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\documentclass{article}% |
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\usepackage{amsfonts} |
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\usepackage{amsmath} |
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\usepackage{amssymb} |
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\usepackage{graphicx}% |
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\setcounter{MaxMatrixCols}{30} |
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%TCIDATA{OutputFilter=latex2.dll} |
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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=Tuesday, January 20, 2004 11:37:59} |
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%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">} |
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%TCIDATA{<META NAME="SaveForMode" CONTENT="1">} |
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%TCIDATA{<META NAME="DocumentShell" CONTENT="Standard LaTeX\Standard LaTeX Article">} |
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%TCIDATA{ComputeDefs= |
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%$H$ |
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%} |
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\newtheorem{theorem}{Theorem} |
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\newtheorem{acknowledgement}[theorem]{Acknowledgement} |
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\newtheorem{algorithm}[theorem]{Algorithm} |
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\newtheorem{axiom}[theorem]{Axiom} |
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\newtheorem{case}[theorem]{Case} |
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\newtheorem{claim}[theorem]{Claim} |
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\newtheorem{conclusion}[theorem]{Conclusion} |
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\newtheorem{condition}[theorem]{Condition} |
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\newtheorem{conjecture}[theorem]{Conjecture} |
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\newtheorem{corollary}[theorem]{Corollary} |
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\newtheorem{criterion}[theorem]{Criterion} |
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\newtheorem{definition}[theorem]{Definition} |
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\newtheorem{example}[theorem]{Example} |
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\newtheorem{exercise}[theorem]{Exercise} |
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\newtheorem{lemma}[theorem]{Lemma} |
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\newtheorem{notation}[theorem]{Notation} |
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\newtheorem{problem}[theorem]{Problem} |
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\newtheorem{proposition}[theorem]{Proposition} |
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\newtheorem{remark}[theorem]{Remark} |
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\newtheorem{solution}[theorem]{Solution} |
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\newtheorem{summary}[theorem]{Summary} |
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\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} |
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\begin{document} |
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\section{\label{Sec:pbc}Periodic Boundary Conditions} |
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|
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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. 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 \textquotedblleft feeling\textquotedblright\ the symmetries of the |
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periodic lattice, surface effects can be ignored. Cubic, orthorhombic and |
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parallelepiped are the available periodic cells In OOPSE. We use a matrix to |
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describe the property of the simulation box. Therefore, both the size and |
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shape of the simulation box can be changed during the simulation. The |
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transformation from box space vector $\mathbf{s}$ to its corresponding real |
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space vector $\mathbf{r}$ is defined by |
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\begin{equation} |
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\mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{s}% |
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\end{equation} |
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|
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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 three sides of the |
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simulation box respectively. |
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|
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To find the minimum image of a vector $\mathbf{r}$, we convert the real vector |
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to its corresponding vector in box space first, \bigskip% |
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\begin{equation} |
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\mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{r}% |
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\end{equation} |
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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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|
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% |
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|
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\begin{equation} |
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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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|
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|
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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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|
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Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by |
