\section{Molecular dynamics simulations}
+\subsection{Introduction to molecular dynamics simulations}
-\subsection{Potentials}
+Basically, molecular dynamics (MD) simulation is a technique to compute a system of particles, referred to as molecules, evolving in time.
+The MD method was introduced by Alder and Wainwright in 1957 \cite{alder1,alder2} to study the interactions of hard spheres.
+The basis of the approach are Newton's equations of motion to describe classicaly the many-body system.
+MD simulation is the numerical way of solving the $N$-body problem ($N > 3$) which cannot be solved analytically.
+Quantum mechanical effects are taken into account by an analytical interaction potential between the nuclei.
+
+By MD simulation techniques a complete description of the system in the sense of classical mechanics on the microscopic level is obtained.
+This microscopic information has to be translated to macroscopic observables by means of statistical mechanics.
+
+The basic idea is to integrate Newton's equations numerically.
+A system of $N$ particles of masses $m_i$ ($i=1,\ldots,N$) at positions ${\bf r}_i$ and velocities $\dot{{\bf r}}_i$ is given by
+\begin{equation}
+m_i \frac{d^2}{dt^2} {\bf r}_i = {\bf F}_i \, \textrm{.}
+\end{equation}
+The forces ${\bf F}_i$ are obtained from the potential energy $U(\{{\bf r}\})$:
+\begin{equation}
+{\bf F}_i = - \nabla_{{\bf r}_i} U({\{\bf r}\}) \, \textrm{.}
+\end{equation}
+Given the initial conditions ${\bf r}_i(t_0)$ and $\dot{{\bf r}}_i(t_0)$ the equations can be integrated by a certain integration algorithm.
+The solution of these equations provides the complete information of a system evolving in time.
+
+The following chapters cover the tools of the trade necessary for the MD simulation technique.
+First a detailed overview of the available integration algorithms is given, including their advantages and disadvantages.
+After that the interaction potentials and their accuracy for describing certain systems of elements are discussed.
+In addition special techniques will be outlined which reduce the complexity of the MD algorithm, though the force/energy evaluation almost inevitably dictates the overall speed.
+
+\subsection{Integration algorithms}
+
+
+\subsection{Interaction potentials}
+
+The potential energy of $N$ interacting atoms can be written in the form
+\begin{equation}
+U(\{{\bf r}\}) = \sum_i U_1({\bf r}_i) + \sum_i \sum_{j>i} U_2({\bf r}_i,{\bf r}_j) + \sum_i \sum_{j>i} \sum_{k>j>i} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) \ldots
+\end{equation}
+where $U$ is the total potential energy.
+$U_1$ is a single particle potential describing external forces.
+This could for instance be the gravitational force or an electric field.
+$U_2$ is a two body pair potential which only depends on the distance $r_{ij}$ between the two atoms $i$ and $j$.
+
+$U_3$ is a three body potential which may have an additional angular dependence describing covalent bonds, plus higher order terms which are expected to be small and thus neglected.
\subsubsection{The Lennard-Jones potential}
\label{eq:lj-d}
\end{equation}
one can easily identify $\sigma$ by the equilibrium distance of the atoms $r_e=\sqrt[6]{2} \sigma$.
-Applying the equilibrium distance into \eqref{eq:lj-p} $\epsilon$ turns out to be half the negative well depth.
+Applying the equilibrium distance into \eqref{eq:lj-p} $\epsilon$ turns out to be the negative well depth.
The $i$th component of the force $F^j$ on particle $j$ is obtained by
\begin{equation}
F_i^j = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.}
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