Ewald sum: Difference between revisions

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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]].
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==
==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/10.1063/1.464397    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>
====Smooth particle mesh (SPME)====
====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/10.1063/1.470117    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>
<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==
==See also==
Line 25: Line 42:
*[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/10.1016/0010-4655(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/10.1063/1.2206581    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/10.1063/1.3599045 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)]
==External resources==
==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)].
*[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: Computer simulation techniques]]
[[category: electrostatics]]
[[category: electrostatics]]

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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

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

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

Self-energy term

Surface term

Particle mesh

[7]

Smooth particle mesh (SPME)

[8] [9]

See also

References

  1. Paul Ewald "Die Berechnung Optischer und Electrostatischer Gitterpotentiale", Annalen der Physik 64 pp. 253-287 (1921)
  2. 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)
  3. 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)
  4. W. Smith "Point Multipoles in the Ewald Summation", CCP5 Newsletter 4 pp. 13-25 (1982)
  5. W. Smith "Point Multipoles in the Ewald Summation (Revisited)", CCP5 Newsletter 46 pp. 18-30 (1998)
  6. 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)
  7. 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)
  8. 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)
  9. 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)

Related reading

External resources