660 Energies and forces
i
rcr +δ
cFig. B.1Building a Verlet list of the neighbors of particlei.be correct, which means that the neighbor list needs to be recomputed. The saved
positionsr 1 (0),...,rN(0) are overwritten byr 1 (n∆t),...,rN(n∆t), and the Verlet list
is regenerated using these positions. At each time step subsequent to the list update,
the displacement test is performed using ∆i=|ri((k+n)∆t)−ri(n∆t)|. If ∆max>
δ/2, then the neighbor list must be generated again. In this way, we can determine
automatically when the neighbor list must be recalculated.
It is worth mentioning in passing that an alternative to the Verlet list isthelink list
orcell list(Hockney and Eastwood, 1981; Allen and Tildesley, 1989; Frenkel and Smit,
2002). This method consists of dividing the system into cells of size equal to or slightly
larger than the cutoff radiusrc. Thus, each particle interacts only with particles in
the same cell or in nearest-neighbor cells. A link list is generally more efficient than a
Verlet list in large systems, as the calculation of the latter scales asO(N^2 ) while the
former scales asO(N). However, the logic to write such a list if more involved. A very
useful intermediate approach uses a Verlet list for computing the forces and energies
and a link list to create the Verlet list. This improves considerably the efficiency of
generating the Verlet list. Finally, fast algorithms for computing short range forces
on massively parallel architectures have been developed (Plimpton,1995) and have
proved highly successful, although we will not discuss such algorithms here.
We now turn to the evaluation of the long-range energy and forcesin eqn. (B.5).
Because erf(αr)/rbehaves as 1/rfor larger, the long-range energy and forces cannot