X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=posic%2Fthesis%2Fbasics.tex;h=fcf920c3938a01027f2aafea98b6c2b9ef704894;hp=3bc9527e8ca4dc9b76042a7727988e52a9a4abfc;hb=ecfd4c7626a37a0607f9e1239d5e98f322776a49;hpb=3ea37b08b6020dd776a7ec70acd4ef7e47c2d8f1

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
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--- 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}