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3.1 Local Density Approximation (LDA)

For slowly varying densities it is reasonable to assume that the macroscopic thermodynamical relations remain valid on a local scale. In particular,
\begin{displaymath}
{\rm\cal{F}}_{ex} [\rho ] = \int_{V} d\vec{r}\; f(\rho(\vec{r}))
\end{displaymath} (3.1)

where $f(n)$ is the free energy density of a homogeneous fluid at density $n$.

Applying the Hohenberg-Kohn variational principle we find the equation
\begin{displaymath}
f'(\rho(\vec{r}))=\mu - \phi(\vec{r})
\end{displaymath} (3.2)

which may be used to evaluate, for given $f(n)$, the local density $\rho(\vec{r})$.

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