About OOPSE OOPSE News Features Download CVS Mailing Lists Documentation Authors Funding Results	Periodic Boundary Conditions Periodic boundary conditions are widely used to simulate bulk properties with a relatively small number of particles. In this method the simulation box is replicated throughout space to form an infinite lattice. During the simulation, when a particle moves in the primary cell, its image in other cells move in exactly the same direction with exactly the same orientation. Thus, as a particle leaves the primary cell, one of its images will enter through the opposite face. If the simulation box is large enough to avoid ``feeling'' the symmetries of the periodic lattice, surface effects can be ignored. The available periodic cells in OOPSE are cubic, orthorhombic and parallelepiped. OOPSE use a $3 \times 3$ matrix, $\mathsf{H}$ , to describe the shape and size of the simulation box. $\mathsf{H}$ is defined: $$\mathsf{H} = ( \mathbf{h}_x, \mathbf{h}_y, \mathbf{h}_z ),$$ (3.23) where $\mathbf{h}_{\alpha}$ is the column vector of the $\alpha$ axis of the box. During the course of the simulation both the size and shape of the box can be changed to allow volume fluctuations when constraining the pressure. A real space vector, $\mathbf{r}$ can be transformed in to a box space vector, $\mathbf{s}$ , and back through the following transformations: $$\mathbf{s} = \mathsf{H}^{-1} \mathbf{r},$$ (3.24) $$\mathbf{r} = \mathsf{H} \mathbf{s}.$$ (3.25) The vector $\mathbf{s}$ is now a vector expressed as the number of box lengths in the $\mathbf{h}_x$ , $\mathbf{h}_y$ , and $\mathbf{h}_z$ directions. To find the minimum image of a vector $\mathbf{r}$ , OOPSE first converts it to its corresponding vector in box space, and then casts each element to lie in the range $[-0.5,0.5]$ : $$s_{i}^{\prime}=s_{i}-\operatorname{round}(s_{i}),$$ (3.26) where $s_i$ is the $i$ th element of $\mathbf{s}$ , and $\operatorname{round}(s_i)$ is given by $$\operatorname{round}(x) = \begin{cases}\lfloor x+0.5 \rfloor & \text{if $x \ge 0$,} \\ \lceil x-0.5 \rceil & \text{if $x < 0$.} \end{cases}$$ (3.27) Here $\lfloor x \rfloor$ is the floor operator, and gives the largest integer value that is not greater than $x$ , and $\lceil x \rceil$ is the ceiling operator, and gives the smallest integer that is not less than $x$ . Finally, the minimum image coordinates $\mathbf{r}^{\prime}$ are obtained by transforming back to real space, $$\mathbf{r}^{\prime}=\mathsf{H}^{-1}\mathbf{s}^{\prime}.%$$ (3.28) In this way, particles are allowed to diffuse freely in $\mathbf{r}$ , but their minimum images, or $\mathbf{r}^{\prime}$ , are used to compute the inter-atomic forces.
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	Updated on January 16, 2006