# Ewald sum

The **Ewald sum** technique ^{[1]} was originally developed by Paul Ewald to evaluate the Madelung constant ^{[2]}. It is now widely used in order to simulate systems with
long range interactions (typically, electrostatic interactions). Its aim is the computation of the interaction of a system with periodic boundary conditions with all its replicas. This is accomplished by the introduction of fictitious "charge clouds" that shield the charges. The interaction is then divided into a shielded part, which may be evaluated by the usual means, and a part that cancels the introduction of the clouds, which is evaluated in Fourier space.

## Derivation[edit]

In a periodic system one wishes to evaluate the internal energy (Eq. 1.1 ^{[3]}):

where one sums over all the simple cubic lattice points . The prime on the first summation indicates that if then the term is omitted. is the length of the side of the cubic simulation box, is the number of particles, and represent the Euler angles.

This internal energy is partitioned into four contributions:

#### Real-space term[edit]

The real space contribution to the electrostatic energy is given by ^{[4]}^{[5]} (Eq. 7a and 7b ^{[6]}):

where is the complementary error function, and is the Ewald screening parameter. Also,

#### Reciprocal-space term[edit]

#### Self-energy term[edit]

#### Surface term[edit]

## Particle mesh[edit]

^{[7]}
^{[8]}
^{[9]}

#### Smooth particle mesh (SPME)[edit]

SPME^{[10]}. Optimisation
^{[11]}
^{[12]}.

## See also[edit]

## References[edit]

- ↑ Paul Ewald "Die Berechnung Optischer und Electrostatischer Gitterpotentiale", Annalen der Physik
**64**pp. 253-287 (1921) - ↑ S. G. Brush, H. L. Sahlin and E. Teller "Monte Carlo Study of a One-Component Plasma. I", Journal of Chemical Physics
**45**pp. 2102-2118 (1966) - ↑ S. W. de Leeuw, J. W. Perram and E. R. Smith "Simulation of Electrostatic Systems in Periodic Boundary Conditions. I. Lattice Sums and Dielectric Constants", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
**373**pp. 27-56 (1980) - ↑ W. Smith "Point Multipoles in the Ewald Summation", CCP5 Newsletter
**4**pp. 13-25 (1982) - ↑ W. Smith "Point Multipoles in the Ewald Summation (Revisited)", CCP5 Newsletter
**46**pp. 18-30 (1998) - ↑ Joakim Stenhammar, Martin Trulsson, and Per Linse "Some comments and corrections regarding the calculation of electrostatic potential derivatives using the Ewald summation technique", Journal of Chemical Physics
**134**224104 (2011) - ↑ Tom Darden, Darrin York, and Lee Pedersen "Particle mesh Ewald: An N·log(N) method for Ewald sums in large systems", Journal of Chemical Physics
**98**pp. 10089-10092 (1993) - ↑ Markus Deserno and Christian Holm "How to mesh up Ewald sums. I. A theoretical and numerical comparison of various particle mesh routines", Journal of Chemical Physics
**109**7678 (1998) - ↑ Han Wang, Jun Fang, and Xingyu Gao "The optimal particle-mesh interpolation basis", Journal of Chemical Physics
**147**124107 (2017) - ↑ Ulrich Essmann, Lalith Perera, Max L. Berkowitz, Tom Darden, Hsing Lee, and Lee G. Pedersen "A smooth particle mesh Ewald method", Journal of Chemical Physics
**103**pp. 8577-8593 (1995) - ↑ Han Wang, Florian Dommert, and Christian Holm "Optimizing working parameters of the smooth particle mesh Ewald algorithm in terms of accuracy and efficiency", Journal of Chemical Physics
**133**034117 (2010) - ↑ Mark J. Abraham and Jill E. Gready "Optimization of parameters for molecular dynamics simulation using smooth particle-mesh Ewald in GROMACS 4.5", Journal of Computational Chemistry
**32**pp. 2031-2040 (2011)

**Related reading**

- L. V. Woodcock and K. Singer "Thermodynamic and structural properties of liquid ionic salts obtained by Monte Carlo computation. Part 1.—Potassium chloride", Transactions of the Faraday Society
**67**pp. 12-30 (1971) - J.W. Weenk and H.A. Harwig "Calculation of electrostatic fields in ionic crystals based upon the Ewald method", Journal of Physics and Chemistry of Solids
**38**pp. 1047-1054 (1977) - S. W. de Leeuw, J. W. Perram and E. R. Smith "Simulation of Electrostatic Systems in Periodic Boundary Conditions. II. Equivalence of Boundary Conditions", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
**373**pp. 57-66 (1980) - W. Smith; D. Fincham "The Ewald Sum in Truncated Octahedral and Rhombic Dodecahedral Boundary Conditions", Molecular Simulation
**10**pp. 67-71 (1993) - Paul E. Smith and B. Montgomery Pettitt "Efficient Ewald electrostatic calculations for large systems", Computer Physics Communications
**91**pp. 339-344 (1995) - Christopher J. Fennell and J. Daniel Gezelter "Is the Ewald summation still necessary? Pairwise alternatives to the accepted standard for long-range electrostatics", Journal of Chemical Physics
**124**234104 (2006) - Robert D. Skeel "An alternative construction of the Ewald sum", Molecular Physics
**114**pp. 3166-3170 (2016) - Shasha Yi, Cong Pan, and Zhonghan Hu "Note: A pairwise form of the Ewald sum for non-neutral systems", Journal of Chemical Physics
**147**126101 (2017)

## External resources[edit]

- Routines to perform the Ewald sum sample FORTRAN computer code from the book M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989).