As a language model AI, I cannot perform Density Functional Theory DFT calculations directly. However, I can provide you with a general approach to solve this problem.The Fischer-Tropsch synthesis FTS is a catalytic process that converts synthesis gas a mixture of CO and H2 into hydrocarbons, including methane. To determine the most favorable reaction pathway for methane formation on a Cu surface using DFT calculations, you should follow these steps:1. Identify possible reaction intermediates and pathways: In FTS, the main reaction intermediates include CO adsorption, hydrogenation, and C-H bond formation. You should consider all possible intermediates and elementary steps that lead to methane formation on the Cu surface.2. Build a model of the Cu surface: To perform DFT calculations, you need to create a model of the Cu surface. This can be done using periodic boundary conditions and a suitable unit cell. You may also need to consider different surface facets and possible defects, as they can influence the catalytic activity.3. Perform DFT calculations for each intermediate and transition state: For each reaction intermediate and transition state, you need to optimize their geometries and calculate their energies using DFT. You can use various DFT functionals and basis sets, but it's essential to choose a suitable level of theory that provides accurate results and is computationally feasible.4. Calculate activation barriers and reaction energies: Using the optimized geometries and energies of the intermediates and transition states, you can calculate the activation barriers and reaction energies for each elementary step in the reaction pathway.5. Determine the most favorable pathway: By comparing the activation barriers and reaction energies of the different pathways, you can identify the most favorable reaction pathway for methane formation on the Cu surface.6. Validate your results: To ensure the reliability of your results, you may want to compare them with experimental data or previous theoretical studies. Additionally, you can perform sensitivity analysis by testing different DFT functionals, basis sets, or surface models to check the robustness of your conclusions.By following these steps, you should be able to determine the most favorable reaction pathway for the Fischer-Tropsch synthesis of methane on a Cu surface using Density Functional Theory calculations.