Periodic Box and Minimum Image Convention
For any size of the simulated system ,the number of atoms N would be negligible as compared with the number of atoms contained in a macroscopic piece of matter (of the order of 1023).Also the ratio between the number of surface atoms and the total number of atoms would be very large, causing surface effects to be dominant. For this Periodic Boundary Condition (PBC) is used in which particles are enclosed in a box, and the box is replicated to infinity by rigid translation in all the three cartesian directions, completely filling the space.
The application of PBC allows us to simulate equilibrium bulk solid and liquid thermodynamic properties with a manageable number of atoms by eliminating surface effects .The basic idea behind the PBC is that if an atom moves in the original simulation box, all its images move in a concerted manner by the same amount and in the same fashion. The computational advantage of this method is that we need to keep track of the original image only as representative of all other images. As the simulation evolves, atoms can move through the boundary of the simulation cells. When this happens, an image atom from one of the neighbouring cell enters to replace the lost particle.
As a result of applying PBC the number of interacting pairs increases enormously. This is because of each particle in the simulation box not only interacts with other particles in the box but also with their images. This problem can be handled by choosing a finite range potential within the criteria of minimum image convention. The essence of the minimum image criteria is that it allows only the nearest neighbours of particle images to interact. Assuming potential range to be short, a minimum image convention is adopted that each atom interacts with the nearest atom or image in the periodic array.In the course of the simulation, if an atom leaves the basic simulation box, attention canbe switched to the incoming image. In practice, the mechanism of doing so is to use the potential in a finite range such that the interaction of two distant particles at or beyond a finite length can be neglected. This maximum length must be equal to or less than the half of the box length used in the simulation. A cutoff distance Rc (half the box length) or potential cut off is defined when a particle is separated by a distance equal or larger than Rc, two particles do not interact with each other (figure 5 and 6).This helps to avoid expensive force calculation.
The time to examine all pair separations is proportional to the number of distinct pairs, ½ N(N-1) in an N-atom system. To avoid this a pair listing is made in which the potential cutoff sphere, or radiusRc, around a particular atom is surrounded by a `skin', to give a larger sphere of radius Rlist. At the first step, a list is constructed of all the neighbors of each atom, for which the pair separation is withinRlist. Over the next few MD time steps, only pairs appearing in the list are checked in the force routine. From time to time the list is reconstructed: it is important to do this before any unlisted pairs have crossed the safety zone and come within interaction range ( Figure 7) .The value of pairlist distance should be chosen such that no atom pair moves more than pairlistdist − cutoff in one cycle (refer to NAMD user guide: http://www.ks.uiuc.edu/Research/namd/2.9/ug.pdf). This ensures energy conservation and efficiency.