Ewald sum - Revision history

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	<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>
	<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/10.1063/1.477414 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)====		====Smooth particle mesh (SPME)====
	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>. Optimisation		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>. Optimisation

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	*[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/10.1080/00268976.2016.1222455 Robert D. Skeel "An alternative construction of the Ewald sum", Molecular Physics '''114''' pp. 3166-3170 (2016)]		*[http://dx.doi.org/10.1080/00268976.2016.1222455 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==		==External resources==

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	*[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/10.1080/00268976.2016.1222455 Robert D. Skeel "An alternative construction of the Ewald sum", Molecular Physics '''114''' pp. 3166-3170 (2016)]

	==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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	==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>
			<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>
	====Smooth particle mesh (SPME)====		====Smooth particle mesh (SPME)====
	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>. Optimisation		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>. Optimisation

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	~~{{stub-general}}~~
	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]].

Carl McBride: /* Smooth particle mesh (SPME) */ Added a publication

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	<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>		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>. Optimisation
	<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>
			<ref>[http://dx.doi.org/10.1002/jcc.21773 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)]</ref>.

	==See also==		==See also==
	*[[Reaction field]]		*[[Reaction field]]

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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==
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	*[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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	'''Related reading'''		'''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/10.1039/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/10.1098/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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	[[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==		==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>):		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>
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	==See also==		==See also==
	*[[Reaction field]]		*[[Reaction field]]
			*[[Wolf method]]
	==References==		==References==
	<references/>		<references/>
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	*[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 <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.~~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- (1966)]~~		*[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.1980.0135 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~~)]
	*[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)]