Derivation of rate expressions for nonadiabatic proton-coupled electron transfer reactions in solution

Abstract

Articles you may be interested inAn analysis of model proton-coupled electron transfer reactions via the mixed quantum-classical Liouville approach Quantum and dynamical effects of proton donor-acceptor vibrational motion in nonadiabatic proton-coupled electron transfer reactions This paper presents a derivation of rate expressions for nonadiabatic proton-coupled electron transfer ͑PCET͒ reactions in solution. The derivation is based on a multistate continuum theory in which the solvent is described by a dielectric continuum, the solute is represented by a multistate valence bond model, and the transferring proton͑s͒ are treated quantum mechanically. In this formulation, a PCET reaction is described as a transition between two sets of diabatic free energy surfaces associated with the two electron transfer states. For PCET reactions involving the transfer of one electron and one proton, these mixed electronic/proton vibrational free energy surfaces are functions of two scalar solvent coordinates corresponding to electron and proton transfer. The Golden Rule is applied to these two-dimensional free energy surfaces in conjunction with a series of well-defined approximations. The contributions from intramolecular solute modes are also included. The final rate expression is similar in form to the standard rate expression for nonadiabatic single electron transfer, but the reorganization energies, equilibrium free energy differences, and couplings are defined in terms of the two-dimensional free energy surfaces. The practical implementation of this rate expression and the calculation of the input quantities are also discussed.where the second term is defined in Eq. ͑14͒. The free energy surfaces U J (x,y) are parabolic along the y coordinate but are not exactly parabolic along the x coordinate due to the complicated x-dependence of E I (q,Q,x) and E II (q,Q,x) and the averaging over the q coordinate for the different vibrational states. For example, in general these surfaces could be double wells along the x coordinate, as found for single PT reactions. For typical PCET reactions, however, these surfaces have been found to be approximately parabolic along the x coordinate at the energies of interest. 13 This approximately parabolic form results from the small reorganization energy for PT relative to the difference in energies for VB states 1a and 1b and for VB states 2a and 2b. Even for symmetric PT interfaces, the transferring electron introduces significant asymmetry between PT states a and b. Moreover, the reorganization energy for PT is decreased by the presence of the relatively large electron donor and acceptor.The derivation in this paper is applicable to systems for which the free energy surfaces U J (x,y) are approximately harmonic in x and y. In this case, these surfaces can be expressed with a Taylor series expansion aswhere Ū J ϭU J (x J ,ȳ J ) corresponds to the minimum of the paraboloid. Neglecting the dependence of the self-energy on q and the nonlinear x-dependence of E I and E II , the quantities ⌳ xx ,...

Theoretical studies of proton-coupled electron transfer: Models and concepts relevant to bioenergetics

“…In this limit, the expression in Eq. (3) becomes equivalent to the non-dynamical rate expressions derived previously with a multistate continuum theory for fixed R. [25] Further approximation that , in addition to α μν = 0, leads to the Marcus theory rate expression for nonadiabatic electron transfer modified by the inclusion of Franck-Condon overlap terms for the transferring hydrogen. [21,22] An alternative derivation of Equation (3) provides additional connections to previous studies.…”

“…Rate expressions based on linear response theory and the Golden Rule have been derived in various well-defined limits. [25,27] Methods for calculating the vibronic couplings in PCET systems have also been developed and implemented. [28,31] These theoretical approaches enable the calculation of rates and deuterium kinetic isotope effects (i.e., the ratio of the rate for hydrogen to the rate for deuterium), as well as their temperature and pH dependences.…”

“…[19,20] A variety of theoretical approaches have been developed to study PCET reactions. [21][22][23][24][25][26][27][28][29][30] In the vibronically nonadiabatic formulation for PCET, [25][26][27] the active electrons and transferring proton are treated quantum mechanically, and the PCET reaction is described in terms of nonadiabatic transitions between pairs of reactant and product mixed electron-proton vibronic states. The dynamics of the proton donor-acceptor mode and the solvent/protein environment can be included in this formulation.…”

Colossal kinetic isotope effects in proton-coupled electron transfer

“…Coupled electron-proton transfer has been treated theoretically by a number of authors (21)(22)(23)(24)(25)(26)(27)(28). The reactions studied here have been examined theoretically by use of a multistate continuum theory that treats the transferring H-atom quantum mechanically (29), developed by Hammes-Schiffer and coworkers (25)(26)(27)(28).…”

“…The reactions studied here have been examined theoretically by use of a multistate continuum theory that treats the transferring H-atom quantum mechanically (29), developed by Hammes-Schiffer and coworkers (25)(26)(27)(28). Important elements in this theory are the extent of electronic coupling between the electron transfer donor and acceptor, coupling with the solvent, and the extent of vibrational overlap for the transferring proton between the largely N-H , S-H , and P-H vibrational levels in the initial state to O-H levels in the final state; note the reaction in Eq.…”