From: hackbard Date: Thu, 19 May 2011 16:29:05 +0000 (+0200) Subject: nearly finished pseudopotentials X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=fcc70f48c064efc50d86a560245aac02789dfe39;p=lectures%2Flatex.git 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}|