Summation, Ewald

Ewald summation is claimed to be one of the most accurate methods for treating electrostatic interactions when PBCs are used. In fact, the electrostatic interactions are calculated between all atoms located in the central box and between all atoms of the central box with their images in the neighboring boxes. The electrostatic interaction energy in a periodic system is given by  

Equation [6] is valid only for a cubic box, and extensions to rectangular boxes are achieved by modifying L. This is a conditionally convergent sum where the resulting energy is dependent on the order in which the double summation is carried out. A convenient mathematical transformation of Eq. [6] into two expressions with better convergence properties results in Eqs. [7] and 

Ewald summation presented above calls for the calculation of AP terms for each of the periodic boxes, a computationally demanding requirement for large biomolecular systems. Recently, Darden et al. proposed an N log N method, called particle mesh Ewald (PME), which incorporates a spherical cutoff R. This method uses lookup tables to calculate the direa space sum and its derivatives. The reciprocal sum is implemented by means of multidimensional piecewise interpolation methods, which permit the calculation of this sum and its first derivative at predefined grids with fast Fourier transform methods. The overhead for this calculation in comparison to Coulomb interactions ranges from 16 to 84% of computer time, depending on the reciprocal sum grid size and the order of polynomial used in calculating this sum. 

In the PME method, now implemented in AMBER 4.1, the value for a is chosen to ensure that the direct space sum vanishes at the specified cutoff. The algorithm requires the user to specify an acceptably small tolerance for the 

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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... 

Belhadj,M., Alper, H.A., Levy, R.M. Molecular dynamics simulations of wa ter with Ewald summation for the long-range electrostatic interactions. Chem. Phys. Lett. 179 (1991) 13-20. 

Ewald summation was invented in 1921 [7] to permit the efl5.cient computation of lattice sums arising in solid state physics. PBCs applied to the unit cell of a crystal yield an infinite crystal of the appropriate. symmetry performing... 

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). 

The summation is over the different types of ion in the unit cell. The summation ca written as an analytical expression, depending upon the lattice structure (the orij Mott-Littleton paper considered the alkali halides, which form simple cubic lattices) evaluated in a manner similar to the Ewald summation this typically involves a summc over the complete lattice from which the explicit sum for the inner region is subtractec... 

Ewald summation has been applied successfully for many years to liquid simulations [35] and is now becoming a standard for macromolecular simulations [36]. For this reason we focus on Ewald summation for the remainder of this section. 

Regardless of which algorithm is used for fast calculation of Ewald sums, the computational cost is now competitive with the cost of cutoff calculations, and there is no longer a need to employ cutoffs for purposes of efficiency. Since Ewald summation is the natural expression of Coulomb s law in periodic boundary conditions, it is the recommended approach if periodic boundary conditions are to be used in a simulation. 

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... 

Pc- (c) Dipole density p. (d) Water contribution to the surface potential x calculated from the charge density Pc by means of Eq. (1). All data are taken from a 150 ps simulation of 252 water molecules between two mercury phases with (111) surface structure using Ewald summation in two dimensions for the long-range interactions. 

Calculated Electrostatic Energy of the Site(eV) Ewald Summation 977 Pt. Ion Cluster... 

Toukmaji AY, Board JA (1996) Ewald summation techniques in perspective a survey. Comput Phys Commun 95(2-3) 73-92... 

Fig. 2.2. Average electrostatic potential mc at the position of the methane-like Lennard-Jones particle Me as a function of its charge q. mc contains corrections for the finite system size. Results are shown from Monte Carlo simulations using Ewald summation with N = 256 (plus) and N = 128 (cross) as well as GRF calculations with N = 256 water molecules (square). Statistical errors are smaller than the size of the symbols. Also included are linear tits to the data with q < 0 and q > 0 (solid lines). The fit to the tanh-weighted model of two Gaussian distributions is shown with a dashed line. Reproduced with permission of the American Chemical Society...

Hu, H. Yun, R. H. Hermans, J., Reversibility of free energy simulations slow growth may have a unique advantage with a note on use of Ewald summation, Mol. Simul. 2002, 28, 67-80... 

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