which corresponds to the chemical hardness of the catalyst for given electronegativities in order to achieve maximum or minimum reaction energy differences. Accordingly, ideal catalysts should match χ C ≈ χ A ≈ χ B or Eqns. ( ) and ( ) as closely as possible and the resulting product yields should get worse whenever the values converge to Eqn. (11) and Eqn. ( ) for given reactant and product chemical hardnesses and electronegativities.