Interlaced P3M algorithm with analytical and ik-differentiation.

The interlacing technique of Hockney and Eastwood is extended to the particle-particle, particle-mesh (P3M) algorithm with analytical and ik-differentiation that computes the approximate Coulomb forces between N point particles in a periodic box. Interlacing means that one makes two separate computations of the reciprocal-space Ewald force, using two grids shifted with respect to each other by half of the diagonal of the grid subcell, and then takes the average of the two forces. The resulting algorithms compare favorably against their own noninterlaced versions and against the interlaced smooth particle-mesh Ewald algorithm. In our tests, the accuracy of the interlaced P3M methods was usually more than an order of magnitude higher than that of the other particle-mesh algorithms with the same parameter values. This accuracy gain can be converted into a speedup if the parameters of the algorithm are changed. Interlacing allows one to increase the grid spacing by up to a factor of 2 while keeping the same accuracy. A priori error estimates for the new algorithms are constructed, and the removal of the spurious self-force term is discussed. The success of interlacing is shown to be due to the fact that it suppresses the aliasing effects in the forces. It should be easy to incorporate the interlaced P3M algorithms into an existing simulation package, since this only requires a minor modification of the particle-mesh Ewald part of the code.

[1]  J. Mccammon,et al.  Effect of artificial periodicity in simulations of biomolecules under Ewald boundary conditions: a continuum electrostatics study. , 1999, Biophysical chemistry.

[2]  Toshiyuki Fukushige,et al.  PPPM and TreePM Methods on GRAPE Systems for Cosmological N-body Simulations , 2005 .

[3]  Edward D Harder,et al.  Efficient multiple time step method for use with Ewald and particle mesh Ewald for large biomolecular systems , 2001 .

[4]  R. Strebel,et al.  An Alternative to Ewald Sums. Part 3: Implementation and Results , 2001 .

[5]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[6]  G. Martyna,et al.  Electrostatic calculations and multiple time scales in molecular dynamics simulation of flexible molecular systems , 1998 .

[7]  Celeste Sagui,et al.  Towards an accurate representation of electrostatics in classical force fields: efficient implementation of multipolar interactions in biomolecular simulations. , 2004, The Journal of chemical physics.

[8]  Laxmikant V. Kalé,et al.  Scalable molecular dynamics with NAMD , 2005, J. Comput. Chem..

[9]  Ilpo Vattulainen,et al.  Lipid Bilayers Driven to a Wrong Lane in Molecular Dynamics Simulations by Subtle Changes in Long-Range Electrostatic Interactions , 2004 .

[10]  T. Darden,et al.  Efficient particle-mesh Ewald based approach to fixed and induced dipolar interactions , 2000 .

[11]  R. Sperb,et al.  An Alternative to Ewald Sums, Part 2: The Coulomb Potential in a Periodic System , 1999 .

[12]  C. Holm,et al.  P3M algorithm for dipolar interactions. , 2008, The Journal of chemical physics.

[13]  R. Dror,et al.  Gaussian split Ewald: A fast Ewald mesh method for molecular simulation. , 2005, The Journal of chemical physics.

[14]  Christian Holm,et al.  How to mesh up Ewald sums. I. A theoretical and numerical comparison of various particle mesh routines , 1998 .

[15]  Philippe H. Hünenberger,et al.  Optimal charge-shaping functions for the particle–particle—particle–mesh (P3M) method for computing electrostatic interactions in molecular simulations , 2000 .

[16]  Mark E. Tuckerman,et al.  A new reciprocal space based treatment of long range interactions on surfaces , 2002 .

[17]  T. Darden,et al.  Generalization of the Gaussian electrostatic model: extension to arbitrary angular momentum, distributed multipoles, and speedup with reciprocal space methods. , 2006, The Journal of chemical physics.

[18]  Philippe H. Hünenberger,et al.  Lattice-sum methods for computing electrostatic interactions in molecular simulations , 1999 .

[19]  J. Perram,et al.  Cutoff Errors in the Ewald Summation Formulae for Point Charge Systems , 1992 .

[20]  R. W. Hockney,et al.  A 10000 particle molecular dynamics model with long range forces , 1973 .

[21]  Carsten Kutzner,et al.  GROMACS 4:  Algorithms for Highly Efficient, Load-Balanced, and Scalable Molecular Simulation. , 2008, Journal of chemical theory and computation.

[22]  J. D. Gezelter,et al.  Is the Ewald summation still necessary? Pairwise alternatives to the accepted standard for long-range electrostatics. , 2006, The Journal of chemical physics.

[23]  Hans-Jörg Limbach,et al.  ESPResSo - an extensible simulation package for research on soft matter systems , 2006, Comput. Phys. Commun..

[24]  T. Darden,et al.  Molecular dynamics simulations of biomolecules: long-range electrostatic effects. , 1999, Annual review of biophysics and biomolecular structure.

[25]  Robert D. Skeel,et al.  Correcting mesh-based force calculations to conserve both energy and momentum in molecular dynamics simulations , 2007, J. Comput. Phys..

[26]  M. Patra,et al.  Molecular dynamics simulations of lipid bilayers: major artifacts due to truncating electrostatic interactions. , 2003, Biophysical journal.

