From: hackbard <hackbard@sage.physik.uni-augsburg.de>
Date: Thu, 19 May 2011 16:29:05 +0000 (+0200)
Subject: nearly finished pseudopotentials
X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=commitdiff_plain;h=fcc70f48c064efc50d86a560245aac02789dfe39

nearly finished pseudopotentials
---

diff --git a/bibdb/bibdb.bib b/bibdb/bibdb.bib
index 9b84389..5c53f66 100644
--- a/bibdb/bibdb.bib
+++ b/bibdb/bibdb.bib
@@ -4003,6 +4003,20 @@
   notes =        "gga pw91 (as in vasp)",
 }
 
+@Article{chadi73,
+  title =        "Special Points in the Brillouin Zone",
+  author =       "D. J. Chadi and Marvin L. Cohen",
+  journal =      "Phys. Rev. B",
+  volume =       "8",
+  number =       "12",
+  pages =        "5747--5753",
+  numpages =     "6",
+  year =         "1973",
+  month =        dec,
+  doi =          "10.1103/PhysRevB.8.5747",
+  publisher =    "American Physical Society",
+}
+
 @Article{baldereschi73,
   title =        "Mean-Value Point in the Brillouin Zone",
   author =       "A. Baldereschi",
@@ -4018,6 +4032,20 @@
   notes =        "mean value k point",
 }
 
+@Article{monkhorst76,
+  title =        "Special points for Brillouin-zone integrations",
+  author =       "Hendrik J. Monkhorst and James D. Pack",
+  journal =      "Phys. Rev. B",
+  volume =       "13",
+  number =       "12",
+  pages =        "5188--5192",
+  numpages =     "4",
+  year =         "1976",
+  month =        jun,
+  doi =          "10.1103/PhysRevB.13.5188",
+  publisher =    "American Physical Society",
+}
+
 @Article{zhu98,
   title =        "Ab initio pseudopotential calculations of dopant
                  diffusion in Si",
diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index bd259d0..231345a 100644
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -358,7 +358,7 @@ The respective Kohn-Sham equations for the effective single-particle wave functi
 \text{ ,}
 \end{equation}
 \begin{equation}
-V_{\text{eff}}=V(\vec{r})+\int\frac{e^2n(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}'
+V_{\text{eff}}(\vec{r})=V(\vec{r})+\int\frac{e^2n(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}'
  + V_{\text{xc}(\vec{r})}
 \text{ ,}
 \label{eq:basics:kse2}
@@ -369,7 +369,7 @@ n(\vec{r})=\sum_{i=1}^N |\Phi_i(\vec{r})|^2
 \label{eq:basics:kse3}
 \end{equation}
 where the local exchange-correlation potential $V_{\text{xc}}(\vec{r})$ is the partial derivative of the exchange-correlation functional $E_{\text{xc}}[n(\vec{r})]$ with respect to the charge density $n(\vec{r})$ for the ground-state $n_0(\vec{r})$.
-The first term in equation \eqref{eq:basics:kse1} corresponds to the kinetic energy of non-interacting electrons and the second term of equation \eqref{eq:basics:kse2} is just the Hartree contribution to the interaction energy.
+The first term in equation \eqref{eq:basics:kse1} corresponds to the kinetic energy of non-interacting electrons and the second term of equation \eqref{eq:basics:kse2} is just the Hartree contribution $V_{\text{H}}(\vec{r})$ to the interaction energy.
 %\begin{equation}
 %V_{\text{xc}}(\vec{r})=\frac{\partial}{\partial n(\vec{r})}
 % E_{\text{xc}}[n(\vec{r})] |_{n(\vec{r})=n_0(\vec{r})}
@@ -441,10 +441,10 @@ Thus, local basis sets enable the implementation of methods that scale linearly
 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.
+In fact, the employed {\textsc vasp} software is solving the KS equations within a plane-wave (PW) 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
+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 PW basis set
 \begin{equation}
 \Phi_i(\vec{r})=\sum_{\vec{G}
 %, |\vec{G}+\vec{k}|<G_{\text{cut}}}
@@ -456,26 +456,67 @@ Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave
 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 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.
-size of matrix to diagonalize determined by cut-off energy, severe 
+Next to their simplicity, plane waves have several advantages.
+The basis set is orthonormal by construction and, as mentioned above, it is simple to check for convergence.
+The biggest advantage, however, is the ability to perform exact calculations by a discrete sum over a numerical grid.
+This is due to the related construction of the grid and the PW basis.
+Ofcourse, exactness is restricted by the fact that the PW basis set is finite.
+The simple form of the PW representation of the KS equations
+\begin{equation}
+\sum_{\vec{G}'} \left[
+ \frac{\hbar^2}{2m}|\vec{k}+\vec{G}|^2 \delta_{\vec{GG}'}
+ + \tilde{V}(\vec{G}-\vec{G}')
+ + \tilde{V}_{\text{H}}(\vec{G}-\vec{G}')
+ + \tilde{V}_{\text{xc}}(\vec{G}-\vec{G}')
+\right] c_{i,\vec{k}+\vec{G}} = \epsilon_i c_{i,\vec{k}+\vec{G}}
+\label{eq:basics:pwks}
+\end{equation}
+reveals further advantages.
+The various potentials are described in terms of their Fourier transforms.
+Equation \eqref{eq:basics:pwks} is solved by diagonalization of the Hamiltonian matrix $H_{\vec{k}+\vec{G},\vec{k}+\vec{G}'}$ given by the terms in the brackets.
+The gradient operator is diagonal in reciprocal space whereas the exchange-correlation potential has a diagonal representation in real space.
+This suggests to carry out different operations in real and reciprocal space, which requires frequent Fourier transformations.
+These, however, can be efficiently achieved by the fast Fourier transformation (FFT) algorithm.
+
+There are likewise disadvantages associated with the PW representation.
+By construction, PW calculations require a periodic system.
+This does not pose a severe problem since non-periodic systems can still be described by a suitable choice of the simulation cell.
+Describing a defect, for instance, requires the inclusion of enough bulk material in the simulation to prevent or reduce the interaction with its periodic, artificial images.
+As a consequence the number of atoms involved in the calculations are increased.
+To describe surfaces, sufficiently thick vacuum layers need to be included to avoid interaction of adjacent crystal slabs.
+Clearly, to appropriately approximate the wave functions and the respective charge density of a system composed of vacuum in addition to the solid in a PW basis, an increase of the cut-off energy is required.
+According to equation \eqref{eq:basics:pwks} the size of the Hamiltonian depends on the cut-off energy and, therefore, the computational effort is likewise increased.
+For the same reason, the description of tightly bound core electrons and the respective, highly localized charge density is hindered.
+However, a much more profound problem exists whenever wave functions for the core as well as the valence electrons need to be calculated within a PW basis set.
+Wave functions of the valence electrons exhibit rapid oscillations in the region occupied by the core electrons near the nuclei.
+The oscillations maintain the orthogonality between the wave functions of the core and valence electrons, which is compulsory due to the exclusion principle.
+Accurately approximating these oscillations demands for an extremely large PW basis set, which is too large for practical use.
+Fortunately, the impossibility to model the core in addition to the valence electrons is eliminated in the pseudopotential approach discussed in the next section.
 
