This alternative formulation of the forces can be extended to obtain the second-order derivatives of the energy with respect to the atomic displacements \cite{Andrade2010thesis}, which are required to calculate vibrational properties as discussed in section \ref{sec:sternheimer}. In general, the perturbation operator associated with an ionic displacement can be written as

\[\label{eq:ionicpertmod} \frac{\partial {v}_\alpha(\vec{r}- \vec{R}_\alpha)}{\partial R_{i\alpha}} = -{v}_{\alpha}(\vec{r}- \vec{R}_\alpha)\frac{\partial}{\partial r_i} -\frac{\partial}{\partial r_i}{v}_\alpha(\vec{r}- \vec{R}_\alpha)\ .\]

Using this expression, the terms of the dynamical matrix, eq. (\ref{eq:dynmatrix}), are evaluated as \[\begin{gathered} \left< \varphi_n \left| \frac{\partial \hat{v}_{\alpha}}{\partial R_{i\alpha}} \right| \frac{\partial \varphi_n}{\partial R_{j \beta}} \right> = -\left[ \left< \varphi_n \left| \hat{v}_{\alpha} \right| \frac{\partial^2 \varphi_n}{\partial R_{j \beta} \partial r_i} \right> \right.\\ + \left. \left< \frac{\partial \varphi_n}{\partial r_i}\left| \hat{v}_{\alpha} \right| \frac{\partial \varphi_n}{\partial R_{j \beta}} \right>\right] + {\rm c.c.}\ ,\end{gathered}\] and \[\begin{gathered} \left< \varphi_n \left| \frac{\partial^2 \hat{v}_{\alpha}}{\partial R_{i\alpha} \partial R_{j\alpha}} \right| \varphi_n \right> = \left[ \left< \frac{\partial^2 \varphi_n}{\partial r_i \partial r_j} \left| \hat{v}_{\alpha} \right| \varphi_n \right>\right.\\ + \left. \left< \frac{\partial \varphi_n}{\partial r_i} \left| \hat{v}_{\alpha}\right| \frac{\partial \varphi_n}{\partial r_j} \right>\right] + {\rm c.c.}\ .\end{gathered}\]