Expressing $n(\tilde{\vec{r}})$ in a Taylor series, $\epsilon_{\text{xc}}$ can be thought of as a function of coefficients, which correspond to the respective terms of the expansion.
Neglecting all terms of order $\mathcal{O}(\nabla n(\vec{r}))$ results in the functional equal to LDA, which requires the function of variable $n$.
Including the next element of the Taylor series introduces the gradient correction to the functional, which requires the function of variables $n$ and $|\nabla n|$.
-This is called the generalized gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density
+This is called the generalized-gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density
\begin{equation}
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{ .}
More importantly, less accuracy is required compared to all-electron calculations to determine energy differences among ionic configurations, which almost totally appear in the energy of the valence electrons that are typically a factor $10^3$ smaller than the energy of the core electrons.
\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.
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.
where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure.
Clearly, for a single atom species equation \eqref{eq:basics:ef2} is equivalent to equation \eqref{eq:basics:ef1} since $NE_{\text{coh}}^{\text{defect}}$ is equal to the total energy of the defect structure and $NE_{\text{coh}}^{\text{defect-free}}$ corresponds to $N\mu$, provided the structure is fully relaxed at zero temperature.
+However, there is hardly ever only one defect in a crystal, not even only one kind of defect.
+Again, energetic considerations can be used to investigate the existing interaction of two defects.
+The binding energy $E_{\text{b}}$ of a defect pair is given by the difference of the formation energy of the defect combination $E_{\text{f}}^{\text{comb}} $ and the sum of the two separated defect configurations $E_{\text{f}}^{1^{\text{st}}}$ and $E_{\text{f}}^{2^{\text{nd}}}$.
+This can be expressed by
+\begin{equation}
+E_{\text{b}}=
+E_{\text{f}}^{\text{comb}}-
+E_{\text{f}}^{1^{\text{st}}}-
+E_{\text{f}}^{2^{\text{nd}}}
+\label{eq:basics:e_bind}
+\end{equation}
+where the formation energies $E_{\text{f}}^{\text{comb}}$, $E_{\text{f}}^{1^{\text{st}}}$ and $E_{\text{f}}^{2^{\text{nd}}}$ are determined as discussed above.
+Accordingly, energetically favorable configurations result in binding energies below zero while unfavorable configurations show positive values for the binding energy.
+The interaction strength, i.e. the absolute value of the binding energy, approaches zero for increasingly non-interacting isolated defects.
+Thus, $E_{\text{b}}$ indeed can be best thought of a binding energy, which is required to bring the defects to infinite separation.
+
The methods presented in the last two chapters can be used to investigate defect structures and energetics.
Therefore, a supercell containing the perfect crystal is generated in an initial process.
If not by construction, the system should be fully relaxed.
\begin{figure}[t]
\begin{center}
-\subfigure[]{\label{fig:basics:crto}
-\includegraphics[width=0.5\textwidth]{crt_orig.eps}}
-\subfigure[]{\label{fig:basics:crtm}
-\includegraphics[width=0.5\textwidth]{crt_mod.eps}}
+\subfigure[]{\label{fig:basics:crto}\includegraphics[width=0.45\textwidth]{crt_orig.eps}}
+\subfigure[]{\label{fig:basics:crtm}\includegraphics[width=0.45\textwidth]{crt_mod.eps}}
\end{center}
\caption{Schematic of the constrained relaxation technique (a) and of a modified version (b) used to obtain migration pathways and corresponding configurational energies.}
\label{fig:basics:crt}
Structures of maximum configurational energy do not necessarily constitute saddle point configurations, i .e. the method does not guarantee to find the true minimum energy path.
Whether a saddle point configuration and, thus, the minimum energy path is obtained by the CRT method, needs to be verified by caculating the respective vibrational modes.
-
Modifications used to add the CRT feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
-Due to these constraints obtained activation energies can effectively be higher.
% todo
% advantages of pw basis with respect to hellmann feynman forces / pulay forces
+% crt sketch needs increased text
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