Eq. 5: two questions/comments

1) this expression is OK if the calculation is performed with the (non-local) HF exchange potential (not the DFT one). If you use e_Hx^w(n) in your eDFT calculation, you should use the expressions in Killian's manuscript.

2) Eq. 4: it seems awkward to me to use the notation v= n*e. I would use epsilon instead of v to avoid any confusion with the potential (?). Eq. 6 should be updated accordingly.
 
3) Third term on the RHS of Eq. 5: I am not sure about the expression for v_c^w given in Eq. 4. We should have delta E^w_c[n]/delta n(r). If we write

E^w_c[n]=\int dr n(r) e^w_c(n(r))   <--- I denoted this quantity U^w_c[n] in my handwritten notes

then

delta E^w_c[n]/delta n(r)= e^w_c(n(r)) + n(r)*de^w_c/dn(n=n(r)).

Eq 6: 

1) The LZ shift expression is not OK.
2) The DD expression is fine :-)
  

Eq. (10):   

1) Third term on the RHS: same comment as for Eq. 5
2) Caution ! The LZ contribution is NOT the same for all states. It depends on the number of electrons (unless you shift the one-electron potential instead of shifting individual energies)
3) The DD expression is in principle different from the one used in Eq. 6 (I guess).