Particles particle mesh method

Brock A. Luty and Wilfried F. van Gunsteren. Calculating electrostatic interactions using the particle-particle particle-mesh method with nonperiodic long-range interactions. J. Phys. Chem., 100 2581-2587, 1996. 

We note, however, in the present context that as discussed in Chapter 4, there are two alternative techniques to the Ewald sum method for evaluating the long range Coulomb interactions. One is the Particle-Particle/Particle-Mesh method (PPPM) (Eastwood et al., 1980) and the other is the Cell Multipole Method (CMM) (Greengard and Rokhlin, 1987). The computational cost for both PPPM and CMM scale as N, the number of particles, while for the Ewald sum the cost scales as Ni>2 (Fincham, 1994). Of the two alternative techniques, the CMM is gaining more popularity mainly because it is applicable to non-periodic and inhomogeneous systems as well and it is more amenable to parallelization. CMM is slower than the Ewald sum for small systems but it is faster for very large systems. However, it is not certain yet at which value of N the crossover occurs. Values between 300 and 30000 have been quoted (Fincham, 1994). 

A variety of methods are available for computing electrostatic energies (and forces), including the Fast Multipole Method [38], the Particle-Particle-Particle-Mesh Method [179], and methods based on the technique of Ewald Summation [125, 367] we only discuss a particular variant of the latter approach here and not... 

Short-range contributions P< P) are obtained by direct summation over nearby particles with a radius rc (dashed circle). Long-range contributions (Pi- M) are obtained from the particle-mesh method... 

If the interaction potential u is long range, all terms in the infinite sum should be taken into account. The brute force approach here is to truncate the summation at some large enough values of ny, rix. Efficient ways to do this for Coulombic interactions—the Ewald summation, fast multipole, and particle-mesh methods— will be described in the following chapters. 

Darden T, York D and Pedersen L 1993 Particle mesh Ewald—an N.log(N) method for Ewald sums in large systems J. Chem. Phys. 98 10089-92... 

Essmann U, Perera L, Berkowitz M L, Darden T, Lee H and Pedersen L G 1995 A smooth particle mesh Ewald method J. Chem. Phys. 103 8577-93... 

Luty, B.A., Davis, M.E., Tironi, I.G., Van Gunsteren, W.F. A comparison of particle-particle particle-mesh and Ewald methods for calculating interactions in periodic molecular systems. Mol. Simul. 14 (1994) 11-20. 

U. Essmann, L. Perera, M. L. Berkowitz, T. Darden, H. Lee, and L. G. Pedersen. The smooth particle mesh ewald method. J. Chem. Phys., 103 8577, 1995. Brock A. Luty, Ilario G. Tironi, and Wilfried F. van Gunsteren. Lattice-sum methods for calculating electrostatic interactions in molecular simulations. J. Chem. Phys., 103 3014-3021, 1995. 

There are three different algorithms for the calculation of the electrostatic forces in systems with periodic boundary conditions (a) the (optimized) Ewald method, which scales like (b) the Particle Mesh... 

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

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]. 

They compared the PME method with equivalent simulations based on a 9 A residue-based cutoflF and found that for PME the averaged RMS deviations of the nonhydrogen atoms from the X-ray structure were considerably smaller than in the non-PME case. Also, the atomic fluctuations calculated from the PME dynamics simulation were in close agreement with those derived from the crystallographic temperature factors. In the case of DNA, which is highly charged, the application of PME electrostatics leads to more stable dynamics trajectories with geometries closer to experimental data [30]. A theoretical and numerical comparison of various particle mesh routines has been published by Desemo and Holm [31]. 

Cheatham T E III, J L Miller, T Fox, T A Darden and P A Kollman 1995. Molecular Dynamics Simulations on Solvated Biomolecular Systems The Particle Mesh Ewald Method Leads to Stable Trajectories of DNA, RNA and Proteins. Journal of the American Chemical Society 117 4193-4194. 

Luty B A, M E David, I G Tironi and W F van Gunsteren 1994. A Comparison of Particle-Particle, Particle-Mesh and Ewald Methods for Calculating Electrostatics Interactions in Periodic Molecular Systems. Molecular Simulation 14 11-20. 

In the case of the reciprocal sum, two methods have been implemented, smooth particle mesh Ewald (SPME) [65] and fast Fourier Poisson (FFP) [66], SPME is based on the realization that the complex exponential in the structure factors can be approximated by a well behaved function with continuous derivatives. For example, in the case of Hermite charge distributions, the structure factor can be approximated by... 

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