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transforming back to real space,% |
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|
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\begin{equation} |
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\mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
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\end{equation} |
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|
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\end{document} |
| 1 |
> |
\documentclass{article}% |
| 2 |
> |
\usepackage{amsfonts} |
| 3 |
> |
\usepackage{amsmath} |
| 4 |
> |
\usepackage{amssymb} |
| 5 |
> |
\usepackage{graphicx}% |
| 6 |
> |
\setcounter{MaxMatrixCols}{30} |
| 7 |
> |
%TCIDATA{OutputFilter=latex2.dll} |
| 8 |
> |
%TCIDATA{Version=5.00.0.2552} |
| 9 |
> |
%TCIDATA{CSTFile=40 LaTeX article.cst} |
| 10 |
> |
%TCIDATA{Created=Friday, September 19, 2003 08:29:53} |
| 11 |
> |
%TCIDATA{LastRevised=Tuesday, January 20, 2004 11:37:59} |
| 12 |
> |
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">} |
| 13 |
> |
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">} |
| 14 |
> |
%TCIDATA{<META NAME="DocumentShell" CONTENT="Standard LaTeX\Standard LaTeX Article">} |
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%TCIDATA{ComputeDefs= |
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%$H$ |
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%} |
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\newtheorem{theorem}{Theorem} |
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> |
\newtheorem{acknowledgement}[theorem]{Acknowledgement} |
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> |
\newtheorem{algorithm}[theorem]{Algorithm} |
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\newtheorem{axiom}[theorem]{Axiom} |
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\newtheorem{case}[theorem]{Case} |
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\newtheorem{claim}[theorem]{Claim} |
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\newtheorem{conclusion}[theorem]{Conclusion} |
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\newtheorem{condition}[theorem]{Condition} |
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\newtheorem{conjecture}[theorem]{Conjecture} |
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\newtheorem{corollary}[theorem]{Corollary} |
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\newtheorem{criterion}[theorem]{Criterion} |
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\newtheorem{definition}[theorem]{Definition} |
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\newtheorem{example}[theorem]{Example} |
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\newtheorem{exercise}[theorem]{Exercise} |
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\newtheorem{lemma}[theorem]{Lemma} |
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\newtheorem{notation}[theorem]{Notation} |
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\newtheorem{problem}[theorem]{Problem} |
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\newtheorem{proposition}[theorem]{Proposition} |
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\newtheorem{remark}[theorem]{Remark} |
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\newtheorem{solution}[theorem]{Solution} |
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\newtheorem{summary}[theorem]{Summary} |
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\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} |
| 40 |
> |
\begin{document} |
| 41 |
> |
\section{\label{Sec:pbc}Periodic Boundary Conditions} |
| 42 |
> |
|
| 43 |
> |
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 44 |
> |
macroscopic systems with a relatively small number of particles. The |
| 45 |
> |
simulation box is replicated throughout space to form an infinite lattice. |
| 46 |
> |
During the simulation, when a particle moves in the primary cell, its image in |
| 47 |
> |
other boxes move in exactly the same direction with exactly the same |
| 48 |
> |
orientation.Thus, as a particle leaves the primary cell, one of its images |
| 49 |
> |
will enter through the opposite face.If the simulation box is large enough to |
| 50 |
> |
avoid \textquotedblleft feeling\textquotedblright\ the symmetries of the |
| 51 |
> |
periodic lattice, surface effects can be ignored. Cubic, orthorhombic and |
| 52 |
> |
parallelepiped are the available periodic cells In OOPSE. We use a matrix to |
| 53 |
> |
describe the property of the simulation box. Therefore, both the size and |
| 54 |
> |
shape of the simulation box can be changed during the simulation. The |
| 55 |
> |
transformation from box space vector $\mathbf{s}$ to its corresponding real |
| 56 |
> |
space vector $\mathbf{r}$ is defined by |
| 57 |
> |
\begin{equation} |
| 58 |
> |
\mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{s}% |
| 59 |
> |
\end{equation} |
| 60 |
> |
|
| 61 |
> |
|
| 62 |
> |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
| 63 |
> |
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the three sides of the |
| 64 |
> |
simulation box respectively. |
| 65 |
> |
|
| 66 |
> |
To find the minimum image of a vector $\mathbf{r}$, we convert the real vector |
| 67 |
> |
to its corresponding vector in box space first, \bigskip% |
| 68 |
> |
\begin{equation} |
| 69 |
> |
\mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{r}% |
| 70 |
> |
\end{equation} |
| 71 |
> |
And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
| 72 |
> |
\begin{equation} |
| 73 |
> |
s_{i}^{\prime}=s_{i}-round(s_{i}) |
| 74 |
> |
\end{equation} |
| 75 |
> |
where |
| 76 |
> |
|
| 77 |
> |
% |
| 78 |
> |
|
| 79 |
> |
\begin{equation} |
| 80 |
> |
round(x)=\left\{ |
| 81 |
> |
\begin{array} |
| 82 |
> |
[c]{c}% |
| 83 |
> |
\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\ |
| 84 |
> |
\lceil{x-0.5}\rceil & \text{otherwise}% |
| 85 |
> |
\end{array} |
| 86 |
> |
\right. |
| 87 |
> |
\end{equation} |
| 88 |
> |
|
| 89 |
> |
|
| 90 |
> |
For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$. |
| 91 |
> |
|
| 92 |
> |
Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by |
| 93 |
> |
transforming back to real space,% |
| 94 |
> |
|
| 95 |
> |
\begin{equation} |
| 96 |
> |
\mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
| 97 |
> |
\end{equation} |
| 98 |
> |
|
| 99 |
> |
\end{document} |
