Ab initio methods have been employed to study the oxidation of the CH bonds in homo- (1a-d) and heterosubstituted alkanes (2a-d, 3a,b) by the parent dioxirane as a model for the dioxirane oxidation of proteins. The study involved methane (1a), ethane (1b), propane (1c), isobutane (1d), methylamine (2a), methanol (2b), ethanol (2c), acetaldehyde (2d), glycolaldehyde (3a), and a peptide model, N-formylglycine amide (3b). Geometries were optimized at DFT (B3LYP) and MP2 levels of theory using 6-31G* and 6-311+G** basis sets. Stationary points were characterized by vibrational frequency analysis. Final energies for the oxidation of la were obtained at the MP4-(SDTQ)/6-311+G** and QCISD(T)/6-311+G** levels. A new mechanism of the oxidation reconciling the apparently contradictory experimental data was found. The reaction proceeds via a highly polar asynchronous transition state, which is common for either concerted oxygen insertion into the CH bond and formation of a radical pair (alkyl radical + alpha-hydroxyalkoxyl radical). These channels appear at a bifurcation point on the potential energy surface after the common transition state, which corresponds to formation of the new O-H bond, and ruptures of the C-H bond in the substrate and of the O-O bond in the dioxirane. The B3LYP theoretical model gives substantial hydride transfer character to the transition state and describes adequately the selectivity of the oxidation of hydrocarbons, alcohols, 1,2-diols, and leucine derivatives. The agreement with experimental data is further improved by taking into account the influence of a dielectric medium (IPCM model). The electrophilicity of dioxiranes in the oxidations of CH bonds implies that side groups of protected amino acids and proteins are more probable points for the attack of these oxidants than the weak alpha-CH bonds. It is of interest that selectivities of dioxirane oxidation are incorrectly predicted by the MP2 method, which overestimates the proton-transfer character of the transition state.