From: hackbard <hackbard@sage.physik.uni-augsburg.de>
Date: Fri, 13 May 2011 17:49:30 +0000 (+0200)
Subject: finished (?) bo
X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=0937be7d4a2ab73d70a6ee6dad0a686be0347d35;p=lectures%2Flatex.git

finished (?) bo
---

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index 135f2ac..65048e7 100644
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -291,8 +291,20 @@ As will be shown, DFT can formally be regarded as an exactification of the Thoma
 
 Born and Oppenheimer proposed a simplification enabling the effective decoupling of the electronic and ionic degrees of freedom \cite{born27}.
 Within the Born-Oppenheimer (BO) approximation the light electrons are assumed to move much faster and, thus, follow adiabatically to the motion of the heavy nuclei, if the latter are only slightly deflected from their equilibrium positions.
-Thus, on the timescale of electronic motion the ions appear at fixed positions and, on the other way round, for the nuclei the electrons appear blurred in space adding an extra term to the ion-ion potential.
-The simplified Schr\"odinger equation is rewritten without the kinetic energy of the ions and its positions enter as fixed parameters.
+Thus, on the timescale of electronic motion the ions appear at fixed positions.
+On the other way round, on the timescale of nuclear motion the electrons appear blurred in space adding an extra term to the ion-ion potential.
+The simplified Schr\"odinger equation no longer contains the kinetic energy of the ions.
+The momentary positions of the ions enter as fixed parameters and, therefore, the ion-ion interaction may be regarded as a constant added to the electronic energies.
+The Schr\"odinger equation describing the remaining electronic problem reads
+\begin{equation}
+\left[-\frac{\hbar^2}{2m}\sum_j\nabla^2_j-
+\sum_{j,l} \frac{Z_le^2}{|\vec{r}_j-\vec{R}_l|}+
+\frac{1}{2}\sum_{j\neq j'}\frac{e^2}{|\vec{r}_j-\vec{r}_{j'}|}
+\right] \Psi = E \Psi
+\text{ ,}
+\end{equation}
+where $Z_l$ are the atomic numbers of the nuclei and $\Psi$ is the many-electron wave function, which depends on the positions and spins of the electrons.
+There is only a parametrical dependence on the ionic coordinates $\vec{R}_l$.
 
 \subsection{Bloch theorem}