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... 

Since the summation method allows absolute partial ionization cross sections to be determined, it is straightforward to extract the counting or ion production cross section from the data. Since this is the quantity which is given by many of the current... 

Method of Craven [50] This average bond energy summation method (ABES) is a simplification of the method described by Sanderson [55]. The reaction enthalpy is calculated by subtracting the total bond energies present before the reaction from the total bond energies of the products. 

Average bond energy summation method of Craven [50]... 

Unlike Eq. (8.27), the Fourier summation method for evaluation of the electrostatic potential and its derivatives leads to the potential of the thermally averaged distribution. However, the differences are likely to be small relative to other uncertainties, especially when low-temperature data are used in the analysis. 

The TEB value may be obtained by either the sum of the individually measured cations or by evaporating and igniting a portion of the ammonium ethanoate leachate to convert the metallic cations to oxides and carbonates, followed by addition of excess acid (to convert carbonates to chlorides) and back-titration with alkali. The latter method is difficult if the soil is insufficiently base-rich to provide an adequate amount of bases for the titration. On the other hand, the calcium carbonate in calcareous soils may be partially leached by the ammonium ethanoate at pH 7.0 in addition to the exchangeable bases and thus give an exaggerated TEB value and a percentage base saturation in excess of 100%. The TEB by ignition/titration can serve as a check on the values from the summation method. 

Hamaker (1932) and de Boer (1936) calculated van der Waals forces between macroscopic bodies using the summation method. 

Vol. 53 G.A. Arteca, F.M. Fernandez, E.A. Castro, Large Order Perturbation Theory and Summation Methods in Quantum Mechanics. XI, 644 pages. 1990. 

The summation method is based on relation (16) together with [2,11]... 

J. Cizek, F. Vinette, and E. J. Weniger, Int. ]. Quantum Chem., Quantum Chem. Symp., S25, 209 (1991). Examples on the Use of Symbolic Computation in Physics and Chemistry Applications of the Inner Projection Technique and of a New Summation Method for Divergent Series. 

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