In principle if you solve the many body Schrodinger equation you will get the whole physics and nature of the problem. With the hamiltonian, in the most general case (without Born-Oppenheimer approximation), taking into account the protons, electrons, electron-electron interaction, proton-proton interaction, and the proton-electron interaction.
In DFT you have the exchange and correlation term. I get why they have to be added and what they correct... the fermionic nature of the particles give rise to the exchange term, and the Coulomb nature of charged particles give rise to correlation term.
My question is, if in a hypothetical scenario you would be able to solve any system (regardless of its size) with the Schrodinger equation, the physics that the exchange-correlation term plays in DFT will inherently be in the solutions, right?
Another question if the answer to the above is yes. In which part of the whole DFT approximations does that term gets lost? Which approximation or postulate looses those inherent properties you would get by solving the many body Schrodinger equation.
I had read the density had the same information as the wave function but I'm thinking that's where the information gets lost.
Went to talk to a teacher today to ask this but I am still a bit lost. I told him that given the definition of the exchange-correlation term (the remainder of the total energy with the rest of the known energies) and how DFT accounts for the terms in the hamiltonian it should be zero (since there is a term that accounts for electron-electron interaction, their kinetic energy, and the potential of the nuclei)... so I asked him where in the approximations does information get lost so that we need to add an extra term. I asked if we were to put the proper wave function in the exchange-correlation term (hypothetical exact solution to the many body Schrodinger equation) it would be zero, and he said yes. So, if given an exact solution it's zero, then where does the information get lost? Are the formulas of the kinetic energy and potential energy not exact?