\chapter{Basics}
+\section{Molecular dynamics simulations}
+
+\subsection{Introduction to molecular dynamics simulations}
+
+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.
+
+
+
+\subsection{Integration algorithms}
+
+\subsection{Interaction potentials}
+
+\subsubsection{The Lennard-Jones potential}
+
+The L-J potential is a realistic two body pair potential and is of the form
+\begin{equation}
+U^{LJ}(r) = 4 \epsilon \Big[ \Big( \frac{\sigma}{r} \Big)^{12} - \Big( \frac{\sigma}{r} \Big)^6 \Big] \, \textrm{,}
+\label{eq:lj-p}
+\end{equation}
+where $r$ denotes the disatnce between the two atoms.
+
+The attractive tail for large separations $(\sim r^{-6})$ is essentially due to correlations between electron clouds surrounding the atoms. The attractive part is also known as {\em van der Waals} or {\em London} interaction.
+It can be derived classically by considering how two charged spheres induce dipol-dipol interactions into each other, or by considering the interaction between two oscillators in a quantum mechanical way.
+
+The repulsive term $(\sim r^{-12})$ captures the non-bonded overlap of the electron clouds.
+It does not have a true physical motivation, other than the exponent being larger than $6$ to get a steep rising repulsive potential wall at short distances.
+Chosing $12$ as the exponent of the repulsive term it is just the square of the attractive term which makes the potential evaluable in a very efficient way.
+
+The constants $\epsilon$ and $\sigma$ are usually determined by fitting to experimental data.
+$\epsilon$ accounts to the depth of the potential well, where $\sigma$ is regarded as the radius of the particle, also known as the van der Waals radius.
+
+Writing down the derivation of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector $\vec{r}$)
+\begin{equation}
+\frac{\partial}{\partial x_i} U^{LJ}(r) = 4 \epsilon x_i \Big( -12 \frac{\sigma^{12}}{r^{14}} + 6 \frac{\sigma^6}{r^8} \Big)
+\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 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{.}
+\label{eq:lj-f}
+\end{equation}
+
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