THE JOURNAL OF CHEMICAL PHYSICS 128, 214106 共2008兲
On mesh-based Ewald methods: Optimal parameters
for two differentiation schemes
Harry A. Sterna兲 and Keith G. Calkinsb兲
Department of Chemistry, University of Rochester, Rochester, New York 14607, USA
共Received 31 March 2008; accepted 1 May 2008; published online 6 June 2008兲
The particle-particle particle-mesh Ewald method for the treatment of long-range electrostatics
under periodic boundary conditions is reviewed. The optimal Green’s function for exact 共real-space
differentiation兲, which differs from that for reciprocal-space differentiation, is given. Simple
analytic formulas are given to determine the optimal Ewald screening parameter given a
differentiation scheme, a real-space cutoff, a mesh spacing, and an assignment order. Simulations of
liquid water are performed to examine the effect of the accuracy of the electrostatic forces on
calculation of the static dielectric constant. A target dimensionless root-mean-square error of 10−4 is
sufficient to obtain a well-converged estimate of the dielectric constant. © 2008 American Institute
of Physics. 关DOI: 10.1063/1.2932253兴
I. INTRODUCTION
Many molecular simulations use electrostatic models
given by a set of point charges at atomic sites, and are performed under conditions of periodic boundary conditions.1
Although there is still debate in literature,2–4 some degree of
consensus has emerged that for such a system, the Ewald
sum is the most reliable method for minimizing artifacts due
to finite system size.5–7 In the past few decades several fast
methods have been proposed for performing the reciprocalspace part of the Ewald sum that take advantage of the fast
Fourier transform 共FFT兲 algorithm by discretization onto a
mesh.8–13 These methods are conceptually similar, but differ
in the particular way that charges are mapped onto the mesh,
the way that the electrostatic potential is obtained from the
mesh-based charges and the way that the potential is differentiated in order to obtain the electric field and the forces.
Papers by Deserno and Holm give an excellent discussion of
the subtleties of different methods.14,15
The present work focuses on one particular set of algorithms, variously called the particle-particle particle-mesh
Ewald 共P3ME兲 method or smooth particle-mesh Ewald
method. In this method, charges are interpolated smoothly
onto mesh points. The electrostatic potential is obtained not
from the “true” Coulomb Green’s function but from a
Green’s function that is optimal 共in the sense discussed below兲 given the particular discretization scheme. Differentiation may be performed either exactly in real space 共by differentiating the weight functions used to map charges onto
the mesh兲 or approximately in reciprocal space 共by multiplying the reciprocal-space potential by −ik兲.
The goals of this paper are 共i兲 to give the optimal
Green’s function for exact, real-space differentiation, which
differs from that for reciprocal-space differentiation, and
does not seem to have appeared in literature; 共ii兲 to give
a兲
Electronic mail: [email protected].
Present address: Department of Mathematics, Andrews University, Berrien
Springs, MI 49104. Electronic mail: [email protected].
b兲
0021-9606/2008/128共21兲/214106/8/$23.00
simple 共fitted兲 formulas for the optimal Ewald screening parameter and error estimate given a differentiation scheme, a
real-space cutoff, a mesh spacing, and assignment order; and
共iii兲 to study the relation between the accuracy of the electrostatic forces and the accuracy of a calculated physical observable, the static dielectric constant. A recent paper by Ballenegger et al.16 focuses on optimal Green’s functions for
electrostatic energies; in this paper, we focus on the accuracy
of the force calculation, as that is most directly relevant for
molecular dynamics simulations.
II. THEORY
Consider a neutral system of point charges qi at locations
ri, under periodic boundary conditions defined by primitive
translation vectors a1 , a2 , a3. The energy of the system is
given by
U=
1
兺 qi␾共ri兲,
2 i
共1兲
where the electrostatic potential is given by
␾共ri兲 = 兺 ⬘
j艎
qj
.
兩r j − ri + 艎兩
共2兲
Here the sum is over all lattice vectors 艎 = n1a1 + n2a2 + n3a3,
where n1 , n2 , n3 are integers, and the prime denotes the omission of the j = i term when 艎 = 0. This sum is only conditionally convergent and is conventionally taken to be the limit of
a 共roughly兲 spherical collection of neighboring boxes. The
limit depends not only on the order in which the interactions
with neighboring boxes are summed but on the medium surrounding the collection of boxes. The most common choice
is to consider “tin-foil” or conducting boundary conditions,
and that is the choice adopted here.
