ice

diff --git a/bibdb/bibdb.bib b/bibdb/bibdb.bib
index f3d64c0..9b84389 100644 (file)
--- a/bibdb/bibdb.bib
+++ b/bibdb/bibdb.bib
@@ -16,6 +16,23 @@
   year =         "1926",
 }
 
   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
 @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",
 }
 
   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",
 @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 (file)
--- 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 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.
 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}
 
 
 \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}
 
 
 
 \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}
 \subsection{Brillouin zone sampling}
 
 \subsection{Hellmann-Feynman forces}