\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}

