Editing Ewald sum

The '''Ewald sum''' technique <ref>[http://dx.doi.org/10.1002/andp.19213690304  Paul Ewald "Die Berechnung Optischer und Electrostatischer Gitterpotentiale", Annalen der Physik '''64''' pp. 253-287 (1921)]</ref> was originally developed by Paul Ewald to evaluate the Madelung constant <ref>[http://dx.doi.org/10.1063/1.1727895 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)]</ref>. It is now widely used in order to simulate systems with
[[long range interactions]] (typically, [[Electrostatics |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_analysis | Fourier space]].
==Derivation==
In a periodic system one wishes to evaluate the [[internal energy]] <math>U</math> (Eq. 1.1 <ref>[http://dx.doi.org/10.1098/rspa.1980.0135 	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)]</ref>):

:<math>U = \frac{1}{2} {\sum_{\mathbf n}}^{'} \left[ \sum_{i=1}^N \sum_{j=1}^N \phi \left({\mathbf r}_{ij} + L{\mathbf n}, {\mathbf \Omega_i}, {\mathbf \Omega_j} \right)   \right] </math> 

where one sums over all the [[Building up a simple cubic lattice | simple cubic lattice]] points <math>{\mathbf n} = (l,m,n)</math>. The prime on the first summation indicates that if <math>i=j</math> then the <math>{\mathbf n} = 0</math> term is omitted. <math>L</math> is the length of the side of the cubic simulation box, <math>N</math> is the number of particles, and <math>{\mathbf \Omega}</math> represent the [[Euler angles]].

This internal energy is partitioned into four contributions:

:<math>U_{\mathrm total} =  U_{\mathrm real~space} + U_{\mathrm reciprocal~space} + U_{\mathrm self~energy} + U_{\mathrm surface}  </math>

====Real-space term====
The real space contribution to the electrostatic energy is given by <ref>[http://www.ccp5.ac.uk/ftpfiles/ccp5.newsletters/4/pdf/smith.pdf W. Smith  "Point Multipoles in the Ewald Summation", CCP5  Newsletter '''4''' pp. 13-25 (1982)]</ref><ref>[http://www.ccp5.ac.uk/ftpfiles/ccp5.newsletters/46/pdf/smith.pdf W. Smith "Point Multipoles in the Ewald Summation (Revisited)", CCP5  Newsletter '''46''' pp. 18-30 (1998)]</ref>  (Eq. 7a and 7b <ref>[http://dx.doi.org/10.1063/1.3599045 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)]</ref>):

:<math>\widehat{\frac{1}{r}}  =  \frac{\mathrm {erfc}(\alpha r)}{r}</math>

where <math>{\mathrm {erfc}}()</math> is the [http://mathworld.wolfram.com/Erfc.html complementary error function], and <math>\alpha</math> is the Ewald screening parameter. Also,

:<math>\widehat{ \frac{1}{r^{2n+1}} } =  r^{-2} \left[ \widehat{ \frac{1}{r^{2n-1}} } +  \frac{(2\alpha^2)^n}{ \sqrt{\pi} \alpha (2n-1)!! } \exp(-\alpha^2r^2) \right] </math>
====Reciprocal-space term====
====Self-energy term====
====Surface term====

==Particle mesh==
<ref>[http://dx.doi.org/10.1063/1.464397 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)]</ref>
<ref>[http://dx.doi.org/10.1063/1.477414  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)]</ref>
<ref>[http://dx.doi.org/10.1063/1.4994857 Han Wang, Jun Fang, and Xingyu Gao "The optimal particle-mesh interpolation basis", Journal of Chemical Physics '''147''' 124107 (2017)]</ref>
====Smooth particle mesh (SPME)====
<ref>[http://dx.doi.org/10.1063/1.470117     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)]</ref>
<ref>[http://dx.doi.org/10.1063/1.3446812  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)]</ref>
==See also==
*[[Reaction field]]
*[[Wolf method]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1039/TF9716700012  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)]
*[http://dx.doi.org/10.1016/0022-3697(77)90209-8 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)]
*[http://dx.doi.org/10.1098/rspa.1980.0136 	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)]
*[http://dx.doi.org/10.1080/08927029308022499 W. Smith; D. Fincham "The Ewald Sum in Truncated Octahedral and Rhombic Dodecahedral Boundary Conditions", Molecular Simulation '''10''' pp. 67-71 (1993)]
*[http://dx.doi.org/10.1016/0010-4655(95)00058-N  Paul E. Smith and B. Montgomery Pettitt  "Efficient Ewald electrostatic calculations for large systems", Computer Physics Communications  '''91''' pp. 339-344 (1995)]
*[http://dx.doi.org/10.1063/1.2206581     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)]
*[http://dx.doi.org/10.1080/00268976.2016.1222455 Robert D. Skeel "An alternative construction of the Ewald sum", Molecular Physics '''114''' pp. 3166-3170 (2016)]
*[http://dx.doi.org/10.1063/1.4998320 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==
*[ftp://ftp.dl.ac.uk/ccp5/ALLEN_TILDESLEY/F.22    Routines to perform the Ewald sum] sample FORTRAN computer code from the book [http://www.oup.com/uk/catalogue/?ci=9780198556459 M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989)].
[[category: Computer simulation techniques]]
[[category: electrostatics]]