||DFT studies on the catalytic mechanism involved in the catalytic oxidation of C-H bonds of hydrocarbons by osmium catalyst and the formation of the first Schrock carbene complex are reported in this thesis. This study can on one way shed a light on the further optimization of the catalyst, and on the other hand, provide decisive evidence for a long disputed pathway of the formation of Schrock carbene (CH2But)3Ta=CHBut. The traditionally proposed mechanisms of oxidation of methane and aromatic C-H bonds are tested respectively by density functional theory at the level of B3LYP. The effect of proximal basal ligand in the osmium catalyst has also been studied. The proximal basal ligand would facilitate proton-shuttle mechanism of the oxidation of aromatic C-H bonds. Most importantly, one recently proposed method for the detection of NIH shift has found to be problematic. That is to say, the H-D shift may well take place through other pathways other than NIH shift. One possible pathway of this kind is developed in this work. The mechanism for the formation of the first reported Schrock carbene has also been explored with density functional calculations at the B3LYP level. It has been found that the reaction go through a Ta(CH2But)5 intermediate, and then an α-elimination takes place to give the Schrock carbene. Further, the experimentally observed high KIE (kinetic isotope effect) value cannot be reproduced by DFT calculation when no tunneling effect is considered.