X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Fthesis%2Fbasics.tex;h=f6a8eda37e41458fa71099fe61ac63fe811dcbe8;hb=e9b904cd12213f534e64f904e48071337f1f6cb6;hp=cba8c5626362e9512c93717412d7a88c16e6a85e;hpb=e3a0b64c27ef578fffd44e26b43953917b858e86;p=lectures%2Flatex.git

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index cba8c56..f6a8eda 100644
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -421,7 +421,7 @@ for the exchange-correlation energy, where $\epsilon_{\text{xc}}(\vec{r};[n(\til
 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{ .}
@@ -532,6 +532,7 @@ Using PPs the rapid oscillations of the wave functions near the core of the atom
 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.
@@ -631,10 +632,8 @@ The path exhibiting the minimal energy difference determines the diffusion path
 
 \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}
@@ -655,10 +654,9 @@ In the modified version respective energies could be higher than the real ones d
 
 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