  The electronic properties of two spinels Fe$_3$O$_4$ and Fe$_2$SiO$_4$ are
studied by the density functional theory. The local Coulomb repulsion $U$ and
the Hund's exchange $J$ between the $3d$ electrons on iron are included. For
$U=0$, both spinels are half-metals, with the minority $t_{2g}$ states at the
Fermi level. Magnetite remains a metal in a cubic phase even at large values of
$U$. The metal-insulator transition is induced by the $X_3$ phonon, which
lowers the total energy and stabilizes the charge-orbital ordering.
Fe$_2$SiO$_4$ transforms to a Mott insulating state for $U>2$ eV with a gap
$\Delta_g\sim U$. The antiferromagnetic interactions induce the tetragonal
distortion, which releases the geometrical frustration and stabilizes the
long-range order. The differences of electronic structures in the high-symmetry
cubic phases and the distorted low-symmetry phases of both spinels are
discussed.