In the Ewald method, the slowly and only conditionally
converging sum in Eq. 共2兲 is evaluated by adding a screening
Gaussian charge distribution of opposite sign to each point
charge. The interactions between screened charges are then
128, 214106-1
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214106-2
J. Chem. Phys. 128, 214106 共2008兲
H. A. Stern and K. G. Calkins
short ranged, and only neighbors within a given radius of
each screened charge need to be considered. In order to recover the original potential, a canceling Gaussian charge distribution must be added as well. The potential due to this
canceling Gaussian charge distribution is evaluated in Fourier space. A correction due to the interaction of a point
charge with its own Gaussian screening charge must also be
included. Using the Ewald method, the electrostatic potential
is written as a sum of a real-space term, a self-energy term,
and a reciprocal-space term, ␾共ri兲 = ␾real共ri兲 + ␾self共ri兲
+ ␾reciprocal共ri兲, where
␾real共ri兲 = 兺 ⬘q j
j艎
␾self共ri兲 = −
erfc共␩兩ri − r j + 艎兩兲
,
兩ri − r j + 艎兩
2␩
冑␲ q i ,
␾reciprocal共ri兲 =
1
兺 eik·riĜk兺j e−ik·r jq j .
V k⫽0
共3兲
共4兲
共5兲
Here ␩ is the width of the screening Gaussian and V
= a1 · 共a2 ⫻ a3兲 is the volume of the basic simulation box. The
reciprocal-space term is a sum over reciprocal lattice vectors
k, which are given by all linear combinations of primitive
vectors for the reciprocal lattice,
b1 =
2␲
共a2 ⫻ a3兲,
V
共6兲
b2 =
2␲
共a3 ⫻ a1兲,
V
共7兲
2␲
共a1 ⫻ a2兲.
b3 =
V
共8兲
4␲ −兩k兩2/4␩2
e
兩k兩2
共9兲
are the Fourier coefficients for the Green’s function associated with the screening distribution 共that is, the Fourier coefficients of the potential due to a charge distribution consisting of a periodic array of Gaussians centered at lattice
vectors兲.
The basic idea behind the P3M method is to discretize
the charges onto a mesh and to replace the continuous Fourier transforms in Eq. 共5兲 with discrete Fourier transforms,
which can be evaluated via the FFT algorithm. The simplest
way to generate a mesh is to take a simple cubic grid in
“crystallographic space”—that is, to choose a number of
mesh points N1 , N2 , N3 along each primitive lattice vector
and to take the set of points
rn ⬅
n1
n2
n3
a1 + a2 + a3 .
N1
N2
N3
共11兲
The charges at each mesh point qrn are computed from an
assignment function W共r兲:
qrn = 兺 qiW共rn − ri兲.
共12兲
i
The particular form of the assignment function is discussed
below; for now, it is simply assumed that it is even and
differentiable.
Once the charges are assigned to the real-space mesh,
the Fourier coefficients of the mesh-based charge density
␳ˆ kn =
1
兺 e−ikn·rnqrn
V rn
共13兲
may be computed via the FFT algorithm. The Fourier coefficients of the mesh-based potential are then given by multiplying those of the charge density by those of the Green’s
function. A key idea in the P3M method is to replace the
Fourier coefficients of the actual screened Coulomb Green’s
2
2
function Ĝk = 共4␲ / 兩k兩2兲e−兩k兩 /4␩ with different coefficients
Ĝk⬘ , which are chosen to minimize the error resulting from
n
discretization,8,14 so that
␾ˆ kn =
兺
kn⫽0
Ĝk⬘ ␳ˆ kn .
共14兲
n
The potential on the real-space mesh is then computed from
a reverse FFT:
␾rn = 兺 eikn·rn␾ˆ kn .
共15兲
kn
Finally, the mesh-based potential is interpolated back to the
locations of the actual charges using the same assignment
function:
␾共ri兲 = 兺 ␾rnW共ri − rn兲.
The coefficients
Ĝk ⬅
k n ⬅ n 1b 1 + n 2b 2 + n 3b 3 .
