fast feddich!

[lectures/latex.git] / posic / thesis / basics.tex

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index fcf920c..bd259d0 100644 (file)
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -438,7 +438,7 @@ Local basis set functions usually are atomic orbitals, i.e. mathematical functio
 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 ...
+However, these methods rely on the fact that the wave functions are localized and exhibit an exponential decay resulting in a sparse Hamiltonian.
 
 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.
@@ -455,22 +455,14 @@ Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave
 \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.
-
+Convergence with respect to the basis set, however, is easily achieved by increasing $E_{\text{cut}}$ until the respective differences in total energy approximate zero.
+Next to their simplicity, plane waves have several advantages.
+The basis set is orthonormal by construction.
+matrix elements of the Hamiltonian have a simple form (pw rep of ks equations)
+As mentioned above ... simple to check for convergence.
 
 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 ...
-
-
+size of matrix to diagonalize determined by cut-off energy, severe 
 
 \subsection{Pseudopotentials}
 
@@ -478,6 +470,13 @@ Since core electrons tend to be concentrated very close to the atomic nuclei, re
 
 \subsection{Brillouin zone sampling}
 
+Due to the Bloch theorem only a finite number of electronic wave functions need to be calculated for a periodic system.
+However, to calculate quantities like the total energy or charge density, these have to be evaluated in a sum over an infinite number of $\vec{k}$ points.
+Since the values of the wave function within a small interval around $\vec{k}$ are almost identical, it is possible to approximate the infinite sum by a sum over an affordable number of $k$ points, each representing the respective region of the wave function in $\vec{k}$ space. 
+Methods have been derived for obtaining very accurate approximations by an intergration over special sets of $\vec{k}$ points \cite{}.
+If present, symmetries in reciprocal space may further reduce the number of calculations.
+For supercells, i.e. repeating unit cells that contain several primitive cells, sampling of the Brillouin zone restricted to the $\Gamma$ point can be quite accurately used, which is equivalent to calculating a single primitive cell using multiple $\vec{k}$ points.
+
 \subsection{Hellmann-Feynman forces}
 
 \section{Modeling of defects}