X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=physics_compact%2Fsolid.tex;h=c4cf86947ad49fa982159d62d439a116024bc4ee;hp=3cb0480306e4452a81de06e3b821bc21369dc47e;hb=fa167b99a2c520549296e61b92d56bbdd44d3849;hpb=5cf9c85d580fb9d18c1f0ab3b6647edcca0b0cf1

diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex
index 3cb0480..c4cf869 100644
--- 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 +106,97 @@ 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}
+
+Following the idea of orthogonalized planewaves leads to the pseudopotential idea, which --- in describing only the valence electrons --- effectively removes an undesriable subspace from the investigated problem.
+
+Let $\ket{\Psi_\text{V}}$ be the wavefunction of a valence electron with the Schr\"odinger equation 
+\begin{equation}
+H \ket{\Psi_\text{V}} = \left(\frac{1}{2m}p^2+V\right)\ket{\Psi_\text{V}}=
+E\ket{\Psi_\text{V}} \text{ .}
+\end{equation}
+\ldots projection operatore $P_\text{C}$ \ldots
+
+\subsubsection{Semilocal form of the pseudopotential}
+
+Ionic potentials, which are spherically symmteric, suggest to treat each angular momentum $l,m$ separately leading to $l$-dependent non-local (NL) model potentials $V_l(r)$ and a total potential
+\begin{equation}
+V=\sum_{l,m}\ket{lm}V_l(r)\bra{lm} \text{ .}
+\end{equation}
+In fact, applied to a function, the potential turns out to be non-local in the angular coordinates but local in the radial variable, which suggests to call it asemilocal (SL) potential.
+
+Problem of semilocal potantials become valid once matrix elements need to be computed.
+Integral with respect to the radial component needs to be evaluated for each planewave combination, i.e.\  $N(N-1)/2$ integrals.
+\begin{equation}
+\bra{k+G}V\ket{k+G'} = \ldots
+\end{equation}
+
+A local potential can always be separated from the potential \ldots
+\begin{equation}
+V=\ldots=V_{\text{local}}(r)+\ldots
+\end{equation}
+
+\subsubsection{Norm conserving pseudopotentials}
+
+HSC potential \ldots
+
+\subsubsection{Fully separable form of the pseudopotential}
+
+KB transformation \ldots
+
+\subsection{Spin-orbit interaction}
+
+Relativistic effects can be incorporated in the normconserving pseudopotential method up to but not including order $\alpha^2$ with $\alpha$ being the fine structure constant.
+This is advantageous since \ldots
+With the solutions of the all-electron Dirac equations, the new pseudopotential reads
+\begin{equation}
+V(r)=\sum_{l,m}\left[
+\ket{l+\frac{1}{2},m+{\frac{1}{2}}}V_{l,l+\frac{1}{2}}(r)
+\bra{l+\frac{1}{2},m+{\frac{1}{2}}} +
+\ket{l-\frac{1}{2},m-{\frac{1}{2}}}V_{l,l-\frac{1}{2}}(r)
+\bra{l-\frac{1}{2},m-{\frac{1}{2}}}
+\right] \text{ .}
+\end{equation}
+By defining an averaged potential weighted by the different $j$ degeneracies of the $\ket{l\pm\frac{1}{2}}$ states
+\begin{equation}
+\bar{V}_l(r)=\frac{1}{2l+1}\left(
+l V_{l,l-\frac{1}{2}}(r)+(l+1)V_{l,l+\frac{1}{2}}(r)\right)
+\end{equation}
+and a potential describing the difference in the potential with respect to the spin
+\begin{equation}
+V^{\text{SO}}_l(r)=\frac{2}{2l+1}\left(
+V_{l,l+\frac{1}{2}}(r)-V_{l,l-\frac{1}{2}}(r)\right)
+\end{equation}
+the total potential can be expressed as
+\begin{equation}
+V(r)=\sum_l \ket{l}\left[\bar{V}_l(r)+V^{\text{SO}}_l(r)LS\right]\bra{l}
+\text{ ,}
+\end{equation}
+where the first term correpsonds to the mass velocity and Darwin relativistic corrections and the latter is associated with the spin-orbit (SO) coupling.