Multistate continuum theory for multiple charge transfer reactions in solution

Theory for proton-coupled energy transfer

In the recently discovered proton-coupled energy transfer (PCEnT) mechanism, the transfer of electronic excitation energy between donor and acceptor chromophores is coupled to a proton transfer reaction. Herein, we develop a general theory for PCEnT and derive an analytical expression for the nonadiabatic PCEnT rate constant. This theory treats the transferring hydrogen nucleus quantum mechanically and describes the PCEnT process in terms of nonadiabatic transitions between reactant and product electron-proton vibronic states. The rate constant is expressed as a summation over these vibronic states, and the contribution of each pair of vibronic states depends on the square of the vibronic coupling as well as the spectral convolution integral, which can be viewed as a generalization of the Förster-type spectral overlap integral for vibronic rather than electronic states. The convolution integral also accounts for the common vibrational modes shared by the donor and acceptor chromophores for intramolecular PCEnT. We apply this theory to model systems to investigate the key features of PCEnT processes. The excited vibronic states can contribute significantly to the total PCEnT rate constant, and the common modes can either slow down or speed up the process. Because the pairs of vibronic states that contribute the most to the PCEnT rate constant may correspond to spectroscopically dark states, PCEnT could occur even when there is no apparent overlap between the donor emission and acceptor absorption spectra. This theory will assist in the interpretation of experimental data and will guide the design of additional PCEnT systems.

Proton-Coupled Electron Transfer upon Oxidation of Tyrosine in a De Novo Protein: Analysis of Proton Acceptor Candidates

Redox-active residues, such as tyrosine and tryptophan, play important roles in a wide range of biological processes. The α3Y de novo protein, which is composed of three α helices and a tyrosine residue Y32, provides a platform for investigating the redox properties of tyrosine in a well-defined protein environment. Herein, the proton-coupled electron transfer (PCET) reaction that occurs upon oxidation of tyrosine in this model protein by a ruthenium photosensitizer is studied by using a vibronically nonadiabatic PCET theory that includes hydrogen tunneling and excited vibronic states. The input quantities to the analytical nonadiabatic rate constant expression, such as the diabatic proton potential energy curves and associated proton vibrational wave functions, reorganization energy, and proton donor-acceptor distribution functions, are obtained from density functional theory calculations on model systems and molecular dynamics simulations of the solvated α3Y protein. Two possible proton acceptors, namely, water or a glutamate residue in the protein scaffold, are explored. The PCET rate constant is greater when glutamate is the proton acceptor, mainly due to the more favorable driving force and shorter equilibrium proton donor-acceptor distance, although contributions from excited vibronic states mitigate these effects. Nevertheless, water could be the dominant proton acceptor if its equilibrium constant associated with hydrogen bond formation is significantly greater than that for glutamate. Although these calculations do not definitively identify the proton acceptor for this PCET reaction, they elucidate the conditions under which each proton acceptor can be favored. These insights have implications for tyrosine-based PCET in a wide variety of biochemical processes.

Probing Nonadiabaticity of Proton-Coupled Electron Transfer in Ribonucleotide Reductase

The enzyme ribonucleotide reductase, which is essential for DNA synthesis, initiates the conversion of ribonucleotides to deoxyribonucleotides via radical transfer over a 32 Å pathway composed of proton-coupled electron transfer (PCET) reactions. Previously, the first three PCET reactions in the α subunit were investigated with hybrid quantum mechanical/molecular mechanical (QM/MM) free energy simulations. Herein, the fourth PCET reaction in this subunit between C439 and guanosine diphosphate (GDP) is simulated and found to be slightly exoergic with a relatively high free energy barrier. To further elucidate the mechanisms of all four PCET reactions, we analyzed the vibronic and electron-proton nonadiabaticities. This analysis suggests that interfacial PCET between Y356 and Y731 is vibronically and electronically nonadiabatic, whereas PCET between Y731 and Y730 and between C439 and GDP is fully adiabatic and PCET between Y730 and C439 is in the intermediate regime. These insights provide guidance for selecting suitable rate constant expressions for these PCET reactions.