共10兲
Here n = 共n1 , n2 , n3兲 is a triplet of integers, each one chosen
from 0 to N1 − 1 , N2 − 1 , N3 − 1, respectively. Associated with
the lattice of mesh points is a 共finite兲 reciprocal lattice,
共16兲
rn
The P3M reciprocal-space potential is therefore
␾P3M共ri兲 = 兺 W共ri − rn⬘ 兲
rn⬘
1
e−ikn·rn 兺
兺 eikn·rn⬘Ĝk⬘n兺
V kn⫽0
j
rn
⫻W共rn − r j兲q j ,
共17兲
where Ĝk⬘ remains to be determined.
n
In order to determine the forces on the particles, the
potential must be differentiated to obtain the electric field.
There are several methods by which this can be done. This
paper considers two: exact differentiation in real space using
the gradient of the assignment function, which gives
EP3M共ri兲 = 兺 − ⵱W共ri − rn⬘ 兲
rn⬘
1
兺 eikn·rn⬘Ĝk⬘n
V kn⫽0
⫻ 兺 e−ikn·rn 兺 W共rn − r j兲q j ,
rn
共18兲
j
and approximate differentiation on the reciprocal-space mesh
by multiplying the potential by ikn, which gives
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214106-3
J. Chem. Phys. 128, 214106 共2008兲
On mesh-based Ewald methods
lated to the root-mean-square 共rms兲 error in the forces for a
system of N charges qi by15
冋兺 册
N
⌬FP3M ⬇
q2i N−1/2V−2/3␹P3M .
i=1
For differentiation in real space, minimizing Eq. 共20兲 with
respect to Ĝk⬘ yields
n
Ĝk⬘ =
n
兺MÛk2
关兺MÛk2
n+M
n+M
Ĝkn+M兩kn+M兩2
兴关兺MÛk2
n+M
兩kn+M兩2兴
.
共21兲
For differentiation on the reciprocal-space mesh, a different
expression is obtained,
FIG. 1. Comparison between analytical estimate for the dimensionless rms
error in the reciprocal-space contribution to the electric field ␹ with the
numerical rms error for a randomly distributed unit source charge and target
location in a cubic box of unit volume, ␴E. Circles, real-space differentiation; crosses, reciprocal-space differentiation. P3M calculations were performed with grid spacings ranging from 0.01 to 0.1, assignment orders from
2 to 5, and screening parameters from 4 to 32.
EP3M共ri兲 = 兺 W共ri − rn⬘ 兲
rn⬘
⫻共− ikn兲Ĝk⬘
共19兲
2
␹ P3M
⬅V
−2/3
冕冕
V
Ĝkn+Mkn · kn+M
关兺MÛk2
n+M
兴2兩kn兩2
共22兲
.
N 1N 2N 3
Ŵk ,
V
共23兲
冕
共24兲
where
n
If real-space differentiation is performed, only one reverse
FFT needs to be done. If differentiation on the reciprocalspace mesh is used, a total of four reverse FFTs is needed:
one for the potential and one for each component of the field.
As mentioned above, the coefficients of the reciprocalspace mesh-based Green’s function Ĝkn are chosen to minimize the error in the calculated electric field due to discretization. One estimate for this discretization error is the square
of the error in the field due to a Gaussian distribution of unit
charge, integrated over all positions within the primary unit
cell, and all positions of the distribution:8
n+M
Equations 共21兲 and 共22兲 are derived in the Appendix. In these
equations,
Ûk ⬅
e−ik ·r 兺 W共rn − r j兲q j .
兺
j
r
兺MÛk2
n
1
兺 eikn·rn⬘
V kn⫽0
n n
n
Ĝk⬘ =
Ŵk =
e−ik·rW共r兲d3r
V
are the Fourier coefficients of the assignment function. Since
the assignment function is even, the coefficients are real and
Ŵk = Ŵ−k. M is a triplet of multiples of the number of gridpoints in each dimension; that is,
M ⬅ 共m1N1,m2N2,m3N3兲,
共25兲
where m1 , m2 , m3 are integers. The sums are over all such
triplets, and
kn+M ⬅ 共n1 + m1N1兲b1 + 共n2 + m2N2兲b2 + 共n3 + m3N3兲b3 .
