The basic idea is that, in real systems, the bond order, i.e.\ the strength of the bond, depends upon the local environment~\cite{abell85}.
Atoms with many neighbors form weaker bonds than atoms with only a few neighbors.
Although the bond strength intricately depends on geometry, the focus on coordination, i.e.\ the number of neighbors forming bonds, is well motivated qualitatively from basic chemistry since for every additional formed bond the amount of electron pairs per bond and, thus, the strength of the bonds is decreased.
-If the energy per bond decreases rapidly enough with increasing coordination the most stable structure will be the dimer.
+If the energy per bond decreases rapidly enough with increasing coordination, the most stable structure will be the dimer.
In the other extreme, if the dependence is weak, the material system will end up in a close-packed structure in order to maximize the number of bonds and likewise minimize the cohesive energy.
This suggests the bond order to be a monotonously decreasing function with respect to coordination and the equilibrium coordination being determined by the balance of bond strength and number of bonds.
Based on pseudopotential theory, the bond order term $b_{ijk}$ limiting the attractive pair interaction is of the form $b_{ijk}\propto Z^{-\delta}$ where $Z$ is the coordination number and $\delta$ a constant~\cite{abell85}, which is $\frac{1}{2}$ in the second-moment approximation within the tight binding scheme~\cite{horsfield96}.
\label{subsection:statistical_ensembles}
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 particles, 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.
+Providing a stable integration algorithm, the total energy $E$, i.e.\ the kinetic and configurational energy of the particles, 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.
\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.
+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
\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.
+Berendsen~et~al.\ proposed a method~\cite{berendsen84}, 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}
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.
+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.
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 determinants to be evaluated massively increases.
+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 determinants 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.
\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.
-On the other way round, on the timescale of nuclear motion the electrons appear blurred in space adding an extra term to the ion-ion potential.
+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.
+On the other way round, on the timescale of nuclear motion, the electrons appear blurred in space adding an extra term to the ion-ion potential.
The simplified Schr\"odinger equation no longer contains the kinetic energy of the ions.
The momentary positions of the ions enter as fixed parameters and, therefore, the ion-ion interaction may be regarded as a constant added to the electronic energies.
The Schr\"odinger equation describing the remaining electronic problem reads
\end{equation}
where $Z_l$ are the atomic numbers of the nuclei and $\Psi$ is the many-electron wave function, which depends on the positions and spins of the electrons.
Accordingly, there is only a parametric dependence on the ionic coordinates $\vec{R}_l$.
-However, the remaining number of free parameters is still too high and need to be further decreased.
+However, the remaining number of free parameters is still too high and needs to be further decreased.
\subsection{Hohenberg-Kohn theorem and variational principle}
This raised the question how to establish a connection between TF expressed in terms of $n(\vec{r})$ and the exact Schr\"odinger equation expressed in terms of the many-electron wave function $\Psi({\vec{r}})$ and whether a complete description in terms of the charge density is possible in principle.
The answer to this question, whether the charge density completely characterizes a system, became the starting point of modern DFT.
-Considering a system with a nondegenerate ground state there is obviously only one ground-state charge density $n_0(\vec{r})$ that corresponds to a given potential $V(\vec{r})$.
-In 1964 Hohenberg and Kohn showed the opposite and far less obvious result~\cite{hohenberg64}.
-Employing no more than the Rayleigh-Ritz minimal principle it is concluded by {\em reductio ad absurdum} that for a nondegenerate ground state the same charge density cannot be generated by different potentials.
+Considering a system with a nondegenerate ground state, there is obviously only one ground-state charge density $n_0(\vec{r})$ that corresponds to a given potential $V(\vec{r})$.
+In 1964, Hohenberg and Kohn showed the opposite and far less obvious result~\cite{hohenberg64}.
+Employing no more than the Rayleigh-Ritz minimal principle, it is concluded by {\em reductio ad absurdum} that for a nondegenerate ground state the same charge density cannot be generated by different potentials.
Thus, the charge density of the ground state $n_0(\vec{r})$ uniquely determines the potential $V(\vec{r})$ and, consequently, the full Hamiltonian and ground-state energy $E_0$.
-In mathematical terms the full many-electron ground state is a unique functional of the charge density.
+In mathematical terms, the full many-electron ground state is a unique functional of the charge density.
In particular, $E$ is a functional $E[n(\vec{r})]$ of $n(\vec{r})$.
