From: hackbard Date: Sun, 15 May 2011 20:37:10 +0000 (+0200) Subject: started ks system ... X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=2cfbb5d8e375f3f740f35b6f02dc45a45a236a26;p=lectures%2Flatex.git started ks system ... --- diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index 46f871c..6c03878 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -277,7 +277,7 @@ It provides a stable algorithm that allows smooth changes of the system to new v \section{Denstiy functional theory} \label{section:dft} -Dirac declared that chemistry has come to an end, its content being entirely contained in the powerul equation published by Schr\"odinger in 1926 \cite{schroeder26} marking the beginning of wave mechanics. +Dirac declared that chemistry has come to an end, its content being entirely contained in the powerul equation published by Schr\"odinger in 1926 \cite{schroedinger26} marking the beginning of wave mechanics. Following the path of Schr\"odinger the problem in quantum-mechanical modeling of describing the many-body problem, i.e. a system of a large amount of interacting particles, is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of all nuclei and electrons. The Schr\"odinger equation contains the kinetic energy of the ions and electrons as well as the electron-ion, ion-ion and electron-electron interaction. This cannot be solved exactly and finding approximate solutions requires several layers of simplification in order to reduce the number of free parameters. @@ -331,11 +331,12 @@ E_0=\min_{n(\vec{r})} \text{ ,} \end{equation} where $F[n(\vec{r})]$ is a universal functional of the charge density $n(\vec{r})$, which is composed of the kinetic energy functional $T[n(\vec{r})]$ and the interaction energy functional $U[n(\vec{r})]$. -The challenging problem of determining the exact ground-state is now formally reduced to the determination of the $3$-dimensional function $n(\vec{r})$ via a well-defined but not explicitly known functional of the charge density. +The challenging problem of determining the exact ground-state is now formally reduced to the determination of the $3$-dimensional function $n(\vec{r})$, which minimizes the energy functional. +However, the complexity associated with the many-electron problem is now relocated in the task of finding the well-defined but, in contrast to the potential energy, not explicitly known functional $F[n(\vec{r})]$. -It is worth to note, that this minimal principle may be regarded as exactification of TF theory, which is rederived by the approximations +It is worth to note, that this minimal principle may be regarded as exactification of the TF theory, which is rederived by the approximations \begin{equation} -T=\int n(\vec{r})\frac{3}{10}k_{\text{F}}^2[n(\vec{r})]d\vec{r} +T=\int n(\vec{r})\frac{3}{10}k_{\text{F}}^2(n(\vec{r}))d\vec{r} \text{ ,} \end{equation} \begin{equation} @@ -347,6 +348,8 @@ U=\frac{1}{2}\int\frac{n(\vec{r})n(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}d\vec{r Now find $F[n]$ ... +As in the last section, the complex many-electron effects are relocated, this time into the exchange-correlation functional. + \subsection{Approximations for exchange and correlation} \subsection{Pseudopotentials}