As a language model AI, I cannot perform density functional theory DFT calculations directly. However, I can provide you with some guidance on how to approach this problem.To calculate the activation energy of the reaction mechanism for the catalytic conversion of methane to methanol using DFT, you would need to follow these general steps:1. Choose an appropriate DFT method and basis set: Select a suitable functional and basis set for your calculations. Popular functionals include B3LYP, PBE, and M06, while common basis sets are 6-31G d,p , 6-311G d,p , and cc-pVTZ. The choice depends on the balance between computational cost and accuracy needed for your specific problem.2. Model the catalytic system: Construct a model of the catalyst and the reactants methane and oxygen in their initial state. This may involve creating a cluster model or using a periodic boundary condition approach for heterogeneous catalysts, or modeling the active site of an enzyme or homogeneous catalyst.3. Optimize the geometry: Perform a geometry optimization of the reactants, transition state, and products to find the minimum energy structures. This may involve using a constrained optimization or a transition state search algorithm, such as the nudged elastic band NEB method or the synchronous transit-guided quasi-Newton STQN method.4. Calculate the energy: Compute the electronic energy of the optimized structures, including any necessary corrections for basis set superposition error BSSE or dispersion interactions.5. Determine the activation energy: Calculate the activation energy Ea as the difference between the energy of the transition state and the energy of the reactants. This can be expressed as Ea = E TS - E Reactants .6. Analyze the results: Compare your calculated activation energy with experimental values or other theoretical studies to assess the accuracy of your model and method.Keep in mind that DFT calculations can be computationally demanding, and you may need access to high-performance computing resources to perform these calculations. Additionally, you may need to use specialized software packages, such as Gaussian, ORCA, or VASP, to carry out DFT calculations.