author hackbard Fri, 20 May 2011 23:41:41 +0000 (01:41 +0200) committer hackbard Fri, 20 May 2011 23:41:41 +0000 (01:41 +0200)

index f469237..f78de50 100644 (file)
@@ -53,10 +53,12 @@ A system of $N$ particles of masses $m_i$ ($i=1,\ldots,N$) at positions ${\bf r} m_i \ddot{\bf r}_i = {\bf F}_i \Leftrightarrow m_i \dot{\bf r}_i = {\bf p}_i\textrm{, } \dot{\bf p}_i = {\bf F}_i\textrm{ .} +\label{eq:basics:newton} The forces${\bf F}_i$are obtained from the potential energy$U(\{{\bf r}\})$: {\bf F}_i = - \nabla_{{\bf r}_i} U({\{\bf r}\}) \, \textrm{.} +\label{eq:basics:force} Given the initial conditions${\bf r}_i(t_0)$and$\dot{\bf r}_i(t_0)$the equations can be integrated by a certain integration algorithm. The solution of these equations provides the complete information of a system evolving in time. @@ -540,20 +542,49 @@ For supercells, i.e. repeating unit cells that contain several primitive cells, In fact, with respect to BZ sampling, calculating wave functions of a supercell containing$n$primitive cells for only one$\vec{k}$point is equivalent to the scenario of a single primitive cell and the summation over$n$points in$\vec{k}$space. In general, finer$\vec{k}$point meshes better account for the periodicity of a system, which in some cases, however, might be fictious anyway. -\subsection{Structural relaxation and the Hellmann-Feynman theorem} +\subsection{Structural relaxation and Hellmann-Feynman theorem} -what is relaxed, ions! -if md based relaxaion followed the respective equations of motion are valid ... -of course better cg than simple moveing ions according to force ... -force as a derivative of total energy with respect to positions of the ions ... +Up to this point, the system is in the ground state with respect to the electronic subsystem, while the positions of the ions as well as size and shape of the supercell are fixed. +To investigate equilibrium structures, however, the ionic subsystem must also be allowed to relax into a minimum energy configuration. +Local minimum configurations can be easily obtained in a MD-like way by moving the nuclei over small distances along the directions of the forces, as discussed in the MD chapter above. +Clearly, the conjugate gradient method constitutes a more sophisticated scheme, which will locate the equilibrium positions of the ions more rapidly. +To find the global minimum, i.e. the absolute ground state, methods like simulated annealing or the Monte Carlo technique, which allow the system to escape local minima, have to be used for the search. +The force on an ion is given by the negative derivative of the total energy with respect to the position of the ion. +However, moving an ion, i.e. altering its position, changes the wave functions to the KS eigenstates corresponding to the new ionic configuration. +Writing down the derivative of the total energy$E$with respect to the position$\vec{R}_i$of ion$i$+ +\frac{dE}{d\vec{R_i}}= + \sum_j \Phi_j^* \frac{\partial H}{\partial \vec{R}_i} \Phi_j ++\sum_j \frac{\partial \Phi_j^*}{\partial \vec{R}_i} H \Phi_j ++\sum_j \Phi_j^* H \frac{\partial \Phi_j}{\partial \vec{R}_i} +\text{ ,} + +indeed reveals a contributon to the chnage in total energy due to the change of the wave functions$\Phi_j$. +However, provided that the$\Phi_j$are eigenstates of$H$, it is easy to show that the last two terms cancel each other and in the special case of$H=T+V\$ the force is given by
+
+\vec{F}_i=-\sum_j \Phi_j^*\Phi_j\frac{\partial V}{\partial \vec{R}_i}
+\text{ .}
+
+This is called the Hellmann-Feynman theorem \cite{feynman39}, which enables the calculation of forces, called the Hellmann-Feynman forces, acting on the nuclei for a given configuration, without the need for evaluating computationally costly energy maps.

-\section{Modeling of defects}
+\section{Modeling of point defects}
\label{section:basics:defects}

+Point defects are defects that affect a single lattice site.
+At this site the crystalline periodicity is interrupted.
+An empty lattice site, which would be occupied in the perfect crystal structure, is called a vacancy defect.
+If an additional atom is incorporated into the perfect crystal, this is called interstitial defect.
+The disturbance caused by these defects may result in the distortion of the surrounding atomic structure and is accompanied by an increase in configurational energy.
+Thus, next to the structure of the defect, the energy needed to create such a defect, i.e. the defect formation energy, is an important value characterizing the defect and its probability of occurence.
+
+The methods presented in the last two chapters enable the investigation of defects.
...
constructing defect configuration intuitively followed by relaxation procedure

\section{Migration paths and diffusion barriers}
\label{section:basics:migration}

+% todo
+% advantages of pw basis with respect to hellmann feynman forces / pulay forces
+