How to Conceptualize Catalytic Cycles? The Energetic Span Model

A computational study of a catalytic cycle generates state energies (the E-representation), whereas experiments lead to rate constants (the k-representation). Based on transition state theory (TST), these are equivalent representations. Nevertheless, until recently, there has been no simple way to calculate the efficiency of a catalytic cycle, that is, its turnover frequency (TOF), from a theoretically obtained energy profile. In this Account, we introduce the energetic span model that enables one to evaluate TOFs in a straightforward manner and in affinity with the Curtin-Hammett principle. As shown herein, the model implies a change in our kinetic concepts. Analogous to Ohm's law, the catalytic chemical current (the TOF) can be defined by a chemical potential (independent of the mechanism) divided by a chemical resistance (dependent on the mechanism and the nature of the catalyst). This formulation is based on Eyring's TST and corresponds to a steady-state regime. In many catalytic cycles, only one transition state and one intermediate determine the TOF. We call them the TOF-determining transition state (TDTS) and the TOF-determining intermediate (TDI). These key states can be located, from among the many states available to a catalytic cycle, by assessing the degree of TOF control (X(TOF)); this last term resembles the structure-reactivity coefficient in classical physical organic chemistry. The TDTS-TDI energy difference and the reaction driving force define the energetic span (δE) of the cycle. Whenever the TDTS appears after the TDI, δE is the energy difference between these two states; when the opposite is true, we must also add the driving force to this difference. Having δE, the TOF is expressed simply in the Arrhenius-Eyring fashion, wherein δE serves as the apparent activation energy of the cycle. An important lesson from this model is that neither one transition state nor one reaction step possess all the kinetic information that determines the efficiency of a catalyst. Additionally, the TDI and TDTS are not necessarily the highest and lowest states, nor do they have to be adjoined as a single step. As such, we can conclude that a change in the conceptualization of catalytic cycles is in order: in catalysis, there are no rate-determining steps, but rather rate-determining states. We also include a study on the effect of reactant and product concentrations. In the energetic span approximation, only the reactants or products that are located between the TDI and TDTS accelerate or inhibit the reaction. In this manner, the energetic span model creates a direct link between experimental quantities and theoretical results. The versatility of the energetic span model is demonstrated with several catalytic cycles of organometallic reactions.