+Using the above mentioned algorithms the most basic type of MD is realized by simply integrating the equations of motion of a fixed number of particles ($N$) in a closed volume $V$ realized by periodic boundary conditions (PBC).
+Providing a stable integration algorithm the total energy $E$, i.e. the kinetic and configurational energy of the paticles, is conserved.
+This is known as the $NVE$, or microcanonical ensemble, describing an isolated system composed of microstates, among which the number of particles, volume and energy are held constant.
+
+However, the successful formation of SiC dictates precise control of temperature by external heating.
+While the temperature of such a system is well defined, the energy is no longer conserved.
+The microscopic states of a system, which is in thermal equilibrium with an external thermal heat bath, are represented by the $NVT$ ensemble.
+In the so-called canonical ensemble the temperature $T$ is related to the expactation value of the kinetic energy of the particles, i.e.
+\begin{equation}
+\langle E_{\text{kin}}\rangle = \frac{3}{2}Nk_{\text{B}}T \text{, }
+E_{\text{kin}}=\sum_i \frac{\vec{p}^2_i}{2m_i} \text{ .}
+\label{eq:basics:ts}
+\end{equation}
+The volume of the synthesized material can hardly be controlled in experiment.
+Instead the pressure can be adjusted.
+Holding constant the pressure in addition to the temperature of the system its states are represented by the isothermal-isobaric $NpT$ ensemble.
+The expression for the pressure of a system derived from the equipartition theorem is given by
+\begin{equation}
+pV=Nk_{\text{B}}T+\langle W\rangle\text{, }W=-\frac{1}{3}\sum_i\vec{r}_i\nabla_{\vec{r}_i}U
+\text{, }
+\label{eq:basics:ps}
+\end{equation}
+where $W$ is the virial and $U$ is the configurational energy.
+
+Berendsen~et~al.~\cite{berendsen84} proposed a method, which is easy to implement, to couple the system to an external bath with constant temperature $T_0$ or pressure $p_0$ with adjustable time constants $\tau_T$ and $\tau_p$ determining the strength of the coupling.
+Control of the respective variable is based on the relations given in equations \eqref{eq:basics:ts} and \eqref{eq:basics:ps}.
+The thermostat is achieved by scaling the velocities of all atoms in every time step $\delta t$ from $\vec{v}_i$ to $\lambda \vec{v}_i$, with
+\begin{equation}
+\lambda=\left[1+\frac{\delta t}{\tau_T}(\frac{T_0}{T}-1)\right]^\frac{1}{2}
+\text{ ,}
+\end{equation}
+where $T$ is the current temperature according to equation \eqref{eq:basics:ts}.
+The barostat adjusts the pressure by changing the virial through scaling of the particle positions $\vec{r}_i$ to $\mu \vec{r}_i$ and the volume $V$ to $\mu^3 V$, with
+\begin{equation}
+\mu=\left[1-\frac{\beta\delta t}{\tau_p}(p_0-p)\right]^\frac{1}{3}\text{ ,}
+\end{equation}
+where $\beta$ is the isothermal compressibility and $p$ corresponds to the current pressure, which is determined by equation \eqref{eq:basics:ps}.
+
+Using this method the system does not behave like a true $NpT$ ensemble.
+On average $T$ and $p$ correspond to the expected values.
+For large enough time constants, i.e. $\tau > 100 \delta t$, the method shows realistic fluctuations in $T$ and $p$.
+The advantage of the approach is that the coupling can be decreased to minimize the disturbance of the system and likewise be adjusted to suit the needs of a given application.
+It provides a stable algorithm that allows smooth changes of the system to new values of temperature or pressure, which is ideal for the investigated problem.
+
+\section{Denstiy functional theory}
+\label{section:dft}
+
+Dirac declared that chemistry has come to an end, its content being entirely contained in the powerul equation published by Schr\"odinger in 1926 \cite{schroeder26} marking the beginning of wave mechanics.
+Following the path of Schr\"odinger the problem in quantum-mechanical modeling of describing the many-body problem, i.e. a system of a large amount of interacting particles, is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of all nuclei and electrons.
+The Schr\"odinger equation contains the kinetic energy of the ions and electrons as well as the electron-ion, ion-ion and electron-electron interaction.
+This cannot be solved exactly and finding approximate solutions requires several layers of simplification in order to reduce the number of free parameters.
+Approximations that consider a truncated Hilbert space of single-particle orbitals yield promising results, however, with increasing complexity and demand for high accuracy the amount of Slater determinats to be evaluated massively increases.
+
+In contrast, instead of using the description by the many-body wave function, the key point in density functional theory (DFT) is to recast the problem to a description utilizing the charge density $n(\vec{r})$, which constitutes a quantity in real space depending only on the three spatial coordinates.
+In the following sections the basic idea of DFT will be outlined.
+As will be shown, DFT can formally be regarded as an exactification of the Thomas Fermi theory \cite{thomas27,fermi27} and the self-consistent Hartree equations \cite{hartree28}.
+
+\subsection{Born-Oppenheimer approximation}
+
+Born and Oppenheimer proposed a simplification enabling the effective decoupling of the electronic and ionic degrees of freedom \cite{born27}.
+Within the Born-Oppenheimer (BO) approximation the light electrons are assumed to move much faster and, thus, follow adiabatically to the motion of the heavy nuclei, if the latter are only slightly deflected from their equilibrium positions.
+Thus, on the timescale of electronic motion the ions appear at fixed positions and, on the other way round, for the nuclei the electrons appear blurred in space adding an extra term to the ion-ion potential.
+The simplified Schr\"odinger equation is rewritten without the kinetic energy of the ions and its positions enter as fixed parameters.
+
+\subsection{Bloch theorem}
+
+\subsection{Hohenberg-Kohn theorem}
+
+\subsection{Effective potential}
+
+\subsection{Kohn-Sham system}
+
+\subsection{Approximations for exchange and correlation}
+
+\subsection{Pseudopotentials}
+
+\section{Modeling of defects}
+\label{section:basics:defects}
+
+\section{Migration paths and diffusion barriers}
+\label{section:basics:migration}
+