Recent advances in the theoretical treatment of proton-coupled electron transfer (PCET) reactions are reviewed. These reactions play an important role in a wide range of biological processes, as well as in fuel cells, solar cells, chemical sensors, and electrochemical devices. A unified theoretical framework has been developed to describe both sequential and concerted PCET, as well as hydrogen atom transfer (HAT). A quantitative diagnostic has been proposed to differentiate between HAT and PCET in terms of the degree of electronic nonadiabaticity, where HAT corresponds to electronically adiabatic proton transfer and PCET corresponds to electronically nonadiabatic proton transfer. In both cases, the overall reaction is typically vibronically nonadiabatic. A series of rate constant expressions have been derived in various limits by describing the PCET reactions in terms of nonadiabatic transitions between electron-proton vibronic states. These expressions account for the solvent response to both electron and proton transfer and the effects of the proton donor-acceptor vibrational motion. The solvent and protein environment can be represented by a dielectric continuum or described with explicit molecular dynamics. These theoretical treatments have been applied to numerous PCET reactions in solution and proteins. Expressions for heterogeneous rate constants and current densities for electrochemical PCET have also been derived and applied to model systems.

Building on the previously developed multistate empirical valence bond model [U. W. Schmitt and G. A. Voth, J. Chem. Phys 111, 9361 (1999)] for the dynamics and energetics of an excess proton in bulk phase water, a second generation model is described. This model is shown to produce similar dynamic and structural properties to the previous model, while allowing for the use of the full hydronium charge. This characteristic of the model is required for its implementation in a host of realistic applications beyond bulk water. An improved state selection algorithm is also presented, resulting in a significantly reduced energy drift during microcanonical molecular dynamics simulations. The unusually high self diffusion constant of an excess proton in water due to the proton hopping (Grotthuss) process is observed in the simulation data and is found to be quantitatively in the same range as the experimental value if a quantum correction is taken into consideration. Importantly, a more complete analysis of proton transport process is also presented.

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 ,...

The proton-coupled electron transfer reaction catalyzed by soybean lipoxygenase-1 is studied with a multistate continuum theory that represents the transferring hydrogen nucleus as a quantum mechanical wave function. The inner-sphere reorganization energy of the iron cofactor is calculated with density functional theory, and the outer-sphere reorganization energy of the protein is calculated with the frequency-resolved cavity model for conformations obtained with docking simulations. Both classical and quantum mechanical treatments of the proton donor-acceptor vibrational motion are presented. The temperature dependence of the calculated rates and kinetic isotope effects is in agreement with the experimental data. The weak temperature dependence of the rates is due to the relatively small free energy barrier arising from a balance between the reorganization energy and the reaction free energy. The unusually high deuterium kinetic isotope effect of 81 is due to the small overlap of the reactant and product proton vibrational wave functions and the dominance of the lowest energy reactant and product vibronic states in the tunneling process. The temperature dependence of the kinetic isotope effect is strongly influenced by the proton donor-acceptor distance with the dominant contribution to the overall rate. This dominant proton donor-acceptor distance is significantly smaller than the equilibrium donor-acceptor distance and is determined by a balance between the larger coupling and the smaller Boltzmann probability as the distance decreases. Thus, the proton donor-acceptor vibrational motion plays a vital role in decreasing the dominant donor-acceptor distance relative to its equilibrium value to facilitate the proton-coupled electron transfer reaction.

The dynamical behavior and the temperature dependence of the kinetic isotope effects (KIEs) are examined for the proton-coupled electron transfer reaction catalyzed by the enzyme soybean lipoxygenase. The calculations are based on a vibronically nonadiabatic formulation that includes the quantum mechanical effects of the active electrons and the transferring proton, as well as the motions of all atoms in the complete solvated enzyme system. The rate constant is represented by the time integral of a probability flux correlation function that depends on the vibronic coupling and on time correlation functions of the energy gap and the proton donor-acceptor mode, which can be calculated from classical molecular dynamics simulations of the entire system. The dynamical behavior of the probability flux correlation function is dominated by the equilibrium protein and solvent motions and is not significantly influenced by the proton donor-acceptor motion. The magnitude of the overall rate is strongly influenced by the proton donor-acceptor frequency, the vibronic coupling, and the protein/solvent reorganization energy. The calculations reproduce the experimentally observed magnitude and temperature dependence of the KIE for the soybean lipoxygenase reaction without fitting any parameters directly to the experimental kinetic data. The temperature dependence of the KIE is determined predominantly by the proton donor-acceptor frequency and the distance dependence of the vibronic couplings for hydrogen and deuterium. The ratio of the overlaps of the hydrogen and deuterium vibrational wavefunctions strongly impacts the magnitude of the KIE but does not significantly influence its temperature dependence. For this enzyme reaction, the large magnitude of the KIE arises mainly from the dominance of tunneling between the ground vibronic states and the relatively large ratio of the overlaps between the corresponding hydrogen and deuterium vibrational wavefunctions. The weak temperature dependence of the KIE is due in part to the dominance of the local component of the proton donor-acceptor motion.