[27]  R W Hockney,et al.  Computer Simulation Using Particles , 1966 .

[28]  C. Sagui,et al.  Multigrid methods for classical molecular dynamics simulations of biomolecules , 2001 .

[29]  K. Esselink.  A comparison of algorithms for long-range interactions , 1995 .

[30]  Terry P Lybrand,et al.  Staggered Mesh Ewald: An extension of the Smooth Particle-Mesh Ewald method adding great versatility. , 2009, Journal of chemical theory and computation.

[31]  H. Stern,et al.  On mesh-based Ewald methods: optimal parameters for two differentiation schemes. , 2008, The Journal of chemical physics.

[32]  Richard A Friesner,et al.  Efficient Simulation Method for Polarizable Protein Force Fields:  Application to the Simulation of BPTI in Liquid Water. , 2005, Journal of chemical theory and computation.

[33]  I T Todorov,et al.  DL_POLY_3: the CCP5 national UK code for molecular–dynamics simulations , 2004, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[34]  Bilha Sandak,et al.  Multiscale fast summation of long‐range charge and dipolar interactions , 2001, J. Comput. Chem..

[35]  Philippe P. Brieu,et al.  P4M: a parallel version of P3M , 1998 .

[36]  Robert D. Skeel,et al.  Multiple grid methods for classical molecular dynamics , 2002, J. Comput. Chem..

[37]  Jianpeng Ma,et al.  CHARMM: The biomolecular simulation program , 2009, J. Comput. Chem..

[38]  Sanjay Banerjee,et al.  Efficient charge assignment and back interpolation in multigrid methods for molecular dynamics , 2005, J. Comput. Chem..

[39]  T. Darden,et al.  A smooth particle mesh Ewald method , 1995 .

[40]  P. P. Ewald.  Die Berechnung optischer und elektrostatischer Gitterpotentiale , 1921 .

[41]  S. Feller,et al.  Computational modeling of membrane bilayers , 2008 .

[42]  Axel Arnold,et al.  Efficient methods to compute long-range interactions for soft matter systems , 2005 .

[43]  Piero Procacci,et al.  A Very Fast Molecular Dynamics Method To Simulate Biomolecular Systems with Realistic Electrostatic Interactions , 1996 .

[44]  Celeste Sagui,et al.  Electrostatics in Biomolecular Simulations: Where Are We Now and Where Are We Heading? , 2008 .

[45]  B M Pettitt,et al.  Fast multipole methods for particle dynamics , 2006, Molecular simulation.

[46]  Joseph A Morrone,et al.  Long range interactions on wires: a reciprocal space based formalism. , 2004, The Journal of chemical physics.

[47]  I. Tsukerman.  Efficient computation of long-range electromagnetic interactions without Fourier transforms , 2004, IEEE Transactions on Magnetics.

[48]  O. Lenz,et al.  The optimal P3M algorithm for computing electrostatic energies in periodic systems. , 2007, The Journal of chemical physics.

[49]  Thierry Matthey,et al.  Parallel multigrid summation for the N-body problem , 2005, J. Parallel Distributed Comput..

[50]  J. W. Eastwood.  Optimal particle-mesh algorithms , 1975 .

[51]  Siewert J. Marrink,et al.  Methodological issues in lipid bilayer simulations , 2003 .

[52]  Jim Glosli,et al.  Comments on P3M, FMM, and the Ewald method for large periodic Coulombic systems , 1996 .

[53]  Darrin M. York,et al.  The fast Fourier Poisson method for calculating Ewald sums , 1994 .

[54]  Mark E. Tuckerman,et al.  A reciprocal space based method for treating long range interactions in ab initio and force-field-based calculations in clusters , 1999 .

[55]  Tom Evans,et al.  Large-scale molecular dynamics simulation of DNA: implementation and validation of the AMBER98 force field in LAMMPS , 2004, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[56]  M. Deserno,et al.  HOW TO MESH UP EWALD SUMS. II. AN ACCURATE ERROR ESTIMATE FOR THE PARTICLE-PARTICLE-PARTICLE-MESH ALGORITHM , 1998, cond-mat/9807100.

[57]  Lee G. Pedersen,et al.  Long-range electrostatic effects in biomolecular simulations , 1997 .

[58]  T. Darden,et al.  Particle mesh Ewald: An N⋅log(N) method for Ewald sums in large systems , 1993 .

[59]  Rajiv K. Kalia,et al.  Scalable and portable implementation of the fast multipole method on parallel computers , 2003 .

[60]  K. Kremer,et al.  Advanced Computer Simulation Approaches for Soft Matter Sciences III , 2005 .

[61]  A. Brandt,et al.  Multilevel matrix multiplication and fast solution of integral equations , 1990 .

[62]  R. Sperb,et al.  An Alternative to Ewald Sums part I: Identities for Sums , 1998 .

[63]  K. Schulten,et al.  Difficulties with multiple time stepping and fast multipole algorithm in molecular dynamics , 1997 .

[64]  John Lekner,et al.  Summation of dipolar fields in simulated liquid-vapour interfaces , 1989 .