Ewald algorithm

Darden T A, L Perera, L Li and L Pedersen 1999. New Tricks for Modelers from the Crystallography Toolkit The Particle Mesh Ewald Algorithm and Its Use in Nucleic Acid Simulations. Structure with Folding and Design 7 R55-R60. 

Darden T, Perera L, Li LP, Pedersen L (1999) New tricks for modelers from the crystallography toolkit the particle mesh Ewald algorithm and its use in nucleic acid simulations. Struct Fold Des 7(3) R55-R60... 

Barash, D., Yang, L.J., Qian, X.L., SchUck, T. Inherent speedup limitations in multiple time step/particle mesh Ewald algorithms, J. Comput. Chem. 2003,24,77-88. 

Speedup Limitations in Multiple Timestep Particle Mesh Ewald Algorithms. 

But the methods have not really changed. The Verlet algorithm to solve Newton s equations, introduced by Verlet in 1967 [7], and it s variants are still the most popular algorithms today, possibly because they are time-reversible and symplectic, but surely because they are simple. The force field description was then, and still is, a combination of Lennard-Jones and Coulombic terms, with (mostly) harmonic bonds and periodic dihedrals. Modern extensions have added many more parameters but only modestly more reliability. The now almost universal use of constraints for bonds (and sometimes bond angles) was already introduced in 1977 [8]. That polarisability would be necessary was realized then [9], but it is still not routinely implemented today. Long-range interactions are still troublesome, but the methods that now become popular date back to Ewald in 1921 [10] and Hockney and Eastwood in 1981 [11]. 

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... 

Deserno M and C Holm 1998b. How to Mesh Up Ewald Sums. II. An Accurate Error Estimate for the Particle-Particle-Particle-Mesh Algorithm. Journal of Chemical Physics 109 7694-7701. 

Regardless of which algorithm is used for fast calculation of Ewald sums, the computational cost is now competitive with the cost of cutoff calculations, and there is no longer a need to employ cutoffs for purposes of efficiency. Since Ewald summation is the natural expression of Coulomb s law in periodic boundary conditions, it is the recommended approach if periodic boundary conditions are to be used in a simulation. 

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