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 . 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
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