The hybrid density functional (DFT) method B3LYP was used to study the mechanism of the methane hydroxylation reaction catalyzed by a non-heme diiron enzyme, methane monooxygenase (MMO). The key reactive compound Q of MMO was modeled by (NH2)(H2O)Fe(mu-O)(2)(eta(2)-HCOO)(2)Fe(NH2) (H2O), I. The reaction is shown to take place via a bound-radical mechanism and an intricate change of the electronic structure of the Fe core is associated with the reaction process. Starting with I, which has a diamond-core structure with two Fe-IV atoms, L4FeIV(mu-O)(2)(FeL4)-L-IV, the reaction with methane goes over the rate-determining H-abstraction transition state III to reach a bound-radical intermediate IV, L4FeIV(mu-O)(mu-OH(... CH3))(FeL4)-L-III, which has a bridged hydroxyl ligand interacting weakly with a methyl radical and is in an Fe-III-Fe-IV mixed valence state. This short-lived intermediate IV easily rearranges intramolecularly through a low barrier at transition state V for addition of the methyl radical to the hydroxyl ligand to give the methanol complex VI, L4FeIII(OHCH3)(mu-O)(FeL4)-L-III, which has an Fe-III-Fe-III core. The barrier of the rate-determining step, methane H-abstraction, was calculated to be 19 kcal/mol. The overall CH4 oxidation reaction to form the methanol complex, I + CH4 --> VI, was found to be exothermic by 39 kcal/mol.