As a language model AI, I cannot perform quantum chemical calculations directly. However, I can provide you with a general understanding of how the reaction mechanism for the decomposition of hydrogen peroxide H2O2 differs in the presence and absence of a catalyst, and how it affects the activation energy and reaction rate.In the absence of a catalyst, the decomposition of hydrogen peroxide is a relatively slow process. The reaction mechanism involves the homolytic cleavage of the O-O bond, which results in the formation of two hydroxyl radicals OH . These radicals can then react with other hydrogen peroxide molecules to form water H2O and oxygen O2 . The activation energy for this uncatalyzed reaction is relatively high, which leads to a slow reaction rate.H2O2 2 OHOH + H2O2 H2O + HO2HO2 + HO2 H2O2 + O2In the presence of a catalyst, the reaction mechanism for the decomposition of hydrogen peroxide changes, and the activation energy is significantly reduced. This results in a faster reaction rate. Common catalysts for this reaction include manganese dioxide MnO2 , potassium iodide KI , and catalase an enzyme found in living organisms .For example, in the presence of manganese dioxide MnO2 , the reaction mechanism involves the formation of an intermediate complex between the catalyst and hydrogen peroxide. This complex lowers the activation energy required for the O-O bond cleavage, making it easier for the reaction to proceed. The catalyst is not consumed in the reaction and can be reused for multiple decomposition events.MnO2 + H2O2 MnO2H2O2 intermediate complex MnO2H2O2 MnO2 + H2O + O2Quantum chemical calculations, such as density functional theory DFT or ab initio methods, can be used to determine the activation energy and reaction rate for both the uncatalyzed and catalyzed decomposition of hydrogen peroxide. By comparing these values, it can be demonstrated that the presence of a catalyst significantly lowers the activation energy and increases the reaction rate for the decomposition of hydrogen peroxide.