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I would actually like to go through the Kohn-Sham equation derivations to see how the equivalence can be guaranteed. I’m also curious to see more explanations on ground state and excited states, and how DFT can be used to explored/is limited on these aspects.
For sure the derivation is given in the book by Dreizler & Gross (https://www.springer.com/gp/book/9783642861055). Maybe also in the somewhat less mathematical book of Richard Martin (https://doi.org/10.1017/CBO9780511805769).
If you google for ‘derivation of Kohn-Shame equations’, there are a few promising hits as well, for instance this one: https://www.unige.ch/sciences/chifi/wesolowski/public_html/dft_epfl_2016/part_I/dftepfl_part_IIIj.pdf
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