Cell lists (also sometimes referred to as cell linked-lists) is a data structure in molecular dynamics simulations to find all atom pairs within a given cut-off distance of each other. These pairs are needed to compute the short-range non-bonded interactions in a system, such as Van der Waals forces or the short-range part of the electrostatic interaction when using Ewald summation.
Algorithm.
Cell lists work by subdividing the simulation domain into cells with an edge length greater than or equal to the cut-off radius of the interaction to be computed. The particles are sorted into these cells and the interactions are computed between particles in the same or neighbouring cells.
In its most basic form, the non-bonded interactions for a cut-off distance formula_1 are computed as follows:
for all neighbouring cell pairs formula_2 do
for all formula_3 do
for all formula_4 do
formula_5
if formula_6 then
Compute the interaction between formula_7 and formula_8.
end if
end for
end for
end for
Since the cell length is at least formula_1 in all dimensions, no particles within formula_1 of each other can be missed.
Given a simulation with formula_11 particles with a homogeneous particle density, the number of cells formula_12 is proportional to formula_11 and inversely proportional to the cut-off radius (i.e. if formula_11 increases, so does the number of cells). The average number of particles per cell formula_15 therefore does not depend on the total number of particles. The cost of interacting two cells is in formula_16. The number of cell pairs is proportional to the number of cells which is again proportional to the number of particles formula_11. The total cost of finding all pairwise distances within a given cut-off is in formula_18, which is significantly better than computing the formula_19 pairwise distances naively.
Periodic boundary conditions.
In most simulations, periodic boundary conditions are used to avoid imposing artificial boundary conditions. Using cell lists, these boundaries can be implemented in two ways.
Ghost cells.
In the ghost cells approach, the simulation box is wrapped in an additional layer of cells. These cells contain periodically wrapped copies of the corresponding simulation cells inside the domain.
Although the data—and usually also the computational cost—is doubled for interactions over the periodic boundary, this approach has the advantage of being straightforward to implement and very easy to parallelize, since cells will only interact with their geographical neighbours.
Periodic wrapping.
Instead of creating ghost cells, cell pairs that interact over a periodic boundary can also use a periodic correction vector formula_20. This vector, which can be stored or computed for every cell pair formula_21, contains the correction which needs to be applied to "wrap" one cell around the domain to neighbour the other. The pairwise distance between two particles formula_3 and formula_4 is then computed as
formula_24.
This approach, although more efficient than using ghost cells, is less straightforward to implement (the cell pairs need to be identified over the periodic boundaries and the vector formula_20 needs to be computed/stored).
Improvements.
Despite reducing the computational cost of finding all pairs within a given cut-off distance from formula_19 to formula_27, the cell list algorithm listed above still has some inefficiencies.
Consider a computational cell in three dimensions with edge length equal to the cut-off radius formula_1. The pairwise distance between all particles in the cell and in one of the neighbouring cells is computed. The cell has 26 neighbours: 6 sharing a common face, 12 sharing a common edge and 8 sharing a common corner. Of all the pairwise distances computed, only about 16% will actually be less than or equal to formula_1. In other words, 84% of all pairwise distance computations are spurious.
One way of overcoming this inefficiency is to partition the domain into cells of edge length smaller than formula_1. The pairwise interactions are then not just computed between neighboring cells, but between all cells within formula_1 of each other (first suggested in and implemented and analysed in and ). This approach can be taken to the limit wherein each cell holds at most one single particle, therefore reducing the number of spurious pairwise distance evaluations to zero. This gain in efficiency, however, is quickly offset by the number of cells formula_32 that need to be inspected for every interaction with a cell formula_33, which, for example in three dimensions, grows cubically with the inverse of the cell edge length. Setting the edge length to formula_34, however, already reduces the number of spurious distance evaluations to 63%.
Another approach is outlined and tested in Gonnet, in which the particles are first sorted along the axis connecting the cell centers. This approach generates only about 40% spurious pairwise distance computations, yet carries an additional cost due to sorting the particles.