This document discusses the Ewald summation method, which is a technique for efficiently calculating long-range electrostatic interactions in systems with periodic boundary conditions. It separates interactions into short-range and long-range parts for faster calculation. The method decomposes the lattice sum into real space and reciprocal space contributions. It also discusses applications of the Ewald summation method, including for uniformly charged surfaces, using fast Fourier transforms to improve computation time, and calculating interactions between molecules.

1. A term project on
by
Shatrudhan Palsaniya
shatrudhan@iitg.ernet.in
Roll No. 146153010
4/20/2015
Ewald Summation Method
Center for Nanotechnology
Indian Institute of Technology Guwahati India

2. Contents
Introduction The Ewald Sum
 Physical meaning of Ewald Sum
 Ewald Methods
 Ewald Applications
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3. Introduction
 The Ewald sum is a technique for efficiently summing the
interaction between an ion and all its periodic images.
It developed algorithm to calculate long range interaction
for uniformly periodic surface in 2-D and 3-D.
 It developed algorithm to find n body for system with PBC.
 We compute, force exerted on particles for various system.
 The basic idea of Ewald sum is to separate interactions into
a short range part and a long range part.
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4. Introduction
 Consider N atoms/ions in vacuum, at locations r1, r2...rn, and possessing point charges q1,
q2...qn, respectively. The total Coulomb interaction energy is given by the following expression:
Direct Sum Method
 In MD uses PBC. Characteristic length L= |x|=|y|=|z|, x, y, z are repeat vectors when a point
charge qi at location ri, so existing point charge is ri + ax +by +cz, a,b,c lattice constant.
So ax +by+cz =nL
The above equation converges very slowly, and conditionally convergent in 3D space, means
value of the sum is not well defined.
 This summation is just distribution of charges in a certain way, only with respect to a single
cell. As the number of cells grow, there is always a finite contribution.
 The biggest drawback is that the above is an infinite series, and summing over n is not
efficient.
Draw Back of Direct Sum
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5. The Ewald Sum
 Ewald sum is a faster method to compute energies or forces.
 Ewald sum splitting in two series equations. Fast series(for
small r) which decay fast and negligible beyond some cutoff
distance, and the small part (for large r),i.e. represented by
Fourier transform.
 The short-range contribution calculated in real space, and
long-range contribution is calculated by Fourier transform.
 The Ewald sum recast in form of potential energy
written as sum of three parts,
Real (direct) space sum (Ur),
Reciprocal (imaginary, or Fourier) sum (Urn)
Constant term (U°), known as the self-term,
U Ewald=Ur+Um+U0
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n=Cell coordinate vector, n=(n1,n2,n3) =
nLx+nLy+nLz
U0 it is a self correction term, it cancels
during interaction
m(i,j,k) is reciprocal space vector
α is positive parameter that
determines the width of the
distribution, and r is the center of
distribution.
erfc(x) = 1 - erf(x) =1 -(2/√π)
The complimentary error function decreases monotonically as x increases and is defined by
In neutral charge system ewald sum transform P.E. in real and reciprocal space of form like below.
reciprocal space
 A physical interpretation of decomposition of the lattice sum follows. Each point charge in
the system surrounded by a Gaussian charge distribution of equal magnitude and opposite
sign with charge density
ρi(r) = Qi α3 exp(

7.  The choice of
charge distribution
conveniently take to
Gaussian as arbitrary.
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 Ewald sum
components of a 1-D
point-charge system.
The vertical lines are
(+/-) unit charges, with
Gaussians normalized
unity.
 In MD simulation
force quantity needed
for simulation progress.
So resulting force
equations using Um and
Ur are
Fr
p(i) = Qi [erfc(αr(i,j,n)+ exp(-αr(i,j,n))²]
Fm
p(i) is reciprocal space force.
Fm
p(i) = exp( )² sin( )
Where Fr
p(i) is real space force
 The force exerted on particle i is the sum of the direct-space
force, F ' ( i ) , and the reciprocal-space force, Fro(i), for all
components p = x,y,z. Self correction term U ° is constant and
does not contribute to the forces in a simulation, however it adds
a significant contribution to the potential energy.

