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[lectures/latex.git] / posic / thesis / basics.tex

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index d80a4d0..249614a 100644 (file)
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -558,15 +558,15 @@ However, moving an ion, i.e.\ altering its position, changes the wave functions
 Writing down the derivative of the total energy $E$ with respect to the position $\vec{R}_i$ of ion $i$
 \begin{equation}
 \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}
+ \sum_j \langle \Phi_j | \frac{\partial H}{\partial\vec{R}_i} | \Phi_j \rangle
++\sum_j \langle \frac{\partial \Phi_j}{\partial\vec{R}_i} | H \Phi_j \rangle
++\sum_j \langle \Phi_j H | \frac{\partial \Phi_j}{\partial \vec{R}_i} \rangle
 \text{ ,}
 \end{equation}
 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 \Phi_j^*\Phi_j\frac{\partial V}{\partial \vec{R}_i}
+\vec{F}_i=-\sum_j \langle \Phi_j | \Phi_j\frac{\partial V}{\partial \vec{R}_i} \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.