  The Verwey phase transition in magnetite has been analyzed using the group
theory methods. It is found that two order parameters with the symmetries $X_3$
and $\Delta_5$ induce the structural transformation from the high-temperature
cubic to the low-temperature monoclinic phase. The coupling between the order
parameters is described by the Landau free energy functional. The electronic
and crystal structure for the cubic and monoclinic phases were optimized using
the {\it ab initio} density functional method. The electronic structure
calculations were performed within the generalized gradient approximation
including the on-site interactions between 3d electrons at iron ions -- the
Coulomb element $U$ and Hund's exchange $J$. Only when these local interactions
are taken into account, the phonon dispersion curves, obtained by the direct
method for the cubic phase, reproduce the experimental data. It is shown that
the interplay of local electron interations and the coupling to the lattice
drives the phonon order parameters and is responsible for the opening of the
gap at the Fermi energy. Thus, it is found that the metal-insulator transition
in magnetite is promoted by local electron interactions, which significantly
amplify the electron-phonon interaction and stabilize weak charge order
coexisting with orbital order of the occupied $t_{2g}$ states at Fe ions. This
provides a scenario to understand the fundamental problem of the origin of the
Verwey transition in magnetite.
