Ewald summation computer time

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. 

Thus, for both the ionic and the dipolar systems, the actual use of the rigorously derived Ewald summation for slab systems loads to a substantial increase in computer time. One way of dealing with this problem would be to employ precalculated tables [252] for potential energies (and forces) on a three-dimensional spatial grid amended by a suitable interpolation scheme. Another strategy is to employ approximate methods such as the one presented in the subsequent Section 6.3.2. 

Minimize real-space evaluation in Ewald summation Sagui et al. have quantified the impact of the real-space cutoff used in their MTP Ewald implementation on the energy drift of a constant-energy simulation [54]. Restricting the real-space evaluation to a minimal amount and performing the rest in reciprocal space can lead to significant improvements. The authors showed minimal energy and force errors down to 4.25 A of direct interaction cutoff. They report an increase in computational time due to MTP interactions of only 8.5 with respect to simple PCs. 

Except for the Coulombic contributions, which are computed by an Ewald-type summation, as described in Section 2.3, a potential cutoff distance is imposed to avoid unnecessary computing time calculating negligible contributions by short-range interactions from most of the V 2N N— 1) atom pairs in the system. In the commonly used nearest-image convention, if the cutoff obeys the condition re < VtJL, Then atom pair interactions included are between atom i in the central box and either atomj in the same box or one of its imagesf in an adjacent one, depending on whichever distance x/ - Xy or x/ - f is least (see Fig. 5.4). 

In the case of 5 < 5 x 10", Eq. 105 could be used instead of Eq. 100 to show the dependence of the accuracy on the parameters more clearly. Equations 119 and 120 provide the qualitative function relations of Tc and kc with a as rda) - AVln S/a and kc a) - uVln 6a. Inserting them into Eq. 118 and differentiating it with respect to a yields a cx and thus rc a and kc oc The minimized computation time is then proportional to with the proportionality constant depending on the accuracy. The same results can be found for the Coulomb Ewald method in Refs. [9,35,46]. This can be easily understood by comparing Eqs. 100 and 112 in Sect. 3 of this paper with Eqs. 18 and 32 in Ref. [35], and finding the same exponential dependences of the cutoff errors on a, rc, and kc for the dipolar and Coulomb Ewald summations. 

The trusted technique for determining the total energy of a crystal is Ewald summation (2). This method is used for amorphous systems as well, except that it is very slow. The boxes used for simulating amorphous systems usually contain thousands of atoms rather than the handful of atoms in the imit cell of a crystal, and this slows down the method tremendously. The routine application of Ewald summation to determine the energies of these large systems is very wasteful of computer time, and many modelers prefer to use faster methods of provable accuracy for amorphous phases. 

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. 

Computer simulations have been applied to studies of the structure of molten salts along two lines one is the fi ee standing application of the computer simulation to obtain the partial pair correlation functions, the other is the refining of x-ray and neutron diffraction and EXAFS measurements by means of a suitable model. In both cases a suitable potential function for the interactions of the ions must be employed, as discussed in Sect. 3.2.4. Such potential functirms are employed in both the Monte Carlo (MC) and the molecular dynamics (MD) simulation methods. A further aspect that has been considered in the case of molten salts is the long range coulombic interaction that exceeds the limits of the periodic simulation boxes usually involved (for 1000 ions altogether), requiring the Ewald summation that is expensive in computation time and is prone to truncation errors if not applied carefully. 

The past 20 years have seen a renewal of interest in lattice summation methods, catalyzed by the advances in high-performance computing and the ability thereby provided to approach molecular dynamics and condensed-phase structural problems that had previously seemed inaccessible. In this respect, an important development was the so-called fast multipole method (FMM) [5]. With its help, the electrostatic energies of arrays of charged particles can be evaluated in computing times that are nearly linear in the number of particles. One of the strengths of FMM is that the charge distribution need not be periodic, and methods of Ewald character can be combined with FMM concepts for studies of periodic systems [6]. 

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