Cutoffs and Boundary Conditions

As a spherical system increases in size, its volume grows as the cube of the radius while its surface grows as the square. Thus, in a truly macroscopic system, surface effects may play little role in the chemistry under study (there are, of course, exceptions to this). However, in a typical simulation, computational resources inevitably constrain the size of the system to be so small that surface effects may dominate tlie system properties. Put more succinctly, the modeling of a cluster may not tell one much about tlie behavior of a macroscopic system. This is particularly true when electrostatic interactions are important, since the energy associated witli tliese interactions has an r dependence. 

PBCs function to preserve mass, particle number, and, it can be shown, total energy in the simulation cell. In an MD simulation, PBCs also conserve linear momentum since linear momentum is not conserved in real contained systems, where container walls disrupt the property, this is equivalent to reducing the number of degrees of freedom by 3. However, this effect on system properties is typically negligible for systems of over 100 atoms. Obviously, PBCs do not conserve angular momentum in the simulation cell of an MD simulation, but over time the movement of atoms in and out of each wall of the cell will be such that 

Of course, concerns about periodicity only relate to systems that are not periodic. The discussion above pertains primarily to the simulations of liquids, or solutes in liquid solutions, where PBCs are a useful approximation that helps to model solvation phenomena more realistically than would be the case for a small cluster. If the system truly is periodic, e.g., a zeolite crystal, tlien PBCs are integral to the model. Moreover, imposing PBCs can provide certain advantages in a simulation. For instance, Ewald summation, which accounts for electrostatic interactions to infinite length as discussed in Chapter 2, can only be carried out within the context of PBCs. 

As already noted in Chapter 2, for electrostatic interactions, Ewald sums are generally to be preferred over cut-offs because of tlie long-range nature of the interactions. For van der Waals type terms, cut-offs do not introduce significant artifacts provided they are reasonably large (typically 8-12 A). 

Because of the cost of computing interatomic distances, the evaluation of non-bonded terms in MD is often handled with the aid of a pairlist , which holds in memoiy all pairs of atoms within a given distance of one another. The pairlist is updated periodically, but less often than every MD step. Note that a particular virtue of MC compared to MD is that the only changes in the potential energy are those associated with a moved panicle - all other interactions remain constant. This makes evaluation of the total energy a much simpler process in MC. 

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