The ground-state charge density $n_0(\vec{r})$ minimizes the energy functional $E[n(\vec{r})]$, its value corresponding to the ground-state energy $E_0$, which enables the formulation of a minimal principle in terms of the charge density~\cite{hohenberg64,levy82}
In addition to the Hartree-Fock (HF) method, KS theory includes the difference of the kinetic energy of interacting and non-interacting electrons as well as the remaining contributions to the correlation energy that are not part of the HF correlation.
The self-consistent KS equations \eqref{eq:basics:kse1}, \eqref{eq:basics:kse2} and \eqref{eq:basics:kse3} are non-linear partial differential equations, which may be solved numerically by an iterative process.
-Starting from a first approximation for $n(\vec{r})$ the effective potential $V_{\text{eff}}(\vec{r})$ can be constructed followed by determining the one-electron orbitals $\Phi_i(\vec{r})$, which solve the single-particle Schr\"odinger equation for the respective potential.
+Starting from a first approximation for $n(\vec{r})$, the effective potential $V_{\text{eff}}(\vec{r})$ can be constructed followed by determining the one-electron orbitals $\Phi_i(\vec{r})$, which solve the single-particle Schr\"odinger equation for the respective potential.
The $\Phi_i(\vec{r})$ are used to obtain a new expression for $n(\vec{r})$.
These steps are repeated until the initial and new density are equal or reasonably converged.
Again, it is worth to note that the KS equations are formally exact.
-Assuming exact functionals $E_{\text{xc}}[n(\vec{r})]$ and potentials $V_{\text{xc}}(\vec{r})$ all many-body effects are included.
+Assuming exact functionals $E_{\text{xc}}[n(\vec{r})]$ and potentials $V_{\text{xc}}(\vec{r})$, all many-body effects are included.
Clearly, this directs attention to the functional, which now contains the costs involved with the many-electron problem.
\subsection{Approximations for exchange and correlation}
As discussed above, the HK and KS formulations are exact and so far no approximations except the adiabatic approximation entered the theory.
However, to make concrete use of the theory, effective approximations for the exchange and correlation energy functional $E_{\text{xc}}[n(\vec{r})]$ are required.
-Most simple and at the same time remarkably useful is the approximation of $E_{\text{xc}}[n(\vec{r})]$ by a function of the local density~\cite{kohn65}
+Most simple and at the same time remarkably useful, is the approximation of $E_{\text{xc}}[n(\vec{r})]$ by a function of the local density~\cite{kohn65},
\begin{equation}
E^{\text{LDA}}_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}))n(\vec{r}) d\vec{r}
\text{ ,}
E^{\text{GGA}}_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}),|\nabla n(\vec{r})|)n(\vec{r}) d\vec{r}
\text{ .}
\end{equation}
-These functionals constitute the simplest extensions of LDA for inhomogeneous systems.
-At modest computational costs gradient-corrected functionals very often yield much better results than the LDA with respect to cohesive and binding energies.
+This functional constitutes the simplest extension of the LDA for inhomogeneous systems.
+At modest computational costs, gradient-corrected functionals very often yield much better results than the LDA with respect to cohesive and binding energies.
\subsection{Plane-wave basis set}
Finally, a set of basis functions is required to represent the one-electron KS wave functions.
-With respect to the numerical treatment it is favorable to approximate the wave functions by linear combinations of a finite number of such basis functions.
+With respect to the numerical treatment, it is favorable to approximate the wave functions by linear combinations of a finite number of such basis functions.
Convergence of the basis set, i.e.\ convergence of the wave functions with respect to the amount of basis functions, is most crucial for the accuracy of the numerical calculations.
Two classes of basis sets, the plane-wave and local basis sets, exist.
In fact, the employed \textsc{vasp} software is solving the KS equations within a plane-wave (PW) basis set.
The idea is based on the Bloch theorem~\cite{bloch29}, which states that in a periodic crystal each electronic wave function $\Phi_i(\vec{r})$ can be written as the product of a wave-like envelope function $\exp(i\vec{kr})$ and a function that has the same periodicity as the lattice.
The latter one can be expressed by a Fourier series, i.e.\ a discrete set of plane waves whose wave vectors just correspond to reciprocal lattice vectors $\vec{G}$ of the crystal.
-Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave vector $\vec{k}$ can be expanded in terms of a discrete PW basis set
+Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave vector $\vec{k}$ can be expanded in terms of a discrete PW basis set:
\begin{equation}
\Phi_i(\vec{r})=\sum_{\vec{G}
%, |\vec{G}+\vec{k}|<G_{\text{cut}}}
By construction, PW calculations require a periodic system.
