In the following the simulation methods used within the scope of this study are introduced.
Enabling the investigation of the evolution of structure on the atomic scale, molecular dynamics (MD) simulations are chosen for modeling the behavior and precipitation of C introduced into an initially crystalline Si environment.
To be able to model systems with a large amount of atoms computational efficient classical potentials to describe the interaction of the atoms are most often used in MD studies.
-For reasons of flexibility in executing this non-standard task and in order to be able to use a novel interaction potential \cite{albe_sic_pot} an appropriate MD code called {\textsc posic}\footnote{{\textsc posic} is an abbreviation for {\bf p}recipitation {\bf o}f {\bf SiC}} including a library collecting respective MD subroutines was developed from scratch\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/download/posic/posic.tar.bz2}.
+For reasons of flexibility in executing this non-standard task and in order to be able to use a novel interaction potential \cite{albe_sic_pot} an appropriate MD code called {\textsc posic}\footnote{{\textsc posic} is an abbreviation for {\bf p}recipitation {\bf o}f {\bf SiC}} including a library collecting respective MD subroutines was developed from scratch\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/posic}.
The basic ideas of MD in general and the adopted techniques as implemented in {\textsc posic} in particular are outlined in section \ref{section:md}, while the functional form and derivative of the employed classical potential is presented in appendix \ref{app:d_tersoff}.
An overview of the most important tools within the MD package is given in appendix \ref{app:code}.
Although classical potentials are often most successful and at the same time computationally efficient in calculating some physical properties of a particular system, not all of its properties might be described correctly due to the lack of quantum-mechanical effects.
\section{Molecular dynamics simulations}
\label{section:md}
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\begin{quotation}
\dq We may regard the present state of the universe as the effect of the past and the cause of the future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.\dq{}
\noindent
Pierre Simon de Laplace phrased this vision in terms of a controlling, omniscient instance - the {\em Laplace demon} - which would be able to look into the future as well as into the past due to the deterministic nature of processes, governed by the solution of differential equations.
Although Laplace's vision is nowadays corrected by chaos theory and quantum mechanics, it expresses two main features of classical mechanics, the determinism of processes and time reversibility of the fundamental equations.
-This understanding was one of the first ideas for doing molecular dynamics simulations, considering an isolated system of particles, the behaviour of which is fully determined by the solution of the classical equations of motion.
+This understanding may be regarded as the basic principle of molecular dynamics, considering an isolated system of particles, the behaviour of which is fully determined by the solution of the classical equations of motion.
\subsection{Introduction to molecular dynamics simulations}
The MD method was first introduced by Alder and Wainwright in 1957 \cite{alder57,alder59} to study the interactions of hard spheres.
The basis of the approach are Newton's equations of motion to describe classicaly the many-body system.
MD is the numerical way of solving the $N$-body problem which cannot be solved analytically for $N>3$.
-A potential is necessary describing the interaction of the particles.
-By MD a complete description of the system in the sense of classical mechanics on the microscopic level is obtained.
+A potential is necessary to describe the interaction of the particles.
+By MD, a complete description of the system in the sense of classical mechanics on the microscopic level is obtained.
The microscopic information can then be translated to macroscopic observables by means of statistical mechanics.
The basic idea is to assume that the particles can be described classically by Newton's equations of motion, which are integrated numerically.
{\bf F}_i = - \nabla_{{\bf r}_i} U({\{\bf r}\}) \, \textrm{.}
\label{eq:basics:force}
\end{equation}
-Given the initial conditions ${\bf r}_i(t_0)$ and $\dot{\bf r}_i(t_0)$ the equations can be integrated by a certain integration algorithm.
+Given the initial conditions ${\bf r}_i(t_0)$ and $\dot{\bf r}_i(t_0)$, the equations can be integrated by a certain integration algorithm.
The solution of these equations provides the complete information of a system evolving in time.
The following sections cover the tools of the trade necessary for the MD simulation technique.
Three ingredients are required for a MD simulation:
f_R(r_{ij}) & = & A_{ij} \exp (- \lambda_{ij} r_{ij} ) \\
f_A(r_{ij}) & = & -B_{ij} \exp (- \mu_{ij} r_{ij} )
\end{eqnarray}
-The function $f_C$ is the a cutoff function to limit the range of interaction to nearest neighbors.
+The function $f_C$ is a cutoff function to limit the range of interaction to nearest neighbors.
It is designed to have a smooth transition of the potential at distances $R_{ij}$ and $S_{ij}$.
\begin{equation}
f_C(r_{ij}) = \left\{
\subsection{Verlet integration}
\label{subsection:integrate_algo}
-A numerical method to integrate Newton's equation of motion was presented by Verlet in 1967 \cite{verlet67}.
+A numerical method to integrate Newton's equations of motion was presented by Verlet in 1967 \cite{verlet67}.
The idea of the so-called Verlet and a variant, the velocity Verlet algorithm, which additionaly generates directly the velocities, is explained in the following.
Starting point is the Taylor series for the particle positions at time $t+\delta t$ and $t-\delta t$
\begin{equation}
Whether a saddle point configuration and, thus, the minimum energy path is obtained by the CRT method, needs to be verified by caculating the respective vibrational modes.
Modifications used to add the CRT feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
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-% advantages of pw basis with respect to hellmann feynman forces / pulay forces
-% crt sketch needs increased text
+% todo - advantages of pw basis concenring hf forces
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