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3.2 Square Gradient Approximation (SGA)

A first improvement on the simple LDA is based on a gradient expansion for ${\rm\cal {F}}_{ex}$:
\begin{displaymath}
{\rm\cal{F}}_{ex} [\rho ] =
\int_{V} d\vec{r}\; \left[ f(\r...
...f_{2}(\rho(\vec{r}))
\vert\nabla\rho(\vec{r})\vert^{2} \right]
\end{displaymath} (3.3)

Comparing this with the functional Taylor expansion of ${\rm\cal {F}}_{ex}[\rho ]$ we find
\begin{displaymath}
f_{2}(\rho)=\frac{kT}{12} \int d\vec{r}\; r^{2} c^{(2)}(\rho;r)
\end{displaymath} (3.4)

The variational rule leads to
\begin{displaymath}
f'-f_{2}' \vert\nabla\rho(\vec{r})\vert^{2}
-2f_{2} \nabla^{2}\rho(\vec{r})=\mu- \phi(\vec{r}))
\end{displaymath} (3.5)



Franz J. Vesely Oct 2001
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001