As an AI language model, I cannot perform quantum chemical calculations directly. However, I can provide you with a general overview of the reaction mechanism and rate-determining step for the formation of a C-C bond between two ethene molecules catalyzed by a nickel-based catalyst.The reaction mechanism can be described as follows:1. Coordination of the ethene molecules to the nickel catalyst: The nickel catalyst, typically in the form of a nickel 0 complex, coordinates to the -bond of one of the ethene molecules. This process weakens the C=C bond in the ethene molecule and generates a nickel-alkene complex.2. Oxidative addition: The second ethene molecule coordinates to the nickel center, leading to the oxidative addition of the C=C bond. This step results in the formation of a nickel II complex with two alkyl ligands.3. Migratory insertion: The alkyl ligands on the nickel center undergo a migratory insertion, forming a new C-C bond between the two alkyl groups. This step generates a four-membered metallacycle.4. Reductive elimination: The metallacycle undergoes reductive elimination, releasing the product 1-butene and regenerating the nickel 0 catalyst.The rate-determining step in this reaction is typically the migratory insertion step, where the new C-C bond is formed. This step involves the highest energy transition state and is therefore the slowest step in the reaction.To determine the activation energy, transition state geometry, and reaction energetics, you would need to perform quantum chemical calculations using computational chemistry software such as Gaussian, ORCA, or NWChem. These programs allow you to model the reaction pathway and calculate the energy barriers and geometries of the transition states involved in the reaction.