From ecfd4c7626a37a0607f9e1239d5e98f322776a49 Mon Sep 17 00:00:00 2001 From: hackbard Date: Wed, 18 May 2011 18:45:31 +0200 Subject: [PATCH] ice --- bibdb/bibdb.bib | 34 +++++++++++++++++++++++++++++ posic/thesis/basics.tex | 47 ++++++++++++++++++++++++++++++++++++++--- 2 files changed, 78 insertions(+), 3 deletions(-) diff --git a/bibdb/bibdb.bib b/bibdb/bibdb.bib index f3d64c0..9b84389 100644 --- a/bibdb/bibdb.bib +++ b/bibdb/bibdb.bib @@ -16,6 +16,23 @@ year = "1926", } +@Article{bloch29, + author = "Felix Bloch", + affiliation = "Institut d. Universität f. theor. Physik Leipzig", + title = "Über die Quantenmechanik der Elektronen in + Kristallgittern", + journal = "Zeitschrift für Physik A Hadrons and Nuclei", + publisher = "Springer Berlin / Heidelberg", + ISSN = "0939-7922", + keyword = "Physics and Astronomy", + pages = "555--600", + volume = "52", + issue = "7", + URL = "http://dx.doi.org/10.1007/BF01339455", + note = "10.1007/BF01339455", + year = "1929", +} + @Article{albe_sic_pot, author = "Paul Erhart and Karsten Albe", title = "Analytical potential for atomistic simulations of @@ -3010,6 +3027,23 @@ publisher = "American Physical Society", } +@Article{payne92, + title = "Iterative minimization techniques for ab initio + total-energy calculations: molecular dynamics and + conjugate gradients", + author = "M. C. Payne and M. P. Teter and D. C. Allan and T. A. + Arias and J. D. Joannopoulos", + journal = "Rev. Mod. Phys.", + volume = "64", + number = "4", + pages = "1045--1097", + numpages = "52", + year = "1992", + month = oct, + doi = "10.1103/RevModPhys.64.1045", + publisher = "American Physical Society", +} + @Article{levy82, title = "Electron densities in search of Hamiltonians", author = "Mel Levy", diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index 3bc9527..fcf920c 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -381,7 +381,7 @@ The one-electron KS orbitals $\Phi_i(\vec{r})$ as well as the respective KS ener The KS equations may be considered the formal exactification of the Hartree theory, which it is reduced to if the exchange-correlation potential and functional are neglected. In addition to the Hartree-Fock (HF) method, KS theory includes the difference of the kinetic energy of interacting and non-interacting electrons as well as the remaining contributions to the correlation energy that are not part of the HF correlation. -The self-consistent KS equations \eqref{eq:basics:kse1}, \eqref{eq:basics:kse2} and \eqref{eq:basics:kse3} may be solved numerically by an iterative process. +The self-consistent KS equations \eqref{eq:basics:kse1}, \eqref{eq:basics:kse2} and \eqref{eq:basics:kse3} are non-linear partial differential equations, which may be solved numerically by an iterative process. Starting from a first approximation for $n(\vec{r})$ the effective potential $V_{\text{eff}}(\vec{r})$ can be constructed followed by determining the one-electron orbitals $\Phi_i(\vec{r})$, which solve the single-particle Schr\"odinger equation for the respective potential. The $\Phi_i(\vec{r})$ are used to obtain a new expression for $n(\vec{r})$. These steps are repeated until the initial and new density are equal or reasonably converged. @@ -429,12 +429,53 @@ At modest computational costs gradient-corrected functionals very often yield mu \subsection{Plane-wave basis set} -Practically, the KS equations are non-linear partial differential equations that are iteratively solved. -The one-electron KS wave functions can be represented in different basis sets. +Finally, a set of basis functions is required to represent the one-electron KS wave functions. +With respect to the numerical treatment it is favorable to approximate the wave functions by linear combinations of a finite number of such basis functions. +Covergence of the basis set, i.e. convergence of the wave functions with respect to the amount of basis functions, is most crucial for the accuracy of the numerical calulations. +Two classes of basis sets, the plane-wave and local basis sets, exist. + +Local basis set functions usually are atomic orbitals, i.e. mathematical functions that describe the wave-like behavior of electrons, which are localized, i.e. centered on atoms or bonds. +Molecular orbitals can be represented by linear combinations of atomic orbitals (LCAO). +By construction, only a small number of basis functions is required to represent all of the electrons of each atom within reasonable accuracy. +Thus, local basis sets enable the implementation of methods that scale linearly with the number of atoms. +However, these methods rely on ... + +Another approach is to represent the KS wave functions by plane waves. +In fact, the employed {\textsc vasp} software is solving the KS equations within a plane-wave basis set. +The idea is based on the Bloch theorem \cite{bloch29}, which states that in a periodic crystal each electronic wave function $\Phi_i(\vec{r})$ can be written as the product of a wave-like envelope function $\exp(i\vec{kr})$ and a function that has the same periodicity as the lattice. +The latter one can be expressed by a Fourier series, i.e. a discrete set of plane waves whose wave vectors just correspond to reciprocal lattice vectors $\vec{G}$ of the crystal. +Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave vector $\vec{k}$ can be expanded in terms of a discrete plane-wave basis set +\begin{equation} +\Phi_i(\vec{r})=\sum_{\vec{G} +%, |\vec{G}+\vec{k}|