Ewald sum: Difference between revisions

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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
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]].
[[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]].
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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>):
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>  
:<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]].
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]].

Revision as of 13:23, 13 June 2014

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)

SPME[8]. Optimisation [9] [10].

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)
  10. 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

External resources