Ewald sum

The Ewald sum technique was originally developed by Paul Ewald to evaluate the Madelung constant . 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 \(U\) (Eq. 1.1 ):

\[U = \frac{1}{2} {\sum_{\mathbf n}}^' \left[ \sum_{i=1}^N \sum_{j=1}^N \phi \left({\mathbf r}_{ij} + L{\mathbf n}, {\mathbf \Omega_i}, {\mathbf \Omega_j} \right) \right] \]

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

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

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

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

\[\widehat{ \frac{1}{r^{2n+1}} } = r^{-2} \left[ \widehat{ \frac{1}{r^{2n-1}} } + \frac{(2\alpha^2)^n}{ \sqrt{\pi} \alpha (2n-1)!! } \exp(-\alpha^2r^2) \right] \]

See also

• Reaction field
• Wolf method

References

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)
• 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)
• 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).