1549380323-Statistical Mechanics Theory and Molecular Simulation

Energies and forces 657

in real space and one that is short ranged in reciprocal space. Given two rapidly
convergent terms, the Coulomb interaction should be easier to evaluate and exhibit
better scaling.
We begin by introducing a simple identity that pertains to a function known as
the error function erf(x) defined by

erf(x) =

2

√

π

∫x

0

dte−t

2

. (B.2)

In the limitx→ ∞, erf(x)→1. Note also that erf(0) = 0. The error function has a
complement erfc(x) defined by

erfc(x) = 1−erf(x) =

2

√

π

∫∞

x

dte−t

2

. (B.3)

Both functions are defined forx≥0. Note that asx→ ∞, erfc(x)→ 0. From
these definitions, it is clear that erf(x) + erfc(x) = 1. This identity can now be used
to divide up the Coulomb interaction into short-range and long-range components.
Consider writing 1/ras
1
r

=

erfc(αr)

r

+

erf(αr)

r

, (B.4)

where the parameterαhas units of inverse length. Since erfc(αr) decays rapidly asr
increases, the first term in eqn. (B.4) is short ranged. The parameterαcan be used to
tune the range over which erfc(αr)/ris nonnegligible. The second term in eqn. (B.4) is
long ranged and behaves asymptotically as 1/r. Introducing eqn. (B.4) into eqn. (B.1),
we can write the nonbonded forces in terms of short- and long-ranged components as

Unb(r 1 ,...,rN) =Ushort(r 1 ,...,rN) +Ulong(r 1 ,...,rN)

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

∑

S

∑

i>j∈nb

{

4 ǫij

[(

σij

rij,S

) 12

−

(

σij

rij,S

) 6 ]

+

qiqjerfc(αrij,S)

rij,S

}

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

∑

S

∑

i>j∈nb

qiqjerf(αrij,S)

rij,S

, (B.5)

whererij,S=|ri−rj+S|, withS=mLfor a periodic cubic box of sideL, andS=hm
for a general box matrixh. Heremis a vector of integers. Note that if a system is
sufficiently large, then the sums over the lattice vectorsSreduce to the single term
m= (0, 0 ,0). As we will soon see, the use of the potential-energy contributions in eqn.
(B.5) has a distinct advantage over eqn. (B.1).
Let us consider first evaluating the short-range forces in eqn. (B.5). For notational
convenience, we denote this term compactly asUshort=

∑

S

∑

i>jushort(rij,S), where
ushort(r) consists of the Lennard-Jones and complementary error function terms.. Be-
cause the range of these interactions is finite, we can reduce the computational over-
head needed to evaluate them by introducing a cutoff radiusrc, beyond which we
assume that the forces are negligible. The cutoffrcis important for determiningα.