started ks system ...

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

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index 46f871c..6c03878 100644 (file)
--- 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}