Go ahead and read my two most famous publications:

Pierre Hohenberg, Walter Kohn: Inhomogeneous Electron Gas. In: Physical Review B 136 , 864-871 (1964) [journals.aps.org]

Walter Kohn, Lu Jeu Sham: Self-Consistent Equations Including Exchange and Correlation Effects. In: Physical Review A 140 , 1133-1138 (1965) [journals.aps.org]

The 1st paper shows via the variational principle and a reductio ad absurdum , that two external potentials differing by more than a trivial additive constant do not yield the same ground state electron density, therefore justifying a maping of all the system's properties (Hamiltonian, wave function, energy eigenvalues) to the density alone as a basic variable (Hohenberg-Kohn Theorem)

The 2nd paper derives Self-Consistent equations including exchange and correlation effects (Kohn-Sham equations) by an ansatz that relates the real system of interacting electrons to an artificial system of non-interacting electrons with the same density and minimizing the energy in a Euler-Lagrange formalism under the constraint, that the number of electrons stays fixed δN=0

Support one of my eager students Leon Walter Redeker [www.linkedin.com].