From: hackbard Date: Tue, 17 Jan 2012 14:29:28 +0000 (+0100) Subject: started dft section X-Git-Url: https://hackdaworld.org/gitweb/?a=commitdiff_plain;h=a9f93985e52272cdecf902c4b559173a80f4b41d;p=lectures%2Flatex.git started dft section --- diff --git a/physics_compact/Makefile b/physics_compact/Makefile new file mode 100644 index 0000000..03c7e27 --- /dev/null +++ b/physics_compact/Makefile @@ -0,0 +1,21 @@ +# Makefile +LATEX = latex +DVIPDF = dvipdf +BIBTEX = bibtex + +SRC = phys_comp.tex +PDF = $(SRC:%.tex=%.pdf) + +all: $(PDF) + +%.dvi: %.tex + $(LATEX) $< + $(BIBTEX) $(SRC:%.tex=%) + $(LATEX) $< + $(LATEX) $< + +%.pdf: %.dvi + $(DVIPDF) $< + +clean: + rm -f *.log *.aux *.blg *.lof *.ps *.pdf *.toc *.bbl diff --git a/physics_compact/ack.tex b/physics_compact/ack.tex index f446c8c..369a074 100644 --- a/physics_compact/ack.tex +++ b/physics_compact/ack.tex @@ -1,4 +1,4 @@ -\chapter*{Danksagung} +\chapter*{Acknowledgements} \addcontentsline{toc}{chapter}{Acknowledgements} Thanks ... diff --git a/physics_compact/app.tex b/physics_compact/app.tex index 5063047..ccee46e 100644 --- a/physics_compact/app.tex +++ b/physics_compact/app.tex @@ -1,4 +1,8 @@ -\chapter{Spherical coordinates} +\part{Appendices} -\chapter{Fourier integrals} +\chapter{Mathematical tools} + +\section{Spherical coordinates} + +\section{Fourier integrals} diff --git a/physics_compact/intro.tex b/physics_compact/intro.tex index c70be6d..5771409 100644 --- a/physics_compact/intro.tex +++ b/physics_compact/intro.tex @@ -3,5 +3,5 @@ As the title suggests, the present work constitutes an attempt to summarize mathematical models and abstractions employed in modern theoretical physics. Focussed on solid state theory, which, however, requires a large amount of tools, the present book tries to additionally include all prerequisites in a hopefully compact way. -A final remark: This is work in progress! +A final remark: This is work in progress and might not be very usefull for the ... diff --git a/physics_compact/mech.tex b/physics_compact/mech.tex index 52cb4c0..e67554c 100644 --- a/physics_compact/mech.tex +++ b/physics_compact/mech.tex @@ -1,2 +1,2 @@ -\chapter{Classical mechanics} +\part{Classical mechanics} diff --git a/physics_compact/qm.tex b/physics_compact/qm.tex index eba2a1f..b33a700 100644 --- a/physics_compact/qm.tex +++ b/physics_compact/qm.tex @@ -1 +1,9 @@ -\chapter{Quantum mechanics} +\part{Quantum mechanics} + +\chapter{Fundamental concepts} + +\section{Variational method} +\label{sec:var_meth} + +\chapter{Quantum dynamics} + diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index f7cf91f..20ecfef 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -1 +1,47 @@ -\chapter{Theory of the solid state} +\part{Theory of the solid state} + +\chapter{Atomic structure} + +\chapter{Electronic structure} + +\section{Noninteracting electrons} + +\subsection{Bloch's theorem} + +\section{Nearly free and tightly bound electrons} + +\subsection{Tight binding model} + +\section{Interacting electrons} + +\subsection{Density functional theory} + +\subsubsection{Hohenberg-Kohn theorem} + +Considering a system with a nondegenerate ground state, there is obviously only one ground-state charge density $n_0(\vec{r})$ that correpsonds to a given potential $V(\vec{r})$. +In 1964, Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}. +For a nondegenerate ground state, the ground-state charge density uniquely determines the external potential in which the electrons reside. +The proof presented by Hohenberg and Kohn proceeds by {\em reductio ad absurdum}. + +Suppose two potentials $V_1$ and $V_2$ exist, which yield the same electron density $n(\vec{r})$. +The corresponding Hamiltonians are denoted $H_1$ and $H_2$ with the respective ground-state wavefunctions $\Psi_1$ and $\Psi_2$ and eigenvalues $E_1$ and $E_2$. +Then, due to the variational principle (see \ref{sec:var_meth}), one can write +\begin{equation} +E_1=\langle \Psi_1 | H_1 | \Psi_1 \rangle < \langle \Psi_2 | H_1 | \Psi_2 \rangle +\end{equation} +Expressing $H_1$ by $H_2+H_1-H_2$ +\begin{equation} +\langle \Psi_2 | H_1 | \Psi_2 \rangle = +\langle \Psi_2 | H_2 | \Psi_2 \rangle + +\langle \Psi_2 | H_1 -H_2 | \Psi_2 \rangle +\end{equation} +and the fact that the two Hamiltonians, which describe the same number of electrons, differ only in the potential +\begin{equation} +H_1-H_2=V_1(\vec{r})-V_2(\vec{r}) +\end{equation} +one obtains +\begin{equation} +E_1