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Based on ab initio density functional theory, we have investigated amicroscopic mechanism that leads to Fe$^{2+}_{B}$/Fe$^{3+}_{B}$ dimerizationand consequent ferroelectricity in charge ordered Fe$_{3}$O$_{4}$ with $P2$symmetry. In addition to the simple inter-site Coulomb repulsion, quantumhybridization effects are invoked to explain the Fe$^{2+}_{B}$/Fe$^{3+}_{B}$bond dimerization. Our results, based on the generalized gradient approximation+ Hubbard $U$ (GGA+$U$) method, indicate that noncentrosymmetric $P2$ magnetiteshows a finite and sizeable ferroelectric polarization along the $b$crystalline axis. From the $U$ dependence of polarization, we conclude that theorigin of ferroelectricity in $P2$ Fe$_{3}$O$_{4}$ lies in the recentlyproposed "intermediate site/bond-centered charge ordering".

Ab initio investigation of Fe$^{2+}$/Fe$^{3+}$ dimerization and ferroelectricity in multiferroic magnetite: role of electronic correlations