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.
 
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