.

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index 24bfc19..2eda8a2 100644 (file)
--- 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.
 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{Integration algorithms}
 

+

 \subsection{Interaction potentials}
 
 \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
 \subsubsection{The Lennard-Jones potential}
 
 The L-J potential is a realistic two body pair potential and is of the form