# Computational Efficiency of Periodic Boundary Conditions

From what I understand, in a molecular dynamics simulation the number of molecules included is limited by the computational power available.

In order to simulate macroscopic properties, one needs to ensure that the particles experience forces as if they were in the bulk, so the system is replicated in all directions through the application of a periodic boundary condition.

How is the creation of this larger, periodic system any more computationally efficient than just simulating a non-periodic system of similar size?

• A non-periodic system will likely suffer from many particles being on the surface of the system. Consider 1000 particles in a cube: almost half of them will be on the surface. Their characteristics will be different from a particle in bulk. Apr 5 '19 at 4:03

Consider a simple 1-dimensional case with three atoms in the unit cell: $$\ce{A-B-C}$$, and lets assume we only need to treat nearest-neighbor-interactions. So the "monomer" has only 2 interactions: AB and BC.
One option to simulate the macroscopic system would be to explicitly deal with $$\ce{A-B-C-A-B-C-A-B-C-A-B-C}$$ (or even longer), but this requires a lot of atoms with accordingly many interactions between them. And even then both ends of the chain would be missing one neighbor.
Using periodic boundary conditions, one explicitly treats $$\ce{A-B-C}$$ only. But this time we will allow $$\ce{C}$$ to interact with $$\ce{A}$$, since the periodic boundary conditions implicitly tell us, there is another $$\ce{C}$$ to the left. Similar, there is an implicit $$\ce{A}$$ to the right of $$\ce{C}$$. Compared to the monomer from above, we have again 3 atoms, but now there are 3 interactions to be considered: AB, BC and AC.