# How to use Density Functional Theory to get the correct energy spectrum

I'm having some trouble understanding how DFT can be used to obtain results for electronic structure calculations. We can assume that we're studying an $$N$$ electron atom with a fixed nucleus and want to get its ground state energy.

In Kohn-Sham DFT, we imagine a fictitious system of non-interacting particles, such that $$\Psi(\mathbf r_1, \ldots, \mathbf r_N) = \prod_i \psi_i(\mathbf r)$$. From this, we can construct the KS equations: $$\left[ -\frac 12 \nabla^2 + V_{ext}(\mathbf r) + \int \frac{ \rho(\mathbf r')}{|\mathbf r- \mathbf r'|} d \mathbf r' + V_{XC}(\mathbf r) \right] \psi_i = E_i \psi_i.$$ The middle term is the Hartree potential. These equations can be solved self consistently. First, guess the form of the electron density and approximate the exchange-correlation potential. Then, at each iteration we solve these equations (I.E by diagonalisation) to get the $$\psi_i$$ and then update the density to construct a new Hamiltonian since $$\rho(\mathbf r) = \sum_{i=1}^N |\psi_i(\mathbf r)|^2$$. Until the eigenvalues converge. This allows us to get the correct ground state charge density (through the variational theorem).

What I don't understand is how to proceed from there. Are the eigenvalues of the KS Hamiltonian equal to the energy spectrum of the true system? If so, why? And if not, how is the correct ground state energy found from here?

Solving the Khon-Sham equations yields the Khon-Sham orbitals $$\psi_i$$ and the KS orbital energies $$\varepsilon_i$$. The groundstate energy of the N-particle system is then obtained as $$E = \sum_i^N\varepsilon_i - E_H[\rho] + E_{xc}[\rho] - \int \frac{\delta E_{xc}[\rho]}{\delta\rho(r)} \rho(r)dr$$
with $$\rho(r)=\sum_i^N|\psi_i(r)|^2$$, $$E_H[\rho]=\frac{e^2}{2}\int dr \int dr' \frac{\rho(r)\rho(r')}{|r-r'|}$$. Check also wikipedia for the definitions.
But the problematic part is actually the exchange correlaction energy functional $$E_{xc}[\rho]$$. The correct expression for this functional is unknown (except for the case of a uniform electron gas) and there exists a whole zoo of exchange-correlation functionals that have to be used as approximations in place of the unknown correct exchange correlation functional. For this reason the groundstate energy obtained via KS-DFT is usually an approximation. Finding/defining new functionals that yield good results over a large range of different systems is still an active field of research.