  Based on ab initio density functional theory, we have investigated a
microscopic mechanism that leads to Fe$^{2+}_{B}$/Fe$^{3+}_{B}$ dimerization
and consequent ferroelectricity in charge ordered Fe$_{3}$O$_{4}$ with $P2$
symmetry. In addition to the simple inter-site Coulomb repulsion, quantum
hybridization 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 $P2$ magnetite
shows a finite and sizeable ferroelectric polarization along the $b$
crystalline axis. From the $U$ dependence of polarization, we conclude that the
origin of ferroelectricity in $P2$ Fe$_{3}$O$_{4}$ lies in the recently
proposed "intermediate site/bond-centered charge ordering".
