From: hackbard Date: Tue, 20 Sep 2011 09:03:19 +0000 (+0200) Subject: force, to be on the save side X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=7fb9069b2ce11bdd32cc2489e98edd632263e944;p=lectures%2Flatex.git force, to be on the save side --- diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index 391af40..5da2eaa 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -566,7 +566,7 @@ Writing down the derivative of the total energy $E$ with respect to the position indeed reveals a contribution to the change 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 \begin{equation} -\vec{F}_i=-\sum_j \langle \Phi_j | \Phi_j\frac{\partial V}{\partial \vec{R}_i} \rangle +\vec{F}_i=-\sum_j \langle \Phi_j | \frac{\partial V}{\partial \vec{R}_i} \Phi_j \rangle \text{ .} \end{equation} 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.