Editing Functional derivative (section)

====von Weizsäcker kinetic energy functional====
In 1935 [[Carl Friedrich von Weizsacker|von Weizsäcker]] proposed to add a gradient correction to the Thomas-Fermi kinetic energy functional to make it better suit a molecular electron cloud:
<math display="block">T_\mathrm{W}[\rho] = \frac{1}{8} \int \frac{\nabla\rho(\mathbf{r}) \cdot \nabla\rho(\mathbf{r})}{ \rho(\mathbf{r}) } d\mathbf{r} = \int t_\mathrm{W}(\mathbf{r}) \ d\mathbf{r} \, ,</math>
where
<math display="block"> t_\mathrm{W} \equiv \frac{1}{8} \frac{\nabla\rho \cdot \nabla\rho}{ \rho } \qquad \text{and} \ \ \rho = \rho(\boldsymbol{r}) \ . </math>
Using a previously derived [[#Formula|formula]] for the functional derivative,
<math display="block">\begin{align}
\frac{\delta T_\mathrm{W}}{\delta \rho}
& = \frac{\partial t_\mathrm{W}}{\partial \rho} - \nabla\cdot\frac{\partial t_\mathrm{W}}{\partial \nabla \rho} \\
& = -\frac{1}{8}\frac{\nabla\rho \cdot \nabla\rho}{\rho^2} - \left ( \frac {1}{4} \frac {\nabla^2\rho} {\rho} - \frac {1}{4} \frac {\nabla\rho \cdot \nabla\rho} {\rho^2} \right ) \qquad \text{where} \ \ \nabla^2 = \nabla \cdot \nabla \ ,
\end{align}</math>
and the result is,<ref name=ParrYangP247A.9>{{harvp|Parr|Yang|1989|loc= p. 247, Eq. A.9}}.</ref>
<math display="block"> \frac{\delta T_\mathrm{W}}{\delta \rho} = \ \ \, \frac{1}{8}\frac{\nabla\rho \cdot \nabla\rho}{\rho^2} - \frac{1}{4}\frac{\nabla^2\rho}{\rho} \ . </math>