summation methods

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]. 

A. Toukmaji and D. Paul and J. A. Board, Jr., Distributed Particle-Mesh Ewald A Parallel Ewald Summation Method, Proceedings, International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA 96), CSREA Press (1996), pp. 33-43. 

In periodic boimdary conditions, one possible way to avoid truncation of electrostatic interaction is to apply the so-called Particle Mesh Ewald (PME) method, which follows the Ewald summation method of calculating the electrostatic energy for a number of charges [27]. It was first devised by Ewald in 1921 to study the energetics of ionic crystals [28]. PME has been widely used for highly polar or charged systems. York and Darden applied the PME method already in 1994 to simulate a crystal of the bovine pancreatic trypsin inhibitor (BPTI) by molecular dynamics [29]. 

In the Ewald summation method the initial set of charges are surrounded by a Gaussian distribution lated in real space) to which a cancelling change distribution must be added (calculated in reciprocal space). 

To date, RNA calculations have been performed on a variety of systems of different topologies including helical duplexes, hairpin loops, and single strands from tRNA, rRNA, and ribozymes. In a simulation of an RNA tetraloop of the GRNA type, which is very common and known to be remarkably stable, it was found that without imposing any external infonnation the simulation found the right confonnation even when it started from the wrong one [72]. Studies have used Ewald summation methods to handle the... 

J. Hautman and M. L. Klein, An Ewald summation method for planar surfaces and interfaces. Mol. Phys. 75 (1992) 379. 

The DLPOLY utilities wateradd and solvadd were employed to add 8 Ca " ions, 20 Na" ions, 12 Cl" ions and 1024 rigid Simple Point Charge (SPC) [95] water molecules to a cell containing 3 PGA chains, each formed by 24 galacturonic units one third of which were taken as deprotonated. The Ewald summation method was employed to evaluate the coulomb interactions with a dielectric constant value of 1.0. A time step of 0.001 ps was adopted in all the simulations. 

The long-range Coulomb interactions were treated using the 3D Ewald summation method with Ewald convergence parameter a = 0.284 A-1 and an Ewald sum precision of 1 10-5 (from the standard in the DL POLY package [35]). 

In modem simulations, various Ewald summation methods are often used in order to correctly describe the long-range electrostatic interactions. The TIPwP potentials were originally parameterised using truncated Coulomb interactions and using these models with Ewald summation results in changes in both the thermodynamic and kinetic properties. 

For two reasons we focus in this chapter on yet a different technique, the. so-called Ewald summation method. First, Ewald sums are nowadays the most widely used and accepted method to handle long-range interactions [140, 238] at least as far as bulk systems are concerned. Second, the formulation of Ewald sums for confined systems is straightforward as we shall demonstrate below and in the accompanying Appendix F to which we refer for a detailed discussion of the derivation of the relevant equations. Moreover, during the last few years, there have been substantial improvements that led to an optimization of those methods such that nowadays they arc... 

Equation (6.40) is our final expression for the energy of a charged system between insulating walls within the slab-adapted three-dimensional Ewald summation method. On the right side, the contribution f/ le =oo is defined by Eqs. (6.15), (6.16), and (6.17b). The reader should also realize that, for the current system, the volume V appearing in the Fourier pjurt of the energy [see Eq. (6.16b)] includes the vacuum space that is, V = = 7S -... 

Because implicit solvent simulations are typically open, non-periodic systems, electrostatic interactions are calculated according to standard cutoff schemes that do not scale as well to large system sizes as the more efficient Ewald summation method [11,12] that is routinely used in explicit solvent simulations. 

Coulomb interactions show critically poor convergence properties as a function of distance (i.e., 1/r interactions). Interaction cutoffs have shown prone to artifacts and motivated the development of long-range electrostatic methods, such as Ewald summation (see, e.g.. Reference [25] and references therein). A number of Ewald summation methods have been extended to MTPs (e.g., [43, 54, 120]), providing a rigorous treatment of electrostatics in MD simulations. 

Long-range Coulomb Forces. - Coulombic interactions are present in many molecular liquids and these play a key role in determining the molecular structure and the physical properties of these systems. It is therefore important to represent them as accurately as possible. The problem is that for charge-charge coulomb interactions, which decay as r simple truncation is not possible and if carried out leads to unrealistic distortions in the structure and the dynamics. Traditionally this has been most often avoided by implementation of the Ewald summation method. The original summation is... 

For a system with box sizes hx, hy and Lz, periodic only in the z direction, the Ewald summation method gives the following expression for the energy [41,42]... 

As in the P M and Ewald summation methods, within the multipole method formalism the force is separated into a long- and short-range interaction, and the short-range component is resolved through a direct summation over the particle-particle interaction. The long-range component of the force on a generic particle i is computed as... 

The treatment of boundary conditions can be incorporated in the EMM scheme easily. Periodic boundary conditions as well as Dirichlet, Neumann, and mixed conditions can be accounted for. The EMM approach has been shown to be more efficient than the Ewald summation method (see the next... 

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