+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 $i$
+\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}
+\text{ ,}
+\end{equation}
+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
+\begin{equation}
+\vec{F}_i=-\sum_j \Phi_j^*\Phi_j\frac{\partial V}{\partial \vec{R}_i}
+\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.