8. Ewald Methods
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 Ewald Summation for uniformly charged surfaces
 Ewald Summation 2 dimensional algorithm
 Cell multipole method for non bonded interactions
 Fast Fourier Transform

9. Ewald Summation for uniformly charged surfaces
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 3D Ewald sum is 10 times faster than the time of 2D Ewald sum.
 In MD force obtained by differentiating electrostatic energy and result is:
Fi=Fi
r+Fi
k+Fi
d (1) Fx=Fy=0 Fi,z=
 Let uniformly charged surface with charge density ρ =
and electric field on particle i around surface is Ei =
 So resulting co-ion distribution on surface is Fsi=Fd
x=Fd
y=0; Fs
i,z=
 Below fig shows the surface charge density is same as force F(r) contributed real surface
charge to force F(p) parallel capacitor surface charge give by:
Fs
i = FP
i = Fi
r,p + Fi
k,p + Fi
d,p

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Ewald Summation for uniformly charged surfaces
Adsorption of co-ions surface charge densities of (a) 0.01 and (b) 0.1
Smooth surface with the charge densities of (c) 0.01 and (d) 0.1.
Neutral ones - blue spheres Co-ions are red spheres charged surface atom - green spheres

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 The computing time for the Ewald summation can be greatly reduced by O(N logN)
using FFTs.
 Lets Dirac delta function which zero except point n∆r multiply with f(r).
δ(r)=
Fast Fourier Transform Ewald Sum
δ(r)f(r)=
=
Taking Fourier transform and integrate gives δ1f(ζ) =
 Integration contain M pieces of information that evaluated at point k=0 ,1…..∞. So a DFT is
contain even & odd terms in following manner.
F(k) = Feven(k) + Fodd(k)
+=
 Advantage of above equation is tells enhances the computational ability.

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Electrostatic interactions Ewald Sum
 Electrostatics interaction always consider partial charges placed on centre of atoms.
 Lets a point charge system with delta function;
Screened by Gaussian Cloud
of charge
Point charge
potential
To compensate screen charges Added
compensating Interaction by
multiplying (-1)
TOTAL SUM = Point Charges + Screening
Gaussian Charges + Compensating
Gaussian Charges
Total energy of point charge will have
three terms as above fig:
(1) Screen charges addition
(2) Excluded self interaction b/w point
charges
(3) Summation term (Potential
distribution of reciprocal charge
space)

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Molecule based pairwise Ewald summation
 Sum of paired & spherical interaction perform overall neutral region sum.
 In crystal & fluids we can groups pair of ions electro neutral dipoles and perform
summation.
 Thus charge based truncation lead to convergence.
 If centre of mass is beyond the cut-off interaction then might be or not paired ion.
 Thus its important have to cut-off radii larger the effective radii as shown in fig.

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Cell Multi Pole Sum interaction
 The cell multipole method (CMM) provides a treatment of the nonbond interactions
for nonperiodic systems .
 Multi pole moments of each cell calculated by summing atoms in side the cell.
 CMM is valid if space b/w interacting particles larger than sum of radii
convergence of multipoles.
 In CMM multipole expansion used for more than one cell distance away & interaction those
within cell ,atomic pair wise interaction method is employed.
A A A A A A
A A B B B B B B A
B B B B B B
A A B B N N N B A
B B N N B
A A B B N N N B A
B B B B B B
A A A A A A
A A A A A A
 27 cells(cell in which atom positioned and surrounded by 26
cells)
 Faraway cell is calculated by using multi pole expansion.
 Potential due to faraway cell will constant for containing avg.
4 atoms.
 Total potential via faraway cell express by taylor series exp.
 If M total cell than M-27 is faraway cells.
 Cell-Cell interaction for entire system order will M(M-27).
* Interaction with atom in cell A&B calculated using Tayler series multipole expansion*

15. References
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