# Functional derivative of meta-generalized gradient approximation (meta-GGA)

I am not able to derive Eq. 21 of this paper

F. Zahariev, S. S. Leang, and Mark S. Gordon, "Functional derivatives of meta-generalized gradient approximation (meta-GGA) type exchange-correlation density functionals", The Journal of Chemical Physics 138, 244108 (2013); https://doi.org/10.1063/1.4811270

$$\tau({\bf r}') = \frac{1}{2}\sum_i|\nabla\psi_i({\bf r'})|^2 =\frac{1}{2}\sum_i \int |\nabla\psi_i({\bf r'})|^2 \delta^3({\bf r}-{\bf r}')d{\bf r}',\tag{2}$$ $$\begin{equation} \begin{split} \frac{\delta \tau({\bf r}')}{\delta \psi_i({\bf r})}&= - \nabla\cdot\frac{\partial}{\partial (\nabla\psi_i({\bf r}))}\Big(\frac{1}{2}|\nabla\psi_i({\bf r'})|^2 \delta^3({\bf r} - {\bf r}')\Big)\\ &= -\nabla \cdot (\nabla\psi_i({\bf r})\delta^3({\bf r} - {\bf r}'))\\ &= - \nabla\psi_i({\bf r})\cdot\nabla\delta^3({\bf r}-{\bf r}') - \nabla^2\psi_i({\bf r})\delta^3({\bf r}-{\bf r}'). \end{split} \end{equation}$$

• – Qmechanic Aug 13 '18 at 8:02
• The site standard for maths and formula is Mathjax and we actively discourage people from posting images. – StephenG Aug 13 '18 at 8:40

Hints: Use that $$\frac{\delta \psi_j({\bf r}^{\prime})}{\delta \psi_i({\bf r})}~=~\delta^i_j ~\delta^3({\bf r}^{\prime}\!-\!{\bf r}),$$
\begin{align} \frac{\delta \tau({\bf r}^{\prime})}{\delta \psi_i({\bf r})} ~=~&\sum_j\nabla^{\prime} \psi_j({\bf r}^{\prime})\cdot \nabla^{\prime}\frac{\delta \psi_j({\bf r}^{\prime})}{\delta \psi_i({\bf r})}\cr ~=~&\nabla^{\prime} \psi_i({\bf r}^{\prime})\cdot \nabla^{\prime}\delta^3({\bf r}^{\prime}\!-\!{\bf r})\cr ~=~&-\nabla^{\prime} \psi_i({\bf r}^{\prime})\cdot \nabla\delta^3({\bf r}\!-\!{\bf r}^{\prime})\cr ~=~&\ldots\cr ~=~&-\nabla \psi_i({\bf r})\cdot \nabla\delta^3({\bf r}\!-\!{\bf r}^{\prime}) -\delta^3({\bf r}\!-\!{\bf r}^{\prime})~\nabla^2 \psi_i({\bf r}) . \end{align}