共26兲
兩EP3M共r1,r2兲
V
− Ereciprocal共r1,r2兲兩2d3r1d3r2 .
共20兲
Here EP3M共r1 , r2兲 and Ereciprocal共r1 , r2兲 denote the electric
field at r1 due to a Gaussian distribution of total unit charge
centered at r2, calculated with P3M and with the ordinary
Ewald reciprocal-space sum, respectively. The factor of V−2/3
is chosen to make ␹P3M dimensionless. Equation 共20兲 is re-
In practice, only a few triplets M need to be included in the
sum for each n, since Ĝk and Ûk decrease rapidly as 兩k兩
becomes large.
It is stressed that the Green’s function coefficients yielding the best accuracy in the forces will depend on the method
of differentiation. Equation 共21兲 gives the best Green’s function coefficients in the case that differentiation is performed
TABLE I. For real-space differentiation: optimal Ewald screening parameter ␩ given in terms of the real-space
cutoff rc and grid spacing s by the formula ␩ = exp关a1rc2 + a2共ln s兲2 + a3rc共ln s兲 + a4rc + a5s + a6兴 for several assignment orders and unit box length.
Assignment Order
2
3
4
5
a1
a2
a3
a4
a5
a6
6.551
6.012
5.551
5.249
−0.031
−0.042
−0.054
−0.061
0.209
0.359
0.460
0.513
−6.371
−5.355
−4.595
−4.113
−0.441
−0.587
−0.728
−0.824
2.143
1.978
1.694
1.477
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214106-4
J. Chem. Phys. 128, 214106 共2008兲
H. A. Stern and K. G. Calkins
TABLE II. For real-space differentiation: dimensionless rms error ␹ for optimal ␩ given in terms of the
real-space cutoff rc and grid spacing s by the formula ␩ = exp关a1rc2 + a2共ln s兲2 + a3rc共ln s兲 + a4rc + a5s + a6兴 for
several assignment orders and unit box length.
a1
a2
a3
a4
a5
a6
9.625
18.590
25.453
31.248
−0.027
−0.010
−0.010
−0.053
0.104
0.123
0.157
0.700
−9.976
−18.494
−26.037
−31.014
0.589
1.611
2.525
2.998
3.596
6.161
9.045
11.187
Assignment order
2
3
4
5
in real space, while Eq. 共22兲 gives the best Green’s function
coefficients in the case that differentiation is performed on
the reciprocal-space mesh.
The mean-square error for real-space differentiation is
2
␹P3MV2/3 =
冉兺
Ĝk2 兩k兩2
k⫽0
−兺
n
冉
冊
关兺MÛk2 Ĝkn+M兩kn+M兩2兴2
n+M
关兺MÛk2
n+M
兴关兺MÛk2
n+M
兩kn+M兩2兴
冊
2
冉兺
Ĝk2 兩k兩2
k⫽0
−兺
n
冉
冊
关兺MÛk2
n+M
Ĝkn+Mkn · kn+M兴2
关兺MÛk2 兴2兩kn兩2
n+M
冋
sin共␬/2兲
␬/2
册
p
共32兲
.
The actual assignment function W共r兲 may then be written as
a product of the displacement along each lattice vector,
scaled by the number of gridpoints:
W共r兲 = w共s1兲w共s2兲w共s3兲,
, 共27兲
共33兲
where
r=
and for reciprocal-space differentiation is
␹P3MV2/3 =
=
s2
s3
s1
a1 + a2 + a3 .
N1
N2
N3
共34兲
Here s1 , s2 , s3 are dimensionless coordinates ranging from 0
to N1 , N2 , N3. Therefore,
冊
Û共kn兲 =
,
共28兲
=
in each case using the optimal Green’s function coefficients
for that method of differentiation.
