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What is the difference between these four approaches?

My current understanding is that Hartree-Fock uses only 1 slater-determine (accounting for antisymmetry) but neglects correlation between electrons. HF orbitals are primarily determined self-consistently.

DFT is the idea that there is a functional (presently unknown) that maps energy in terms of density. Hohenberg and Kohn found a proof that it exists but did not actually find it.

$E[p] = T[p] + E(EE)[P] + E(EN)[P]$

I am rather confused about where kohn-sham theory fits into all of this. Is it's entire purpose simply to define kinetic energy? How does it do this?

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    $\begingroup$ The Kohn-Sham formalism reintroduced the orbital approximation into DFT with a little twist. I recommend reading in a textbook about it, as the topic itself is a little too broad for this site. $\endgroup$ Dec 9 '19 at 1:41
  • $\begingroup$ @Martin-マーチン I do not have a textbook that covers this material because it is not always covered in foundational physical chemistry courses/textbooks. I do not have the funds to purchase an entire textbook for only one unit of material for a course. Any help would be very appreciated. $\endgroup$ Dec 9 '19 at 11:43
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    $\begingroup$ I understand the predicament. I was a student once and did not always have the funds for all books. Especially highly specialised are expensive. Talk to your librarian about the books that are available, that sometimes also are looking for recommendations to buy new stock. You won't learn the specifics of computational chemistry from a physical chemistry book. $\endgroup$ Dec 9 '19 at 12:20
  • $\begingroup$ @Martin-マーチン I really do not need specifics. We covered the very basic in our phys chem course. He wants us to understand the "ingredients" in each. We do not need the nitty gritty details. $\endgroup$ Dec 9 '19 at 17:31
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    $\begingroup$ >My current understanding is that Hartree-Fock uses only 1 slater-determine (accounting for antisymmetry) but neglects correlation between electrons. | Accounting for antisymmetry actually does account for some electron correlation. Not for all of it, but for some of it. $\endgroup$
    – permeakra
    Aug 16 '20 at 12:20
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The Kohn-Sham approach is one on top of the original DFT idea proposed by Hohenberg-Kohn.

They propose – to find the energy functional and its equations, which, as you said, are unknown – to start with what is known: the functional if there is no interactions. So that is what they do: approximate the kinetic energy and Slater determinant from the known, non-interacting system and add all the terms that are additionally known (i.e. Coulomb, kinetic energies and the electron/nucleus potential) until you are left with the Kohn-Sham equation, which has "the smallest unknown", the exchange & correlation potential.

I recommend to read more in a book or on Wikipedia in case you want a more detailed picture on any/all of the methods.

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