 \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.
+As discussed in the last part of the previous section, an extremely large basis set of plane waves would be required to perform an all-electron calculation and a vast amount of computational time would be required to calculate the electronic wave functions.
+It is worth to stress out one more time, that this is due to the orthogonalization wiggles of the wave functions of valence electrons near the nuclei.
+Thus, existing core states practically prevent the use of a PW basis set.
+However, the core electrons, which are tightly bound to the nuclei, do not contribute significantly to chemical bonding or other physical properties of the solid.
+This fact is exploited in the pseudopotential approach \cite{} by removing the core electrons and replacing the atom and the associated strong ionic potential by a pseudoatom and a weaker pseudopotential that acts on a set of pseudo wave functions rather than the true valance wave functions.
+Certain conditions need to be fulfilled by the constructed pseudopotentials and the resulting pseudo wave functions.
+Outside the core region, the pseudo and real wafe functions as well as the generated charge densities need to be identical.
+...
+A pseudopotential is called norm-conserving if the pseudo and real charge contained within the core region match.
+...
 
 \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.
+Following Bloch's 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{}.
+Methods have been derived for obtaining very accurate approximations by a summation over special sets of $\vec{k}$ points with distinct, associated weights \cite{baldereschi73,chadi73,monkhorst76}.
 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.
+For supercells, i.e. repeating unit cells that contain several primitive cells, restricting the sampling of the Brillouin zone (BZ) to the $\Gamma$ point can yield quite accurat results.
+In fact, with respect to BZ sampling, calculating wave functions of a supercell containing $n$ primitive cells for only one $\vec{k}$ point is equivalent to the scenario of a single primitive cell and the summation over $n$ points in $\vec{k}$ space.
+In general, finer $\vec{k}$ point meshes better account for the periodicity of a system, which in some cases, however, might be fictious anyway.
 
 \subsection{Hellmann-Feynman forces}