The vibronic couplings for the phenoxyl/phenol and the benzyl/toluene self-exchange reactions are calculated with a semiclassical approach, in which all electrons and the transferring hydrogen nucleus are treated quantum mechanically. In this formulation, the vibronic coupling is the Hamiltonian matrix element between the reactant and product mixed electronic-proton vibrational wavefunctions. The magnitude of the vibronic coupling and its dependence on the proton donor-acceptor distance can significantly impact the rates and kinetic isotope effects, as well as the temperature dependences, of proton-coupled electron transfer reactions. Both of these self-exchange reactions are vibronically nonadiabatic with respect to a solvent environment at room temperature, but the proton tunneling is electronically nonadiabatic for the phenoxyl/phenol reaction and electronically adiabatic for the benzyl/toluene reaction. For the phenoxyl/phenol system, the electrons are unable to rearrange fast enough to follow the proton motion on the electronically adiabatic ground state, and the excited electronic state is involved in the reaction. For the benzyl/toluene system, the electrons can respond virtually instantaneously to the proton motion, and the proton moves on the electronically adiabatic ground state. For both systems, the vibronic coupling decreases exponentially with the proton donor-acceptor distance for the range of distances studied. When the transferring hydrogen is replaced with deuterium, the magnitude of the vibronic coupling decreases and the exponential decay with distance becomes faster. Previous studies designated the phenoxyl/phenol reaction as proton-coupled electron transfer and the benzyl/toluene reaction as hydrogen atom transfer. In addition to providing insights into the fundamental physical differences between these two types of reactions, the present analysis provides a new diagnostic for differentiating between the conventionally defined hydrogen atom transfer and proton-coupled electron transfer reactions.

This paper presents a general theoretical formulation for proton-coupled electron transfer (PCET) reactions. The solute is represented by a multistate valence bond model, and the active electrons and transferring proton(s) are treated quantum mechanically. This formulation enables the classical or quantum mechanical treatment of the proton donor-acceptor vibrational mode, as well as the dynamical treatment of the proton donor-acceptor mode and the solvent. Nonadiabatic rate expressions are presented for PCET reactions in a number of well-defined limits for both dielectric continuum and molecular representations of the environment. The dynamical rate expressions account for correlations between the fluctuations of the proton donor-acceptor distance and the nonadiabatic PCET coupling. The quantities in the rate expressions can be calculated with a dielectric continuum model or a molecular dynamics simulation of the full system. The significance of the quantum and dynamical effects of the proton donor-acceptor mode is illustrated with applications to model PCET systems.

In this article we present a multistate continuum theory for multiple charge transfer reactions such as proton-coupled electron transfer and multiple proton transfer reactions. The solute is described with a multistate valence bond model, the solvent is represented as a dielectric continuum, and the transferring protons are treated quantum mechanically. This theory provides adiabatic free energy surfaces that depend on a set of scalar solvent variables corresponding to the individual charge transfer reactions. Thus this theory is a multidimensional analog of standard Marcus theory for single charge transfer reactions. For processes involving significant inner-sphere ͑i.e., solute͒ reorganization, the effects of solute intramolecular vibrations can be incorporated into the adiabatic free energy surfaces. The input quantities required for this theory are gas phase valence bond matrix elements fit to standard quantum chemistry calculations and solvent reorganization energy matrix elements calculated with standard continuum electrostatic methods. The goal of this theory is to provide insight into the underlying fundamental physical principles dictating the mechanisms and rates of multiple charge transfer reactions.