658 Energies and forces
Experience has shown that a good balance between the short- andlong-range com-
ponents is achieved ifα= 3. 5 /rc. Ifrcis chosen to be half the lengthLof a cubic
simulation box,rc=L/2, then the valueα= 7/Lshould be used. For large systems,
however, a typical value ofrcis in the range of 10 to 12 ̊A, which is roughly a factor of
3 larger than a typical value ofσij. In most simulations, it is the case that atr=rc,
ushort(r) is not exactly zero, in which case the potential energy exhibits a small jump
discontinuity as the distance between two particles passes throughr=rc. This discon-
tinuity leads to a degradation in energy conservation. One approach for circumventing
this problem is simply to shift the potentialushort(r) by an amountushort(rc), so that
the short-range interaction between two particlesiandjbecomes
u ̃short(rij) =
ushort(rij)−ushort(rc) if rij< rc0 otherwise(B.6)
However, there is a second problem with the use of cutoffs, which is that the forces
also exhibit a discontinuity atr=rc; a simple shift does not remove this discontinuity.
A more robust truncation protocol that ensures continuous energies and forces is to
smoothly switch offushort(r) to zero via a switching functionS(r). When a switch is
used, the potential becomes ̃ushort(r) =ushort(r)S(r). An example of such a function
S(r) was given in eqn. (3.14.5). Switching functions become particularly important
for simulations in the isothermal-isobaric ensemble, as the pressureestimator in eqn.
(4.6.57) is especially sensitive to discontinuities in the force (Martynaet al., 1999).
An important point concerning truncation of the potential is its effect on energies
and pressures. In fact, when the potential is assumed to be zerobeyondrc, many weak
interactions among particle pairs will not be included. However, the contribution of
these neglected interactions can be estimated using eqns. (4.6.46)and (4.6.69). Instead
of integrating these expressions from 0 to∞, if the lower limit is taken to berc, then
the integrals can be performed in the approximation that forr > rc,g(r)≈1. For
example, the corrections to the energy per particle and pressurefor a Lennard-Jones
potential in this approximation would be
∆u= 2πρ∫∞
rcdr r^2 u(r)= 8ǫπρ∫∞
rcdr r^2[(
σ
r) 12
−
(σ
r) 6 ]
=
8 πǫρσ^3
3[
1
3
(
σ
rc) 9
−
(
σ
rc) 3 ]
, (B.7)
∆P=−
2 πρ^2
3∫∞
rcdr r^3 u′(r)= 16πρ^2 ǫ∫∞
rcdr r^2[
2
(σ
r) 12
−
(σ
r