Ewald summation techniques

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

As discussed earlier the charges of the QM atoms are derived by means of a population analysis, whereas the charges of the respective solvent model are used for all MM particles. The reaction field (28) formalism was employed in this equation to account for the error resulting from the Coulombic cutoff given as rc. e is the dielectric constant of the medium beyond the Coulombic cutoff. Alternatively, an Ewald summation technique (29) could be applied as well. 

Now, in the TF-HK equation, all the potential terms can be set up by conventional plane-wave-basis techniques353 371 372 389 with essentially linear scaling. However, for very large systems with more than 5000 nuclei, the computational cost associated with the nuclear-nuclear Coulomb repulsion energy becomes the major bottleneck.116 In this case, linear-scaling Ewald summation techniques should be utilized.390-413... 

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. 

Two standard methods are in common use in the MD community the reaction field method [79,80] and the Ewald summation technique [72,81-83]. There are also various hierarchical algorithms which are quite attractive in principle, but have proved to be difficult to implement efficiently in practice [67,84-87]. An alternative and potentially development interesting complement, is the summation formula developed by Lekner [88,89] which has been given an alternative and more general derivation by Sperb [90]. 

Essential dynamics, 2, 233, 236, 242-244, 247 Euler angles, 3,168 evolutionary determinants, 4, 4, 5 evolvability, 4, 7-9,17 Ewald summation, 2, 265 Ewald summation techniques, 1, 59, 62, 75 exact exchange, 1, 26, 27 exchange repulsion, 3,179,180 excited state structure/dynamics, 1, 24 excretion see ADMET properties explicit-rl2 correlation, 5, 132,140 explicit solvent, 2, 98, 99,101,102,104-106 exponential damping functions, 3,180 extended systems, 1, 26... 

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) ... 

For each point charge qi involved in Eq. (6.4). the resulting electrastatic potential decays in proportion to the inverse distance [see Eq. (6.2)], such that the lattice sum buried in the expression for the total energy Uc [see Eq. (6.1)] converges rather slowly. In view of this dilemma, the central idea of the Ewald summation techniques is to rewrite the d -like charge density in Eq. (6.4) as a sum of three contributions, pf (r ), and p] (r ). Each... 

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. ... 

Moving beyond standard DNA and RNA duplexes, simulations of chemically modified DNA provide a further test of the simulation methods. To this end, we performed unrestrained molecular dynamics simulations on a standard d(CGCGAATTCGCG)2 dodecamer duplex in aqueous solution and its phosphoramidate (N3 -P) analog using the particle mesh Ewald summation technique (5) and recent AMBER force field (10). In the modified dodecamer each phosphodiester has been replaced by a phosphoramidate unit with a N3-P5 intemucleoside linkage. 

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. 

However, the slow decay of the dipole-dipole interaction in (6) (proportional to r ) makes the use of Ewald summation techniques necessary when one uses (6) in a simulation. Thus, the computational effort of working with the full Stockmayer potential (6) is comparable to the effort needed for all-atom models with partial charges at the atoms. 

The computation of the lattice Fourier transform of the is trivial, but that of the has to be done by employing the Ewald summation technique which leads to a formula... 

Do we need the quasi-infinite model (Periodic Boundary Conditions, PBC) What is an appropriate shape and size of the solvation box Should we use the Ewald summation technique to properly calculate electrostatic interactions ... 

Our goal in this chapter is to present a simple and physically meaningful derivation of various Ewald summation techniques. For a mathematically more rigorous presentation, we refer the reader to the original papers by de Leeuw et al. [239-241],... 

Having understood the concepts of Ewald summation techniques for three-dimensional bulk systems, we now turn to systems that are finite in at least one spatial dimension. We focus on a slab-like geometry where the fluid is confined by two plane parallel and structureless solid surfaces separated by a distance s along the z-axis of the coordinate system and of infinite extent in the x-y plane (see also Section 1.3.2). Hence, for the time being, we shall be... 

A. Y. Toukmaji and ]. A. Board, Comput. Phys. Comm ., 95,73 (1996). Ewald Summation Techniques in Perspective A Survey. 

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