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%TCIDATA{Version=5.00.0.2552} |
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%TCIDATA{CSTFile=40 LaTeX article.cst} |
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%TCIDATA{Created=Friday, September 19, 2003 08:29:53} |
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%TCIDATA{LastRevised=Wednesday, January 07, 2004 10:20:42} |
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%TCIDATA{LastRevised=Tuesday, January 13, 2004 10:22:03} |
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%TCIDATA{<META NAME="DocumentShell" CONTENT="Standard LaTeX\Standard LaTeX Article">} |
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\section{\label{Sec:pbc}Periodic Boundary Conditions} |
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\textit{Periodic boundary conditions} are widely used to simulate truly |
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macroscopic systems with a relatively small number of particles. Simulation |
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box is replicated throughout space to form an infinite lattice. During the |
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simulation, when a particle moves in the primary cell, its periodic image |
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particles in other boxes move in exactly the same direction with exactly the |
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same orientation.Thus, as a particle leaves the primary cell, one of its |
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images will enter through the opposite face.If the simulation box is large |
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enough to avoid "feeling" the symmetric of the periodic lattice,the surface |
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effect could be ignored. Cubic and parallelepiped are the available periodic |
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cells. \bigskip In OOPSE, we use the matrix instead of the vector to describe |
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the property of the simulation box. Therefore, not only the size of the |
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simulation box could be changed during the simulation, but also the shape of |
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it. The transformation from box space vector $\overrightarrow{s}$ to its |
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corresponding real space vector $\overrightarrow{r}$ is defined by |
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macroscopic systems with a relatively small number of particles. The |
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simulation box is replicated throughout space to form an infinite lattice. |
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During the simulation, when a particle moves in the primary cell, its image in |
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other boxes move in exactly the same direction with exactly the same |
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orientation.Thus, as a particle leaves the primary cell, one of its images |
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will enter through the opposite face.If the simulation box is large enough to |
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avoid "feeling" the symmetries of the periodic lattice, surface effects can be |
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ignored. Cubic, orthorhombic and parallelepiped are the available periodic |
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cells In OOPSE. We use a matrix to describe the property of the simulation |
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box. Therefore, both the size and shape of the simulation box can be changed |
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during the simulation. The transformation from box space vector $\mathbf{s}$ |
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to its corresponding real space vector $\mathbf{r}$ is defined by |
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\begin{equation} |
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\overrightarrow{r}=H\overrightarrow{s}% |
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\mathbf{r}=\underline{\underline{H}}\cdot\mathbf{s}% |
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\end{equation} |
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|
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where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
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box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the sides of the |
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box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the three sides of the |
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simulation box respectively. |
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To find the minimum image, we need to convert the real vector to its |
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corresponding vector in box space first, \bigskip% |
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To find the minimum image, we convert the real vector to its corresponding |
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vector in box space first, \bigskip% |
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\begin{equation} |
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\overrightarrow{s}=H^{-1}\overrightarrow{r}% |
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\mathbf{s}=\underline{\underline{H}}^{-1}\cdot\mathbf{r}% |
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\end{equation} |
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And then, each element of $\overrightarrow{s}$ is casted to lie between -0.5 |
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to 0.5, |
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And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
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\begin{equation} |
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s_{i}^{\prime}=s_{i}-round(s_{i}) |
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\end{equation} |
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where% |
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where |
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– |
\begin{equation} |
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round(x)=\lfloor{x+0.5}\rfloor\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ |
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}x\geqslant0 |
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\end{equation} |
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% |
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\begin{equation} |
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round(x)=\lceil{x-0.5}\rceil\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ }x<0 |
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round(x)=\left\{ |
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\begin{array} |
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[c]{c}% |
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\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\ |
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\lceil{x-0.5}\rceil & \text{otherwise}% |
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\end{array} |
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\right. |
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\end{equation} |
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For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$. |
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Finally, we could get the minimum image by transforming back to real space,% |
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Finally, we obtain the minimum image coordinates by transforming back to real space,% |
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|
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\begin{equation} |
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\overrightarrow{r^{\prime}}=H^{-1}\overrightarrow{s^{\prime}}% |
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\mathbf{r}^{\prime}=\underline{\underline{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
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\end{equation} |
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