At this point it is still necessary to specify the form of
the assignment function W共r兲, which gives the fraction of
charge assigned to a grid point at a displacement r from a
charge. We use here the set of assignment functions given by
Hockney and Eastwood.8,14 To construct W共r兲 for arbitrary
periodic boundary conditions, an assignment function w共s兲 is
first considered for a one-dimensional mesh with unit spacing between mesh points. A p-order assignment function
共that is, an assignment function that assigns a charge to p
mesh points兲 may be constructed from a p-fold convolution
of the indicator function on the interval 关− 21 , 21 兴:
冋
N 1N 2N 3
V
冕
e−2␲n1s1/N1e−2␲n2s2/N2e−2␲n3s3/N3
V
⫻w共s1兲w共s2兲w共s3兲d3r
sin共␲n1/N1兲
␲n1/N1
册冋
p
sin共␲n2/N2兲
␲n2/N2
册冋
p
sin共␲n3/N3兲
␲n3/N3
册
共35兲
p
. 共36兲
III. NUMERICAL RESULTS
The expressions for the rms error for both differentiation
methods, Eqs. 共27兲 and 共28兲, were checked by comparing
with numerical calculations. Parameters were chosen such
that the accuracy of the calculations would vary over several
orders of magnitude. Results are shown in Fig. 1; close
agreement between the analytical expression and numerical
calculation is obtained.
An estimate for the rms error in the forces due to the
real-space part of the Ewald sum due to Kolafa and Perram17
is
冋兺 册
N
⌬Freal
space ⬇
q2i N−1/2V−2/3␹real
space ,
i=1
共29兲
where
1关−1/2,1/2兴共s兲 =
再
1
2
1 if − 艋 s 艋
0 otherwise.
1
2
冎
␹real space =
共30兲
The Fourier transform of this assignment function is
ŵ p共␬兲 =
冕
w p共s兲e−i␬sds
where
共31兲
2
冑rcV1/3 e
−␩2rc2
.
The dimensionless estimate ␹real space may be added to that
for the reciprocal-space forces in order to obtain a total error
estimate. The total estimate may then be minimized with
respect to the screening parameter ␩ in order to obtain an
optimal value. This was performed for several different values of the real-space cutoff rc, grid spacing s, and assignment
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214106-5
J. Chem. Phys. 128, 214106 共2008兲
On mesh-based Ewald methods
TABLE III. For reciprocal-space differentiation: optimal Ewald screening parameter ␩ given in terms of the
real-space cutoff rc and grid spacing s by the formula ␩ = exp关a1rc2 + a2共ln s兲2 + a3rc共ln s兲 + a4rc + a5s + a6兴 for
several assignment orders and unit box length.
Assignment order
2
3
4
5
a1
a2
a3
a4
a5
a6
5.902
5.434
5.101
4.815
−0.043
−0.055
−0.064
−0.069
0.365
0.473
0.540
0.576
−5.238
−4.461
−3.917
−3.529
−0.603
−0.745
−0.852
−0.925
1.875
1.612
1.376
1.200
order. The optimal screening parameters and errors using
them were fitted to a simple analytical expression, as shown
in Tables I–IV. These expressions allow one to obtain the
optimal screening parameter and associated error for a given
real-space cutoff, grid spacing, and assignment order. The
deviation between the analytic expressions and actual values
was small 共on the order of a few percent兲 for the range of
cutoffs, spacings, and assignment orders.
In order to determine the relationship between the accuracy of the electrostatic forces, as given by the dimensionless
mean-square error described above, and a physical observable which is fairly demanding to calculate 共the static dielectric constant兲, several simulations of liquid water were performed using the simple point charge/extended water
model.18 The system consisted of 216, 343, or 512 water
molecules, under conditions of constant volume, in a cubic
box with size chosen such that the density was 0.997 g / cm3.
Temperature was maintained at 298 K using intermittent resampling of velocities from the Boltzmann distribution. Periodic boundary conditions were used with Lennard-Jones
forces and real-space electrostatic forces smoothly truncated
at 9 Å, fourth-order assignment, and grid spacings varying
from one-hundredth to one-tenth of the box size. For each
grid spacing, 18 independent simulations were run, each consisting of 106 2 fs timesteps for a total duration of 36 ns. The
dielectric constant was calculated from the mean-square
value of the total system dipole moment,
differentiation, which differs from that for reciprocal-space
differentiation. We determined optimal values for the Ewald
screening parameter 共for both methods of differentiation兲 for
a range of real-space cutoffs, grid spacings, and assignment
orders. These and the associated dimensionless rms errors in
the force were fitted to simple analytical forms. These expressions should be a practical tool for choosing appropriate
parameters for the Ewald calculation, given a particular system size and target accuracy. Given that the static dielectric
constant of water exhibits strong dependence on the treatment of long-range electrostatic forces,1,19 it is a good property for determining an appropriate target accuracy for the
P3M calculation. The present simulations show that this target accuracy as given by the dimensionless rms error should
be around 10−4, independent of system size. It should be
noted that a target accuracy, which leads to good convergence of the static dielectric constant for water, should be
more than accurate enough for large-scale simulations of biological molecules,20–22 which are subject to many other
sources of error, most notably due to the potential energy
function.
