equation 23 from VASP website lecture: "Introduction to DFT"

[vasp_user]

This is regarding equation 23 on page 14 of lecture: "Introduction to DFT"

@ https://www.vasp.at/vasp-workshop/slides/dft_introd.pdf


Can someone please give some intuition about the bracket labeled "2" that corresponds to "double-counting corrections".
I'm not sure about the 1/2 the electron-electron interaction term and I'm really not sure about the difference between the functional derivative of the exchange correlation energy wrt density and the exchange correlation of homogenous gas substracted from one another, multiplied by the density and then integrated over. It is not at all obvious to me that the sum over the one-electron energies needed to be corrected by that.

So I have two questions (question 1 is a sanity check and question 2 is icing on the cake):

1) Is the E on the left hand side of equation 23 the same as the total energy from OUTCAR.... in other words: "energy without entropy= -1100.324543 energy(sigma->0) = -1100.324543"

2) Does someone have some intuition about the double-counting corrections mentioned above? (a rigorous place they come from is fine too but something less might do... they just came out of nowhere as I was following the lecture so I can't "go back and build up to them")

[support_vasp]

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