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As I understand Kohn Sham theory, it is proposed that the ground state energy for a molecule can be derived from the ground state of a fictitious set of non-interacting electrons moving in a fictitious potential. This fictitious potential is conventionally written as $E_T+E_V+E_J+E_{XC}$ where we have known formulae for the kinetic energy $E_T$, the potential energy $E_V$ and the Coulomb energy $E_J$ but the "exchange correlation" energy $E_{XC}$ is a priori unknown.

I have two questions:

(1) what theoretical reason do we have to believe that the ground state energy can be derived in this way? Clearly we have practical reasons to know that this approach can work pretty well, but is it possible that there are some molecules where the Kohn Sham approach simply won't work?

(2) is there any theoretical reason to believe that the $E_{XC}$ term has anything in common between different molecules? I have seen it suggested that with heuristic approaches to finding $E_{XC}$ "different models work better in different scenarios" which makes me suspicious that we have simply turned a question of heuristically finding the ground state energy for an individual molecule into a question of heuristically finding the exchange correlation energy for that molecule?

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