The optimal P3M algorithm for computing electrostatic energies in periodic systems

The optimal P3M algorithm for computing electrostatic energies in periodic systems V. Ballenegger Institut UTINAM, Universit´e de Franche-Comt´e, UMR 6213, 16, route de Gray, 25030 Besan¸con cedex France. J. J. Cerda and O. Lenz Frankfurt Inst. for Advanced Studies, J.W. Goethe - Universit¨at, Frankfurt, Germany Ch. Holm Frankfurt Inst. for Advanced Studies, J.W. Goethe - Universit¨at, Frankfurt, Germany and Max-Planck-Institut f¨ur Polymerforschung, Mainz, Germany (Dated: October 26, 2018) Abstract We optimize Hockney and Eastwood’s Particle-Particle Particle-Mesh (P3M) algorithm to achieve maximal accuracy in the electrostatic energies (instead of forces) in 3D periodic charged systems. To this end we construct an optimal influence function that minimizes the RMS errors in the energies. As a by-product we derive a new real-space cut-offcorrection term, give a trans- parent derivation of the systematic errors in terms of Madelung energies, and provide an accurate analytical estimate for the RMS error of the energies. This error estimate is a useful indicator of the accuracy of the computed energies, and allows an easy and precise determination of the optimal values of the various parameters in the algorithm (Ewald splitting parameter, mesh size and charge assignment order). 1 arXiv:0708.0728v1 [physics.comp-ph] 6 Aug 2007 I. INTRODUCTION Long range interactions are ubiquitously present in our daily life. The calculation of these interactions is, however, not an easy task to perform. One needs indeed to resort to specialized algorithms to overcome the quadratic scaling with the number of particles, as soon as the simulated system includes more than a few hundred particles, see for example the review of Arnold and Holm1. In Molecular Dynamics simulations, one is mainly interested in the accuracy of the force computation, since they govern the dynamics of the system. In contrast, in Monte Carlo (MC) simulations, the concern is to compute accurate energies. If the potential is of long range (e.g. a Coulomb potential or dipolar interaction), and one has chosen to use periodic boundary conditions, the computation of both observables is quite time consuming if one uses the traditional Ewald sum. Since the seminal work of Hockney and Eastwood2 it has been common to resort to a faster way of calculating the reciprocal space sum in the Ewald method with the help of Fast-Fourier-Transforms (FFTs). These algorithms are called mesh-based Ewald sums, and various variants exist3. They all scale as N log N with the number of charged particles N, and the algorithms are nowadays routinely used in simulations of bio-systems, charged soft matter, plasmas, and many more areas. The most accurate variant is still the original method of Hockney and Eastwood, which they called particle-particle-particle-mesh (P3M), and into which various other improvements like the analytical differentiation used in other variants of the mesh- based Ewald sum4 can be built in. In addition, an accurate error estimate for P3M exists, so that one can tune the algorithm to a preset accuracy, thus maximizing the computational efficiency before doing any simulations5. While in the standard P3M algorithm2, the lattice Green function, called the “influence function”, is optimized to give the best possible accuracy in the forces, the electrostatic energy is usually calculated with the same force-optimized influence function. However, there are certainly situations where one needs a high precision of the energies, for instance in Monte Carlo simulations, and the natural question arises whether one can optimize the influence function to enhance the accuracy of the P3M energies. The main goal of this paper is to derive the energy-optimized influence function, and to derive an analytical estimate for the error in the P3M energies. This error estimate is a valuable indicator of the accuracy of the calculations and allows a straightforward and precise determination of the optimal 2 values of the various parameters in the algorithm (Ewald splitting parameter, mesh size, charge assignment order). The present derivation of the optimal influence function, and the associated error esti- mate, is concise and entirely self-contained. The present paper can thus also serve as a pedagogical introduction to the main ideas and mathematics of the P3M algorithm. The paper is organized as follows. In Sec. II, we briefly review the ideas of the standard Ewald method and provide the most important formulae. In Sec. III, we derive direct and reciprocal space correction terms which compensate, on average, the effects of cut-off errors in the standard Ewald method. We interpret the formulae in terms of the direct and reciprocal space components of the Madelung energies of the ions. In Sec. IV, the calculation of the reciprocal energy according to the P3M algorithm (i.e. with a fast Fourier transform and an optimized influence function) is presented. The mathematical analysis of the errors intr

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