APPENDIX: ESTIMATES OF THE P3M
DISCRETIZATION ERROR FOR REAL-SPACE
AND RECIPROCAL-SPACE DIFFERENTIATION
The estimate for the discretization error of the P3M
method is
4␲
具M2典.
⑀=1+
3VkBT
2
␹P3M ⬅ V−2/3
Results are shown in Fig. 2. They indicate that a reasonable
target value for the dimensionless rms error is around 10−4
independent of system size.
IV. CONCLUSIONS
In this paper we derived the optimal Green’s function for
the particle-particle P3ME method using exact, real-space
冕冕
V
兩EP3M共r1,r2兲
V
− Ereciprocal共r1,r2兲兩2d3r1d3r2 .
Here EP3M共r1 , r2兲 and Ereciprocal共r1 , r2兲 denote the electric
field at r1 due to a Gaussian distribution of total unit charge
centered at r2, calculated with P3M with the ordinary Ewald
reciprocal-space sum, respectively. By Parseval’s theorem,
TABLE IV. For reciprocal-space differentiation: dimensionless rms error ␹ for optimal ␩ given in terms of the
real-space cutoff rc and grid spacing s by the formula ␩ = exp关a1rc2 + a2共ln s兲2 + a3rc共ln s兲 + a4rc + a5s + a6兴 for
several assignment orders and unit box length.
Assignment order
2
3
4
5
共A1兲
a1
a2
a3
a4
a5
a6
17.483
24.524
30.505
35.825
−0.034
−0.035
−0.072
−0.137
0.344
0.357
0.806
1.573
−16.878
−24.508
−29.898
−33.543
1.314
2.213
2.752
2.963
5.680
8.453
10.702
12.329
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214106-6
J. Chem. Phys. 128, 214106 共2008兲
H. A. Stern and K. G. Calkins
ÊP3M,reciprocal共k1,k2兲 ⬅
冕冕
V
EP3M,reciprocal
V
⫻共r1,r2兲e−ik1·r1e−ik2·r2d3r1d3r2 ,
共A3兲
and the sums are over all reciprocal lattice vectors. The ordinary Ewald reciprocal-space field is given by
1
兺 共− ik兲Ĝkeik·共r1−r2兲 ,
V k⫽0
Ereciprocal共r1,r2兲 =
2
共A4兲
2
where Ĝk ⬅ 共4␲ / 兩k兩2兲e−兩k兩 /4␩ .
For differentiation in real space using the gradient of the
assignment function,
FIG. 2. Dielectric constant for liquid water ⑀ as a function of the dimensionless rms error in the force, ␹. Circles, 216 water molecules; squares, 343
water molecules; crosses, 512 water molecules. Error bars are the standard
error of the mean taken from block averaging.
EP3M共r1,r2兲 = 兺 − ⵱W共r1 − rn⬘ 兲
rn⬘
⫻
␹P3M = V−8/3 兺 兺 兩ÊP3M共k1,k2兲 − Êreciprocal共k1,k2兲兩2 ,
2
k1 k2
1
e−ikn·rnW共rn − r2兲.
