From: hackbard 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} +