## Periodic boundary condition This is a commonly mentioned topic. Periodic boundary condition, PBC, or [[Bloch's theorem and Born-von Karman boundary condition|Born-von Karman boundary condition]] when describing **wave functions** in periodic crystals. The idea is just, a particle leaves from one side wall of a box, then it enters from the other side. This valid for potential, too. In principle, one must include interactions among all boxes (or, as expressed in the slide, all images in periodic boxes). But for short range force, like LJ, we may use a cut-off to limit the interaction only in centre box. This is called “minimum image convention”. The geometry requirement is $d>2R_c$, $d$ is the box length. ![[Drawing 2023-07-27 00.32.03.excalidraw.svg]] >[!note] >PBC also affect the order of complexity of the algorithm: >- when finding Verlet neighbor atom: $O(N^2)$ >- using cell list: $C\cdot O(N)$ >- combined : $O(N)$ >[!notice] >PBC may cause "displacement" related quantity being wrong, like [[Statistical quantities#Mean square displacement (MSD)|mean square displacement]]: >![[Drawing 2023-07-27 00.48.05.excalidraw.svg]] For quantum cases, we describe the potential in the crystal being periodic, this gives $V(\vec{r}+\vec{T})=V(\vec{r})$. The direct result of this is [[Bloch's theorem and Born-von Karman boundary condition#^73ff80|Bloch's theorem]]. ^e466f3 And if we also describe the wave function as periodic, we get [[Bloch's theorem and Born-von Karman boundary condition#^156416|Born-von Karman boundary condition]]. >[!info] >More to read: https://en.wikipedia.org/wiki/Born%E2%80%93von_Karman_boundary_condition >https://en.wikipedia.org/wiki/Periodic_boundary_conditions