Ewald summation Coulomb interaction

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. 

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. 

Two different boundary conditions are usually used for simulation processes. One is an isolated system and the other is a bulk system in which a periodical boundary condition is employed. The Ewald summation (17) is often introduced in the calculation of Coulombic interactions. For liquids and solutions the latter system has been used mostly, but the former has been examined in studying the dynamic behavior of a single molecule interacting with a limited number of particles. 

For the calculation of the normal mode spectra external and internal coordinates were assumed to be dynamically decoupled. Translational and rotational coordinates were extracted from the trajectories while all vibrational coordinates were set to zero. Dynamical matrices were set up for 50 configurations generated by molecular dynamics simulation. Long-range Coulombic interactions were treated using the Ewald summation technique. In Figure 2 the instantaneous normal mode spectra are depicted while in Table 3 some of their integral properties are compiled. 

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. 

Here qi is the effective charge of an atom is a dispersion interaction constant and Ay and bij are parameters of the Born - Mayer atom-atom repulsion potential. To calculate the long range Coulomb term in Eq. (1) one generally has to employ the Ewald summation technique. To obviate this inconvenience, the Coulomb term has been multiplied by the screening factor (and the dispersion term has been neglected) ... 

Intermolecular solvent-reagent forces are usually represented through Lennard-Jones interactions between the component atoms. For dipolar or ionic systems, partial or full charges can be placed on individual atoms to represent Coulomb interactions. (More complex methods exist for dealing with charged systems, such as Ewald summation see ref. 4 for more details.)... 

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. 

Those workers who use periodic boundary conditions must contend with the calculation of the Coulomb energy, which because of its infinitely long range must be summed over all pair interactions in the primary cell and all interactions between a particle in the cell and the infinite number of image particles. The procedure adopted by these workers is to evaluate the electrostatic energy using a Ewald summation technique. ... 

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... 

The Hmit E exists, but differs by some multiple of the square of the dipole moment from the spherical limit as obtained by the Ewald summation [50]. From the physical point of view, the Coulomb interaction is replaced by a screened Coulomb interaction with screening length l/p. E is then the energy in the limit of infinite screening length. But because of the conditional convergence... 

Since the Ewald summation is a well-suited and frequently used method to treat the interactions in strongly coupled Coulomb systems, a brief background will be given and its system size dependence and trimcation error will be examined. Finally, some practical guidehnes will be provided. 

It has been shown that the Ewald summation is an excellent tool for handling Coulomb interactions in highly asymmetric electrolytes, hi particular, (i) the system size convergence, (ii) the truncation error, and (iii) the selection of a for optimization of the CPU time have been discussed, hi practice, issues (i) and (ii) are interrelated values of Rent and cut for a given truncation error depend on system size and simulation results with controlled uncertainties are required to assess the system size dependence. 

Each simulation extended to about 15000 time-steps after a pre-equilibration of approximately the same length. Ewald summation in tabulated form was used for the Coulomb interactions, and the shifted-force method (e.g., Allen and Tildesley 1987) was used for the non-Coulomb parts of the BJH potential. All technical details of the simulations are described in detail elsewhere (Kalinichev and Heinzinger 1992, 1995 Kalinichev 1993). 

Ewald summation techniques are necessary for calculating Coulombic interactions. The non-Coulombic terms contain both attractive and repulsive components and can typically be modeled by using Lennard-Jones, Morse or Buckingham potentials from Eqs. (B5), (B6), and (B7), respectively. 

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