Why do people add H(core)+F in Hartree-Fock?
Actually, $F$ is double-counting the electron-electron interaction when evaluating the total energey. By adding $H^{\text{core}}$ and using a factor of $\frac{1}{2}$, which I suspect is implicit in $D$, one can obtain the correct energy. This is basically an occurence of the HF energy not being the sum of orbital energies (which are a result of $F$).