From: hackbard <hackbard@sage.physik.uni-augsburg.de>
Date: Wed, 18 May 2011 16:45:31 +0000 (+0200)
Subject: ice
X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=commitdiff_plain;h=ecfd4c7626a37a0607f9e1239d5e98f322776a49

ice
---

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}|<G_{\text{cut}}}
+}c_{i,\vec{k}+\vec{G}} \exp\left(i(\vec{k}+\vec{G})\vec{r}\right)
+\text{ .}
+%E_{\text{cut}}=\frac{\hbar^2 G^2_{\text{cut}}}{2m}
+%\text{, }
+\end{equation}
+The basis set, which in principle should be infinite, can be truncated to include only plane waves that have kinetic energies $\hbar^2|\vec{k}+\vec{G}|^2/2m$ less than a particular cut-off energy $E_{\text{cut}}$.
+Although coefficients $c_{i,\vec{k}+\vec{G}}$ corresponding to small kinetic energies are typically more important, convergence with respect to the cut-off energy is crucial for the accuracy of the calculations.
+Convergence, however, is easily achieved by increasing $E_{\text{cut}}$ until the differences in total energy approximate zero.
+
+There are several advantages of plane waves.
+
+
+Disadvantage ... periodic system required, but escapable by respective choice of the supercell.
+
+
+very popular and most natural choice ...
+plane wave, natural ... choice in periodic systems
+can be thought of a fourier series ...
+constructed this way ...
+by definition orthonormal ...
+indeed it has been shown that accuracy ...
+
 
 
 \subsection{Pseudopotentials}
 
+Since core electrons tend to be concentrated very close to the atomic nuclei, resulting in large wavefunction and density gradients near the nuclei which are not easily described by a plane-wave basis set unless a very high energy cutoff, and therefore small wavelength, is used.
+
 \subsection{Brillouin zone sampling}
 
 \subsection{Hellmann-Feynman forces}