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> is 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]]. | ||
==Derivation== | |||
In a periodic system one wishes to evaluate (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]]. | |||
==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> | ||
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<references/> | <references/> | ||
'''Related reading''' | '''Related reading''' | ||
*[http://dx.doi.org/10. | *[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/10.1098/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/10.1098/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/10.1080/08927029308022499 W. Smith; D. Fincham "The Ewald Sum in Truncated Octahedral and Rhombic Dodecahedral Boundary Conditions", Molecular Simulation '''10''' pp. 67-71 (1993)] | ||
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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 (Eq. 1.1 [3]):
![{\displaystyle 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b84b8efc8d23c54ba1b5ffdb151f78740807311)
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.
Particle mesh
Smooth particle mesh (SPME)
See also
References
- ↑ 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)
- ↑ 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)
- ↑ 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)
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)
- 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)
External resources
- 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).