From: 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