\section{Molecular dynamics simulations}
+\subsection{Theory of melecular dynamics simulations}
-\subsection{Potentials}
+Basically molecular dynamics (MD) simulation is a technique to compute a system of particles, referred to as molecules, that evolve 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
+
+\subsection{Interaction potentials}
\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{.}
\addcontentsline{toc}{chapter}{References}
\begin{thebibliography}{99}
+ \bibitem{alder1}
+ B. J. Alder, T.E. Wainwright.
+ J. Chem. Phys. 27 (1957) 1208.
+ \bibitem{alder2}
+ B. J. Alder, T.E. Wainwright.
+ J. Chem. Phys. 31 (1959) 459.
\bibitem{example}
\selectlanguage{german}
F. Zirkelbach, M. H"aberlen, J. K. N. Lindner, B. Stritzker.
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