A Derivation of Generalized Khon-Sham Scheme

Mart1234
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Question dealing with the derivation of the generalized Khon-Sham equations regarding the final eigenvalue.
Hello, I am following the paper: https://www.yumpu.com/en/document/read/42212557/exact-exchange-in-density-functional-calculations and I am confused on page 14 where the generalized Kohn-Sham equations are derived. I follow that the ground state energy is
1700172954424.png


The minimization of this step leads to
1700172984118.png
.
where
1700173019267.png


I'm not sure of the process used to generate this eigenvalue problem. Any insight into how this was done or other resources would be greatly appreciated.
 
From the BCS theory of superconductivity is well known that the superfluid density smoothly decreases with increasing temperature. Annihilated superfluid carriers become normal and lose their momenta on lattice atoms. So if we induce a persistent supercurrent in a ring below Tc and after that slowly increase the temperature, we must observe a decrease in the actual supercurrent, because the density of electron pairs and total supercurrent momentum decrease. However, this supercurrent...
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