1549380323-Statistical Mechanics Theory and Molecular Simulation

Energies and forces 661

be evaluated efficiently in real space. However, because a long-ranged function in real
space is short ranged in Fourier or reciprocal space, if the latter isused, the scaling
can be improved, thereby reducing the computational overhead.Moreover, a Fourier
expansion of erf(αr)/ris consistent with the periodic boundary conditions typically
used in Monte Carlo and molecular dynamics calculations. For simplicity,we will
consider the case of a cubic simulation cell of lengthLand volumeV=L^3. As we saw
in Section 11.2, the reciprocal space of such a cell is composed of allreciprocal-space
vectorsg= 2πn/L, wherenis a vector of integers. Using the Poisson summation rule,
we can expand the error function term in a Fourier series as

∑

S

erf(α|r+S|)

|r+S|

=

1

V

∑

g

Cgeig·r. (B.10)

In eqn. (B.10), the expansion coefficients are

Cg=

∑

S

∫

D(V)

dr

erf(α|r+S|)

|r+S|

e−ig·r

=

∫

all space

erf(αr)

r

e−ig·r

=

4 π

|g|^2

e−|g|

(^2) / 4 α 2

. (B.11)

The second line in eqn. (B.11) again follows from the Poisson summationformula,
which allows the sum overSto be eliminated in favor of an integral over all space.
Note that the coefficient corresponding ton= (0, 0 ,0) is not defined, hence this term
must be excluded from the sum in eqn. (B.10). Moreover, it is not possible, in practice,
to perform a sum over an infinite number of reciprocal-space vectors as required by eqn.
(B.10). However, we note that the factor exp(−|g|^2 / 4 α^2 ) decays to zero very quickly as
|g|→∞, which means that the sum over reciprocal-space vectors can be truncated and
restricted only to a very small part of reciprocal space. We can readily see, therefore,
the advantage of working in Fourier space when faced with the calculation of long-
range forces! BecauseCgonly depends on the magnitude|g|, the most natural way to
truncate the Fourier sum is to restrict the sum to all reciprocal-space vectorsgwith
magnitudes|g|≤gmax, wheregmaxis chosen such that exp(−gmax^2 / 4 α^2 ) is negligible.
The Fourier sum is consequently restricted tog-vectors that lie within a sphere in
reciprocal space of radiusgmax. In addition, because the coefficientsCgare real and
depend only on|g|, they satisfyC−g=Cgand therefore we only need to keep half the
g-vectors in the sphere. For example, we could choose the hemisphereSfor whichgx
is positive or 0. When the sum is performed over half of reciprocal space, we simply
need to multiply the result by 2.
Taking into account the truncation of the Fourier sum and the restriction of the
g-vectors to a single hemisphere, the long-range potential energycan be expressed as

Ulong(r 1 ,...,rN) =

2

V

∑

i>j

qiqj

∑

g∈S

4 π

|g|^2

e−|g|

(^2) / 4 α 2
eig·(ri−rj). (B.12)