From de1b5208eaf3f26a3a2f5efa6041a47a1caf5928 Mon Sep 17 00:00:00 2001 From: hackbard Date: Fri, 13 May 2011 12:53:58 +0200 Subject: [PATCH] started bo --- bibdb/bibdb.bib | 69 ++++++++++++++++++++++++++++++++++------- posic/thesis/basics.tex | 18 ++++++++--- 2 files changed, 71 insertions(+), 16 deletions(-) diff --git a/bibdb/bibdb.bib b/bibdb/bibdb.bib index 4b5e845..c619524 100644 --- a/bibdb/bibdb.bib +++ b/bibdb/bibdb.bib @@ -2,6 +2,20 @@ % bibliography database % +@Article{schroedinger26, + author = "E. Schrödinger", + title = "Quantisierung als Eigenwertproblem", + journal = "Annalen der Physik", + volume = "384", + number = "4", + publisher = "WILEY-VCH Verlag", + ISSN = "1521-3889", + URL = "http://dx.doi.org/10.1002/andp.19263840404", + doi = "10.1002/andp.19263840404", + pages = "361--376", + year = "1926", +} + @Article{albe_sic_pot, author = "Paul Erhart and Karsten Albe", title = "Analytical potential for atomistic simulations of @@ -2872,6 +2886,38 @@ notes = "density functional theory, dft", } +@Article{thomas27, + title = "The calculation of atomic fields", + author = "L. H. Thomas", + journal = "Mathematical Proceedings of the Cambridge + Philosophical Society", + volume = "23", + pages = "542--548", + year = "1927", + doi = "10.1017/S0305004100011683", +} + +@Article{fermi27, + title = "", + author = "E. Fermi", + journal = "Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. Rend.", + volume = "6", + pages = "602", + year = "1927", +} + +@Article{hartree28, + title = "The Wave Mechanics of an Atom with a Non-Coulomb + Central Field. Part {I}. Theory and Methods", + author = "D. R. Hartree", + journal = "Mathematical Proceedings of the Cambridge + Philosophical Society", + volume = "24", + pages = "89--110", + year = "1928", + doi = "10.1017/S0305004100011919", +} + @Article{kohn65, title = "Self-Consistent Equations Including Exchange and Correlation Effects", @@ -2889,17 +2935,18 @@ } @Article{kohn99, - title = {Nobel Lecture: Electronic structure of matter---wave functions and density functionals}, - author = {Kohn, W. }, - journal = {Rev. Mod. Phys.}, - volume = {71}, - number = {5}, - pages = {1253--1266}, - numpages = {13}, - year = {1999}, - month = {Oct}, - doi = {10.1103/RevModPhys.71.1253}, - publisher = {American Physical Society} + title = "Nobel Lecture: Electronic structure of matter---wave + functions and density functionals", + author = "W. Kohn", + journal = "Rev. Mod. Phys.", + volume = "71", + number = "5", + pages = "1253--1266", + numpages = "13", + year = "1999", + month = oct, + doi = "10.1103/RevModPhys.71.1253", + publisher = "American Physical Society", } @Article{ruecker94, diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index 8ab5002..135f2ac 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -277,16 +277,24 @@ It provides a stable algorithm that allows smooth changes of the system to new v \section{Denstiy functional theory} \label{section:dft} -In quantum-mechanical modeling the problem of describing the many-body problem, i.e. a system of a large amount of interacting particles, is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of all nuclei and electrons. +Dirac declared that chemistry has come to an end, its content being entirely contained in the powerul equation published by Schr\"odinger in 1926 \cite{schroeder26} marking the beginning of wave mechanics. +Following the path of Schr\"odinger the problem in quantum-mechanical modeling of describing the many-body problem, i.e. a system of a large amount of interacting particles, is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of all nuclei and electrons. The Schr\"odinger equation contains the kinetic energy of the ions and electrons as well as the electron-ion, ion-ion and electron-electron interaction. -This cannot be solved exactly and there are several layers of approximations to reduce the number of parameters. -The key point in density functional theory (DFT) is to recast the problem to a description using the charge density $n(\vec{r})$ that depends on only three spatial coordinates instead of the many-body wave function. -Formally DFT can be regarded as an exactification of both, the Thomas Fermi and Hartree theory. +This cannot be solved exactly and finding approximate solutions requires several layers of simplification in order to reduce the number of free parameters. +Approximations that consider a truncated Hilbert space of single-particle orbitals yield promising results, however, with increasing complexity and demand for high accuracy the amount of Slater determinats to be evaluated massively increases. + +In contrast, instead of using the description by the many-body wave function, the key point in density functional theory (DFT) is to recast the problem to a description utilizing the charge density $n(\vec{r})$, which constitutes a quantity in real space depending only on the three spatial coordinates. In the following sections the basic idea of DFT will be outlined. +As will be shown, DFT can formally be regarded as an exactification of the Thomas Fermi theory \cite{thomas27,fermi27} and the self-consistent Hartree equations \cite{hartree28}. \subsection{Born-Oppenheimer approximation} -The first approximation employed +Born and Oppenheimer proposed a simplification enabling the effective decoupling of the electronic and ionic degrees of freedom \cite{born27}. +Within the Born-Oppenheimer (BO) approximation the light electrons are assumed to move much faster and, thus, follow adiabatically to the motion of the heavy nuclei, if the latter are only slightly deflected from their equilibrium positions. +Thus, on the timescale of electronic motion the ions appear at fixed positions and, on the other way round, for the nuclei the electrons appear blurred in space adding an extra term to the ion-ion potential. +The simplified Schr\"odinger equation is rewritten without the kinetic energy of the ions and its positions enter as fixed parameters. + +\subsection{Bloch theorem} \subsection{Hohenberg-Kohn theorem} -- 2.20.1