From: hackbard <hackbard@hackdaworld.org>
Date: Sat, 16 Jun 2012 19:44:36 +0000 (+0200)
Subject: pp + so started
X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=eac23ae428984d20c62851681234b206ec1e3dc7;p=lectures%2Flatex.git

pp + so started
---

diff --git a/physics_compact/math_app.tex b/physics_compact/math_app.tex
index ec21dae..4f7822e 100644
--- a/physics_compact/math_app.tex
+++ b/physics_compact/math_app.tex
@@ -180,13 +180,19 @@ holds.
 
 \section{Spherical coordinates}
 
-Cartesian coordinates $\vec{r}(x,y,z)$ are related to spherical coordinates $\vec{r}(r,\theta,\phi)$ by
+Cartesian coordinates $\vec{r}(x,y,z)$ are related to spherical coordinates $\vec{\tilde{r}}(r,\theta,\phi)$ by
 \begin{eqnarray}
-x&=&r\sin\theta\cos\phi\textrm{ ,}\\
-y&=&r\sin\theta\sin\phi\textrm{ ,}\\
-z&=&r\cos\theta\textrm{ .}
+x&=&r\sin\theta\cos\phi\\
+y&=&r\sin\theta\sin\phi\\
+z&=&r\cos\theta
 \end{eqnarray}
-Infinitesimal translations $dq_i$ and $dq'_i$ of the two coordinate systems are related by the partial derivatives.
+and
+\begin{eqnarray}
+r&=&(x^2+y^2+z^2)^{1/2}\\
+\theta&=&\arccos(z/r)\\
+\phi&=&\arctan(y/x)
+\end{eqnarray}
+The total differentials $dq_i$ and $dq'_i$ of two coordinate systems are related by partial derivatives.
 \begin{equation}
 dq_i=\sum_j \frac{\partial q_i}{\partial q'_j}dq'_j
 \end{equation}
@@ -198,7 +204,7 @@ J_{ij}=\frac{\partial q_i}{\partial q'_j}
 is called the Jacobi matrix.
 \end{definition}
 
-For cartesian and spherical coordinates the relation of the translations are presented in detail
+For cartesian and spherical coordinates the relation of the translations are
 \begin{eqnarray}
 dx&=&\frac{\partial x}{\partial r}dr +
      \frac{\partial x}{\partial \theta}d\theta +
@@ -210,17 +216,29 @@ dz&=&\frac{\partial z}{\partial r}dr +
      \frac{\partial z}{\partial \theta}d\theta +
      \frac{\partial z}{\partial \phi}d\phi\\
 \end{eqnarray}
-and the vector consisting of all or using the Jacobi matrix
-
-
+and
+\begin{eqnarray}
+dr&=&\frac{\partial r}{\partial x}dx +
+     \frac{\partial r}{\partial y}dy +
+     \frac{\partial r}{\partial z}dz\\
+d\theta&=&\frac{\partial \theta}{\partial x}dx +
+          \frac{\partial \theta}{\partial y}dy +
+          \frac{\partial \theta}{\partial z}dz\\
+d\phi&=&\frac{\partial \phi}{\partial x}dx +
+        \frac{\partial \phi}{\partial y}dy +
+        \frac{\partial \phi}{\partial z}dz\\
+\end{eqnarray}
+and vectorial translations using the Jacobi matrix are given by matrix multiplications
+\begin{equation}
+d\vec{r}(x,y,z)=Jd\vec{\tilde{r}}(r,\theta,\phi)
+\end{equation}
+and
+\begin{equation}
+d\vec{\tilde{r}}(r,\theta,\phi)=J^{-1}d\vec{r}(x,y,z) \text{ .}
+\end{equation}
+$J$ and $J^{-1}$ are explicitily given by
 \begin{equation}
-   =\sin\theta\cos\phi dr + \\
 \end{equation}
-
-To obtain infinitesimal 
-\begin{definition}[Jacobi matrix]
-
-\end{definition}
 
 \section{Fourier integrals}
 
diff --git a/physics_compact/phys_comp.tex b/physics_compact/phys_comp.tex
index 7a7d0b2..b4d5e66 100644
--- a/physics_compact/phys_comp.tex
+++ b/physics_compact/phys_comp.tex
@@ -12,7 +12,7 @@
 \usepackage{ae}
 \usepackage{aecompl}
 \usepackage[dvips]{graphicx}
-\graphicspath{{../img/}}
+\graphicspath{{./img/}}
 \usepackage{subfigure}
 \usepackage{color}
 \usepackage{pstricks}
@@ -63,6 +63,10 @@
 \newcommand{\dista}[1]{\unit[#1]{\AA}{}}
 \newcommand{\perc}[1]{\unit[#1]{\%}{}}
 
+\newcommand{\bra}[1]{\langle #1 |}
+\newcommand{\ket}[1]{| #1 \rangle}
+\newcommand{\braket}[2]{\langle #1 | #2 \rangle}
+
 % (re)new commands
 \newcommand{\printimg}[5]{%
  \begin{figure}[#1]%
diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 0d5b1b2..e19b7ce 100644
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -109,3 +109,44 @@ E_1 + E_2 < E_2 + E_1 +
 is revealed, which proofs the Hohenberg Kohn theorem.% \qed
 \end{proof}
 
+\section{Electron-ion interaction}
+
+\subsection{Pseudopotential theory}
+
+The basic idea of pseudopotential theory is to only describe the valence electrons, which are responsible for the chemical bonding as well as the electronic properties for the most part.
+
+\subsubsection{Orthogonalized planewave method}
+
+Due to the orthogonality of the core and valence wavefunctions, the latter exhibit strong oscillations within the core region of the atom.
+This requires a large amount of planewaves $\ket{k}$ to adequatley model the valence electrons.
+
+In a very general approach, the orthogonalized planewave (OPW) method introduces a new basis set
+\begin{equation}
+\ket{k}_{\text{OPW}} = \ket{k} - \sum_t \ket{t}\bra{t}k\rangle \text{ ,}
+\end{equation} 
+with $\ket{t}$ being the eigenstates of the core electrons.
+The new basis is orthogonal to the core states $\ket{t}$.
+\begin{equation}
+\braket{t}{k}_{\text{OPW}} =
+\braket{t}{k} - \sum_{t'} \braket{t}{t'}\braket{t'}{k} =
+\braket{t}{k} - \braket{t}{k}=0
+\end{equation}
+The number of planewaves required for reasonably converged electronic structure calculations is tremendously reduced by utilizing the OPW basis set.
+
+\subsubsection{Pseudopotential method}
+
+\subsubsection{Norm conserving pseudopotentials}
+
+\begin{equation}
+V=\ket{lm}V_l(r)\bra{lm}
+\end{equation}
+
+\subsubsection{Fully separable form of the pseudopotential}
+
+\subsection{Spin orbit interaction}
+
+
+\subsubsection{Perturbative treatment}
+
+\subsubsection{Non-perturbative method}
+