Periodic boundary conditions

  What should we do at the boundaries of our simulated system? One possibility is doing nothing special: the system simply terminates, and atoms near the boundary would have less neighbors than atoms inside. In other words, the sample would be surrounded by surfaces.

Unless we really want to simulate a cluster of atoms, this situation is not realistic. No matter how large the simulated system is, its number of atoms N would be negligible compared with the number of atoms contained in a macroscopic piece of matter (of the order of 10²³), and the ratio between the number of surface atoms and the total number of atoms would be much larger than in reality, causing surface effects to be much more important than what they should.

A solution to this problem is to use periodic boundary conditions (PBC). When using PBC, particles are enclosed in a box, and we can imagine that this box is replicated to infinity by rigid translation in all the three cartesian directions, completely filling the space. In other words, if one of our particles is located at position ${\bf r}$in the box, we assume that this particle really represents an infinite set of particles located at

$${\bf r} + \ell {\bf a} + m {\bf b} + n {\bf c} \,\,\, , \,\,\, ( \ell,m,n = -\infty,\infty ) ,$$

where $\ell, m, n$ are integer numbers, and I have indicated with ${\bf a}, {\bf b}, {\bf c}$ the vectors corresponding to the edges of the box. All these ``image'' particles move together, and in fact only one of them is represented in the computer program.

The key point is that now each particle i in the box should be thought as interacting not only with other particles j in the box, but also with their images in nearby boxes. That is, interactions can ``go through'' box boundaries. In fact, one can easily see that (a) we have virtually eliminated surface effects from our system, and (b) the position of the box boundaries has no effect (that is, a translation of the box with respect to the particles leaves the forces unchanged).

Apparently, the number of interacting pairs increases enormously as an effect of PBC. In practice, this is not true because potentials usually have a short interaction range. The minimum image criterion discussed next simplifies things further, reducing to a minimum the level of additional complexity introduced in a program by the use of PBC.