pp + so started

[lectures/latex.git] / physics_compact / solid.tex

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 3cb0480..e19b7ce 100644 (file)
--- a/physics_compact/solid.tex
+++ b/physics_compact/solid.tex
@@ -2,6 +2,10 @@
 
 \chapter{Atomic structure}
 
+\chapter{Reciprocal lattice}
+
+Example of primitive cell ...
+
 \chapter{Electronic structure}
 
 \section{Noninteracting electrons}
@@ -25,14 +29,22 @@ H=T+V+U\text{ ,}
 where
 \begin{eqnarray}
 T & = & \langle\Psi|\sum_{i=1}^N\frac{-\hbar^2}{2m}\nabla_i^2|\Psi\rangle\\
-  & = & \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
+  & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
         \langle \Psi | \vec{r} \rangle \langle \vec{r} |
-        \frac{-\hbar^2}{2m}\nabla_i^2
+        \nabla_i^2
         | \vec{r}' \rangle \langle \vec{r}' | \Psi \rangle\\
+  & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
+        \langle \Psi | \vec{r} \rangle \nabla_{\vec{r}_i}
+        \langle \vec{r} | \vec{r}' \rangle
+        \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\
+  & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
+        \nabla_{\vec{r}_i} \langle \Psi | \vec{r} \rangle
+        \delta_{\vec{r}\vec{r}'}
+        \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\
   & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} \,
-        \nabla_i \Psi^*(\vec{r}) \nabla_i \Psi(\vec{r})
+        \nabla_{\vec{r}_i} \Psi^*(\vec{r}) \nabla_{\vec{r}_i} \Psi(\vec{r})
         \text{ ,} \\
-V & = & V(\vec{r})\Psi^*(\vec{r})\Psi(\vec{r})d\vec{r} \text{ ,} \\
+V & = & \int V(\vec{r})\Psi^*(\vec{r})\Psi(\vec{r})d\vec{r} \text{ ,} \\
 U & = & \frac{1}{2}\int\frac{1}{\left|\vec{r}-\vec{r}'\right|}
         \Psi^*(\vec{r})\Psi^*(\vec{r}')\Psi(\vec{r}')\Psi(\vec{r})
         d\vec{r}d\vec{r}'
@@ -56,7 +68,7 @@ n_0(\vec{r})=\int \Psi_0^*(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N)
 \end{equation}
 In 1964, Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}.
 
-{\begin{theorem}
+\begin{theorem}[Hohenberg / Kohn]
 For a nondegenerate ground state, the ground-state charge density uniquely determines the external potential in which the electrons reside.
 \end{theorem}
 
@@ -94,6 +106,47 @@ E_1 + E_2 < E_2 + E_1 +
 \int n(\vec{r}) \left( V_2(\vec{r})-V_1(\vec{r}) \right) d\vec{r}
 }_{=0}
 \end{equation}
-is revealed, which proofs the Hohenberg Kohn theorem. \qed
+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}
+