This does not pose a severe problem since non-periodic systems can still be described by a suitable choice of the simulation cell.
Describing a defect, for instance, requires the inclusion of enough bulk material in the simulation to prevent or reduce the interaction with its periodic, artificial images.
-As a consequence the number of atoms involved in the calculations are increased.
+As a consequence, the number of atoms involved in the calculations are increased.
To describe surfaces, sufficiently thick vacuum layers need to be included to avoid interaction of adjacent crystal slabs.
Clearly, to appropriately approximate the wave functions and the respective charge density of a system composed of vacuum in addition to the solid in a PW basis, an increase of the cut-off energy is required.
-According to equation \eqref{eq:basics:pwks} the size of the Hamiltonian depends on the cut-off energy and, therefore, the computational effort is likewise increased.
+According to equation \eqref{eq:basics:pwks}, the size of the Hamiltonian depends on the cut-off energy and, therefore, the computational effort is likewise increased.
For the same reason, the description of tightly bound core electrons and the respective, highly localized charge density is hindered.
However, a much more profound problem exists whenever wave functions for the core as well as the valence electrons need to be calculated within a PW basis set.
Wave functions of the valence electrons exhibit rapid oscillations in the region occupied by the core electrons near the nuclei.
%To guarantee transferability of the PP the logarithmic derivatives of the real and pseudo wave functions and their first energy derivatives need to agree outside of the core region.
%A simple procedure was proposed to extract norm-conserving PPs obyeing the above-mentioned conditions from {\em ab initio} atomic calculations~\cite{hamann79}.
-In order to achieve these properties different PPs are required for the different shapes of the orbitals, which are determined by their angular momentum.
+In order to achieve these properties, different PPs are required for the different shapes of the orbitals, which are determined by their angular momentum.
Mathematically, a non-local PP, which depends on the angular momentum, has the form
\begin{equation}
V_{\text{nl}}(\vec{r}) = \sum_{lm} | lm \rangle V_l(\vec{r}) \langle lm |
\subsection{Brillouin zone sampling}
\label{subsection:basics:bzs}
-Following Bloch's theorem only a finite number of electronic wave functions need to be calculated for a periodic system.
+Following Bloch's theorem, only a finite number of electronic wave functions need to be calculated for a periodic system.
However, to calculate quantities like the total energy or charge density, these have to be evaluated in a sum over an infinite number of $\vec{k}$ points.
-Since the values of the wave function within a small interval around $\vec{k}$ are almost identical, it is possible to approximate the infinite sum by a sum over an affordable number of $k$ points, each representing the respective region of the wave function in $\vec{k}$ space.
+Since the values of the wave function within a small interval around $\vec{k}$ are almost identical, it is possible to approximate the infinite sum by a sum over an affordable number of $\vec{k}$ points, each representing the respective region of the wave function in $\vec{k}$ space.
Methods have been derived for obtaining very accurate approximations by a summation over special sets of $\vec{k}$ points with distinct, associated weights~\cite{baldereschi73,chadi73,monkhorst76}.
If present, symmetries in reciprocal space may further reduce the number of calculations.
For supercells, i.e.\ repeating unit cells that contain several primitive cells, restricting the sampling of the Brillouin zone (BZ) to the $\Gamma$ point can yield quite accurate results.
\label{section:basics:defects}
Point defects are defects that affect a single lattice site.
-At this site the crystalline periodicity is interrupted.
+At this site, the crystalline periodicity is interrupted.
An empty lattice site, which would be occupied in the perfect crystal structure, is called a vacancy defect.
If an additional atom is incorporated into the perfect crystal, this is called interstitial defect.
A substitutional defect exists, if an atom belonging to the perfect crystal is replaced with an atom of another species.
If not by construction, the system should be fully relaxed.
The substitutional or vacancy defect is realized by replacing or removing one atom contained in the supercell.
Interstitial defects are created by adding an atom at positions located in the space between regular lattice sites.
-Once the intuitively created defect structure is generated structural relaxation methods will yield the respective local minimum configuration.
+Once the intuitively created defect structure is generated, structural relaxation methods will yield the respective local minimum configuration.
Since the supercell approach applies periodic boundary conditions, enough bulk material surrounding the defect is required to exclude possible interaction of the defect with its periodic image.
\begin{figure}[t]
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