兺 eikn·rn⬘Ĝk⬘n兺
V kn⫽0
rn
共A5兲
共A2兲
Therefore,
where
ÊP3M共k1,k2兲 = −
1
兺
V
rn⬘
=−
冋冕
⵱ W共r1 − rn⬘ 兲e−ik1·r1d3r1
V
1
ik1Ŵk1Ŵk2 兺 Ĝk⬘
n
V
kn⫽0
= − Vik1Ûk1Ûk2
Here
␦k =
再
kn⫽0
eikn·rn⬘ Ĝk⬘
e−ik ·r
兺
r
n n
n
n
⫻
冋冕
V
W共rn − r2兲e−ik2·r2d3r2
册
1
e−ikn·rn关Ŵk2e−ik2·rn兴
兺 关ik1Ŵk1e−ik1·rn⬘兴 k兺⫽0 eikn·rn⬘Ĝk⬘n兺
V
rn
n
rn⬘
=−
册兺
1 if k = 0
0 if k ⫽ 0
兺
kn⫽0
Ĝk⬘
n
冋兺
e−i共k1−kn兲·rn⬘
rn⬘
册冋 兺
e−i共kn+k2兲·rn
rn
册
␦k −k 兺 ␦k +k +M .
兺
M
M
1
1
n+M1
2
n
共A6兲
2
2
冎
共A7兲
is the Kronecker delta function and M1 , M2 are triplets of multiples of the number of gridpoints in each dimension, as in Eq.
共25兲. We have made use of the identity
ei共k−k 兲·r = 兺 ␦k−k
兺
r
M
n
n
n+M
,
n
where the first sum is over all gridpoints, and the second is over all triplets of multiples of the number of gridpoints in each
dimension.
Similarly,
Êreciprocal共k1,k2兲
=−
1
兺 ikĜk
V k⫽0
冋冕
V
ei共k−k1兲·r1d3r1
册冋冕
V
册
ei共k+k2兲·r2d3r2 = − V 兺 ikĜk␦k1−k␦k+k2 = − Vik1Ĝk1␦k1+k2关1 − ␦k1兴.
k⫽0
Since
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共A8兲
214106-7
J. Chem. Phys. 128, 214106 共2008兲
On mesh-based Ewald methods
兺 兩ÊP M共k1,k2兲兩2 = V2兺
兺 兩k1兩2Ûk2 Ûk2
兺
k k
k k
3
1
1
1
2
= V2
2
2
冋
共Ĝk⬘ 兲2 兺 兩kn+M兩2Ûk2
兺
k ⫽0
M
n
n
Ĝk⬘ Ĝk⬘ ⬘ ⫻ 兺 ␦k1−kn+M 兺 ␦k2+kn+M 兺 ␦k1−kn +M 兺 ␦k2+kn +M
n
⬘ 1⬘
⬘ 2⬘
1
2
n
兺
kn⫽0,kn⬘ ⫽0
M1
n+M
册冋兺 册
Ûk2
M
n+M
M2⬘
M1⬘
M2
共A9兲
,
*
␦k +k
共k1,k2兲 = V2 兺 兺 兩k1兩2Ûk Ûk Ĝk ␦k +k 关1 − ␦k 兴 ⫻ 兺 Gk⬘ 兺 ␦k −k
兺
兺
兺 ÊP M共k1,k2兲 · Êreciprocal
k ⫽0
M
M
k k
k k
3
1
2
1
1
2
= V2
2
1
冋
Ĝk⬘ 兺 兩kn+M兩2Ûk2
兺
k ⫽0
M
n
n
1
2
n+M
1
1
n
册
1
n
n+M1
2
2
n+M2
共A10兲
Ĝkn+M ,
and
兩Êreciprocal共k1,k2兲兩2 = V2 兺 Ĝk2 兩k兩2 ,
兺
兺
k k
k⫽0
1
共A11兲
2
then
2
␹P3MV2/3 =
1
1
1
2
*
*
2 兺 兺 兩ÊP3M共k1,k2兲兩 − 2 兺 兺 ÊP3M共k1,k2兲 · Êreciprocal共k1,k2兲 − 2 兺 兺 ÊP3M共k1,k2兲 · Êreciprocal共k1,k2兲
V k1 k2
V k1 k2
V k1 k2
+ 兺 兺 兩Êreciprocal共k1,k2兲兩2
k1 k2
=
冋
共Ĝk⬘ 兲2 兺 兩kn+M兩2Ûk2
兺
k ⫽0
M
n
n
n+M
册冋兺 册
Ûk2
M
−2
n+M
冋
Ĝk⬘ 兺 兩kn+M兩2Ûk2
兺
k ⫽0
M
n
n
n+M
册
Ĝkn+M +
Ĝk2 兩k兩2 .
