From: hackbard <hackbard>
Date: Wed, 21 Jun 2006 15:47:11 +0000 (+0000)
Subject: .
X-Git-Url: https://hackdaworld.org/gitweb/?a=commitdiff_plain;h=cbacb767e889a7a8d7196b37286b9eeec2eb802c;p=lectures%2Flatex.git

.
---

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index 24bfc19..2eda8a2 100644
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -28,13 +28,24 @@ The solution of these equations provides the complete information of a system ev
 The following chapters cover the tools of the trade necessary for the MD simulation technique.
 First a detailed overview of the available integration algorithms is given, including their advantages and disadvantages.
 After that the interaction potentials and their accuracy for describing certain systems of elements are discussed.
-
-
+In addition special techniques will be outlined which reduce the complexity of the MD algorithm, though the force/energy evaluation almost inevitably dictates the overall speed.
 
 \subsection{Integration algorithms}
 
+
 \subsection{Interaction potentials}
 
+The potential energy of $N$ interacting atoms can be written in the form
+\begin{equation}
+U(\{{\bf r}\}) = \sum_i U_1({\bf r}_i) + \sum_i \sum_{j>i} U_2({\bf r}_i,{\bf r}_j) + \sum_i \sum_{j>i} \sum_{k>j>i} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) \ldots
+\end{equation}
+where $U$ is the total potential energy.
+$U_1$ is a single particle potential describing external forces.
+This could for instance be the gravitational force or an electric field.
+$U_2$ is a two body pair potential which only depends on the distance $r_{ij}$ between the two atoms $i$ and $j$.
+
+$U_3$ is a three body potential which may have an additional angular dependence describing covalent bonds, plus higher order terms which are expected to be small and thus neglected.
+
 \subsubsection{The Lennard-Jones potential}
 
 The L-J potential is a realistic two body pair potential and is of the form