Ewald sum

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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[edit]

In a periodic system one wishes to evaluate the internal energy U (Eq. 1.1 [3]):

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]

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

This internal energy is partitioned into four contributions:

U_{\mathrm total} =  U_{\mathrm real~space} + U_{\mathrm reciprocal~space} + U_{\mathrm self~energy} + U_{\mathrm surface}

Real-space term[edit]

The real space contribution to the electrostatic energy is given by [4][5] (Eq. 7a and 7b [6]):

\widehat{\frac{1}{r}}  =  \frac{\mathrm {erfc}(\alpha r)}{r}

where {\mathrm {erfc}}() is the complementary error function, and \alpha is the Ewald screening parameter. Also,

\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]

Reciprocal-space term[edit]

Self-energy term[edit]

Surface term[edit]

Particle mesh[edit]

[7] [8] [9]

Smooth particle mesh (SPME)[edit]

SPME[10]. Optimisation [11] [12].

See also[edit]

References[edit]

  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. 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)
  9. Han Wang, Jun Fang, and Xingyu Gao "The optimal particle-mesh interpolation basis", Journal of Chemical Physics 147 124107 (2017)
  10. 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)
  11. 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)
  12. 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[edit]