兺
k⫽0
共A12兲
Setting ⳵␹P2 3M / ⳵Ĝk⬘ = 0 gives Eq. 共21兲,
n
Ĝk⬘ =
n
兺MÛk2
关兺MÛk2
n+M
n+M
Ĝkn+M兩kn+M兩2
兴关兺MÛk2
n+M
兩kn+M兩2兴
共A13兲
.
Slightly different expressions may be derived for the case of differentiation on the reciprocal-space mesh, where the field
is determined by multiplying the reciprocal-space mesh-based potential by ikn.
EP3M共r1,r2兲 = 兺 W共r1 − rn⬘ 兲
rn⬘
Therefore,
ÊP3M共k1,k2兲 = −
1
兺
V
rn⬘
冋冕
1
e−ikn·rnW共rn − r2兲.
兺 eikn·rn⬘共− ikn兲Ĝk⬘n兺
V kn⫽0
rn
W共r1 − rn⬘ 兲e−ik1·r1d3r1
V
kn⫽0
eikn·rn⬘ iknĜk⬘
e−ik ·r
兺
r
n n
n
n
=−
1
关Ŵk1e−ik1·rn⬘ 兴 兺 eikn·rn⬘ iknĜk⬘ 兺 e−ikn·rn关Ŵk2e−ik2·rn兴
n
V
kn⫽0
rn
=−
1
Ŵk Ŵk 兺 iknĜk⬘
n
V 1 2 kn⫽0
= − VÛk1Ûk2
兺
kn⫽0
iknĜk⬘
n
冋兺
1
1
= V2
rn⬘
1
2
2
n+M1
兺
kn⫽0,kn⬘ ⫽0
冋
兩kn兩2共Ĝk⬘ 兲2 兺 Ûk2
兺
k ⫽0
M
n
n
册冋 兺
e−i共kn+k2兲·rn
rn
1
3
2
e−i共k1−kn兲·rn⬘
␦k −k 兺 ␦k +k
兺
M
M
兺 Ûk2 Ûk2
兺
兺 兩ÊP M共k1,k2兲兩2 = V2兺
k k
k k
1
册兺
2
2
n+M2
共A14兲
⫻
冋冕
W共rn − r2兲e−ik2·r2d3r2
V
册
册
共A15兲
,
kn · kn⬘ Ĝk⬘ Ĝk⬘ ⬘ ⫻ 兺 ␦k1−kn+M 兺 ␦k2+kn+M 兺 ␦k1−kn +M 兺 ␦k2+kn +M
n
⬘ 1⬘
⬘ 2⬘
1
2
n
n+M
册
M1
2
M2
M1⬘
M2⬘
,
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
共A16兲
214106-8
J. Chem. Phys. 128, 214106 共2008兲
H. A. Stern and K. G. Calkins
3
3
1
2
= V2 兺 兺 Ûk1Ûk2Ĝk1␦k1+k2关1 − ␦k1兴
k1 k2
⫻
= V2
兺
kn⫽0
kn · k1Ĝk⬘
冋
n
␦k −k
兺
M
1
1
Ĝk⬘ 兺 kn · kn+MÛk2
兺
k ⫽0
M
n
n
and
2
n+M1
␹P3MV2/3 =
冋
Ûk2
兺 兩kn兩2共Ĝk⬘ 兲2 兺
k ⫽0
M
n
n
−2
M2
n
n+M
兺 Ĝk2 兩k兩2 .
册
2
共A17兲
Ĝkn+M ,
册
2
Ĝk⬘ 兺 kn · kn+MÛk2
兺
k ⫽0
M
n
+
冋
⬍ 兺 ␦k2+kn+M
n+M
n+M
Ĝkn+M
册
共A18兲
k⫽0
Setting ⳵␹P2 3M / ⳵Ĝk⬘ = 0 gives Eq. 共22兲,
n
Ĝk⬘ =
n
1
兺MÛk2 Ĝkn+Mkn
n+M
关兺MÛk2
n+M
· kn+M
兴2兩kn兩2
.
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2
*
共k1,k2兲
兺
兺 ÊP M共k1,k2兲 · Êreciprocal
k k
共A19兲
M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids 共Clarendon, Oxford, 1987兲.
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