GGA Question
(1)
\begin{align} \ \ f(r,r') = \frac{\delta^2 E}{ \delta n(r) \delta n(r')} = \bigg[ \frac{\partial}{\partial n} - \nabla \frac{\partial}{\partial \nabla n} \bigg]\bigg[ \frac{\partial e}{\partial n} - \nabla \frac{\partial e}{\partial \nabla n} \bigg], \end{align}

where

(2)
\begin{align} E = \int e(n, \nabla n) d^3r. \end{align}
(3)
\begin{align} \ \ f(r,r') = \frac{\partial^2 e}{\partial n^2} - \nabla \frac{\partial^2 e}{\partial \nabla n \partial n} - \frac{\partial}{\partial n} \nabla \frac{\partial e}{\partial \nabla n} + \nabla_i\nabla_j \frac{\partial^2 e}{\partial \nabla n_i \nabla n_j }. \end{align}
(4)
\begin{align} \ \ f(r,r') = \frac{\partial^2 e}{\partial n^2} - \nabla \frac{\partial^2 e}{\partial \nabla n \partial n} - \nabla \frac{\partial}{\partial n} \frac{\partial e}{\partial \nabla n} + \nabla_i\nabla_j \frac{\partial^2 e}{\partial \nabla n_i \nabla n_j }. \end{align}
(5)
\begin{align} \ \ f(r,r') = \frac{\partial^2 e}{\partial n^2} - 2 \nabla \frac{\partial^2 e}{\partial \nabla n \partial n} + \nabla_i\nabla_j \frac{\partial^2 e}{\partial \nabla n_i \nabla n_j }. \end{align}
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