Computational Aspects of Single-Molecule Kinetics for Coupled Catalytic Cycles: A Spectral Analysis

Abstract

Catalysis from single active sites is analyzed using methods developed from single-molecule kinetics. Using a stochastic Markov-state description, the observable properties of general catalytic networks of reactions are expressed using an eigenvalue decomposition of the transition matrix for the Markov process. By the use of a sensitivity analysis, the necessary eigenvalues and eigenvectors are related to the energies of controlling barriers and wells located along the reaction routes. A generalization of the energetic span theory allows the eigenvalues to be computed from several activation energies corresponding to distinct barrier-well pairings. The formalism is demonstrated for model problems and for a physically realistic mechanism for an alkene hydrogenation reaction on a single-atom catalyst. The spectral analysis permits a hierarchy of timescales to be identified from the single-molecule signal, which correspond to specific relaxation modes in the network.

Active Site Engineering via Optimizing the Heterogeneous Support Structure for Single-Atom Catalysis

“…N is the number of distinct states of the SAC, and N + 1 index occurs since both the reactant and product states of the AS are included in K . Following the methods introduced in ref , the population of each molecular species involving the vanadium atom is described by a state i with a probability P i . The normalization condition ∑ P i = 1 enforces the conservation of the catalyst.…”

“…This analysis employs the N × N state-to-state transition matrix T ( x ), which is identical to − K ( x ) except that the product-forming reactions are omitted. The exact initial TOF, derived in ref , gives a TOF ν ( x ) = bold1 bold1 T · false( prefix− T ( x ) false) − 1 · P 0 …”

“…The central issue is how to identify the (HTS,LINT) pair that controls the kinetics at each value of x . We have generalized the EST approach to more complex catalytic networks that involve multiple catalytic paths using an eigenvalue sensitivity analysis ,, that is similar to Campbell’s degree of reaction control . The eigenvalues of the transition matrix are found to control the TOF and other kinetic properties of the network.…”

“…38 Based on these findings, a microkinetic model has been proposed that accounts for the hydrogenation process, which involves three interconverting active sites and three coupled catalytic cycles. [25][26][27]39 The proposed microkinetic model, represented in Figure 2C, presumes the dominance of a heterolytic cleavage mechanism involving a tripodal vanadium active site. Three chemically distinct chemical pathways are possible that differ in the participation of the THF species and the ordering of the H 2 or ST addition; these are labeled as paths 5, 6, and 7 in Figure 2C.…”

“…This pathswitching behavior is familiar from the study of product selectivity where two competing reaction paths are controlled by the relative energy of two transition states that lead to different products through a common intermediate. 27,56,57 Optimization of a catalyst for selectivity at an atomic level involves the tuning of relative activation energies controlling different paths in the design space. 58 It is clear that the TOF and selectivity maps exhibit multiple stationary points, and thus, they are not simple convex functions of the design variables.…”