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Adding electron-electron interactions

The exchange correlations and Hartree potenatials are density-dependent and add a self-consistency ingredient to the soup. Now, the Hamiltonian becomes a Kohn-Sham Hamiltonian. After diagonalizing teh Hamiltonian, we obtain teh Fourier compotents of the wave-functions. Then, we can calculate the density in real-space and reciprocal space.

The exchange-correlation potentials is given as the derivative of the energy with respect to the density $n$. It must be calculated in real space, and then Fourier transformed to that it can be added to the Hamiltonian in momentum representation.

The Hartree potential

\begin{displaymath}
V_H({\bf r}) = \int \frac{n({\bf r'})}{\vert{\bf r}-{\bf r}'\vert} d^3r'
\end{displaymath} (255)

can be Fourier transformed to give
\begin{displaymath}
V_H({\bf K}-{\bf K}') = \frac{4\pi}{\vert{\bf K}-{\bf K}'\vert^2} n({\bf K}-{\bf K}').
\end{displaymath} (256)



Adrian E. Feiguin 2009-11-04