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.
\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