A high-level ab initio study on the oxidations of alkanes (methane, propane and isobutane) with dioxirane (DO), dimethyldioxirane (DMDO), difluorodioxirane (DFDO) and methyl(trifluoromethyl)dioxirane (TFDO) has provided a rationale for the formation of products derived from radical intermediates when dioxygen is rigorously excluded and lends strong support to the generally accepted, highly exothermic, concerted oxygen insertion mechanism for the oxidation of alkanes with dioxiranes under typical preparative conditions. At the B3LYP/6-311 +G(3df,2p)//B3LYP/6-311+G(3df,2p) level, the barriers for the oxidation of methane with DO, DMDO, DFDO, and TFDO are 36.5, 41.2, 21.2, and 35.0 kcal/mol. The activation barriers for the oxidation of methane, propane, and isobutane with DMDO are 44.2, 30.3, and 22.4 kcal/mol, respectively, at the CCSD(T)/6-3 1G*//B3LYP/6-31G* level. These barriers are higher than the barriers for epoxidations of ethylene, propene, and isobutene (17.9, 15.9, and 15.4 kcal/mol, respectively, at the B3LYP/6-31G* level). Calculations at the G2 level have shown that C-H bond homolysis is not thermodynamically favorable in hydrogen abstraction reactions of methane, propane, and isobutane with either DMDO, its parent DO, or DFDO. If the sources of the initiating radical species are hydrogen atoms produced by breaking X-H bonds, the energy of the X-H bond should be less than 83.5 kcal/mol in its reaction with DMDO (89.5 and 87.8 kcal/mol for DO and DFDO) for the reaction to be exothermic.