DFT formalism and derivation of the Kohn-Sham equations

We define the energy as a functional of the density as

$$E[n({\bf r})] = \min_{\Psi\vert n}\langle \Psi \vert {H} \vert \Psi \rangle$$ (145)

where we are minimizing with respect to all the possible wavefunctions compatible with the density $n({\bf r})$.

The ground-state can be found by minimizing the functional with respect to the density, subject to the constraint:

$$\int d^3r n({\bf r}) = N$$ (146)

where $N$ is the total number of electrons.