Reconciling Imbalanced and Nonadiabatic Reactivity in Transition Metal-Oxo-Mediated Concerted Proton Electron Transfer (CPET)

Recently, there have been several experimental demonstrations of how the rates of concerted proton electron transfer (CPET) are affected by stepwise thermodynamic parameters of only proton (ΔG°PT) or electron (ΔG°ET) transfer. Semiclassical structure-activity relationships have been invoked to rationalize these linear free energy relationships, but it is not clear how they would manifest in a nonadiabatic reaction. Using density functional theory calculations, we demonstrate how a decrease in ΔG°PT can lead to transition state imbalance in a nonadiabatic framework. We then use these calculations to anchor a theoretical model that reproduces experimental trends with ΔG°PT and ΔG°ET. Our results reconcile predictions from semiclassical transition state theory with models that treat proton transfer quantum mechanically in CPET reactivity, make new predictions about the importance of basicity for uphill CPET reactions, and suggest similar treatments may be possible for other nonadiabatic reactions.

General Kinetic Model for pH Dependence of Proton-Coupled Electron Transfer: Application to an Electrochemical Water Oxidation System

The pH dependence of proton-coupled electron transfer (PCET) reactions, which are critical to many chemical and biological processes, is a powerful probe for elucidating their fundamental mechanisms. Herein, a general, multichannel kinetic model is introduced to describe the pH dependence of both homogeneous and electrochemical PCET reactions. According to this model, a weak pH dependence can arise from the competition among multiple sequential and concerted PCET channels involving different forms of the redox species, such as protonated and deprotonated forms, as well as different proton donors and acceptors. The contribution of each channel is influenced by the relative populations of the reactant species, which often depend strongly on pH, leading to complex pH dependence of PCET apparent rate constants. This model is used to explain the origins of the experimentally observed weak pH dependence of the electrochemical PCET apparent rate constant for a ruthenium-based water oxidation catalyst attached to a tin-doped In2O3 (ITO) surface. The weak pH dependence is found to arise from the intrinsic differences in the rate constants of participating channels and the dependence of their relative contributions on pH. This model predicts that the apparent maximum rate constant will become pH-independent at higher pH, which is confirmed by experimental measurements. Our analysis also suggests that the dominant channels are electron transfer at lower pH and sequential PCET via electron transfer followed by fast proton transfer at higher pH. This work highlights the importance of considering multiple competing channels simultaneously for PCET processes.

Exploring Proton-Coupled Electron Transfer at Multiple Scales

The coupling of electron and proton transfer is critical for chemical and biological processes spanning a wide range of length and time scales and often occurring in complex environments. Thus, diverse modeling strategies, including analytical theories, quantum chemistry, molecular dynamics, and kinetic modeling, are essential for a comprehensive understanding of such proton-coupled electron transfer reactions. Each of these computational methods provides one piece of the puzzle, and all these pieces must be viewed together to produce the full picture.

Understanding the Key Roles of pH Buffer in Accelerating Lignin Degradation by Lignin Peroxidase

pH buffer plays versatile roles in both biology and chemistry. In this study, we unravel the critical role of pH buffer in accelerating degradation of the lignin substrate in lignin peroxidase (LiP) using QM/MM MD simulations and the nonadiabatic electron transfer (ET) and proton-coupled electron transfer (PCET) theories. As a key enzyme involved in lignin degradation, LiP accomplishes the oxidation of lignin via two consecutive ET reactions and the subsequent C-C cleavage of the lignin cation radical. The first one involves ET from Trp171 to the active species of Compound I, while the second one involves ET from the lignin substrate to the Trp171 radical. Differing from the common view that pH = 3 may enhance the oxidizing power of Cpd I via protonation of the protein environment, our study shows that the intrinsic electric fields have minor effects on the first ET step. Instead, our study shows that the pH buffer of tartaric acid plays key roles during the second ET step. Our study shows that the pH buffer of tartaric acid can form a strong H-bond with Glu250, which can prevent the proton transfer from the Trp171-H^•+ cation radical to Glu250, thereby stabilizing the Trp171-H^•+ cation radical for the lignin oxidation. In addition, the pH buffer of tartaric acid can enhance the oxidizing power of the Trp171-H^•+ cation radical via both the protonation of the proximal Asp264 and the second-sphere H-bond with Glu250. Such synergistic effects of pH buffer facilitate the thermodynamics of the second ET step and reduce the overall barrier of lignin degradation by ∼4.3 kcal/mol, which corresponds to a rate acceleration of 10³-fold that agrees with experiments. These findings not only expand our understanding on pH-dependent redox reactions in both biology and chemistry but also provide valuable insights into tryptophan-mediated biological ET reactions.

The nature of proton-coupled electron transfer in a blue light using flavin domain

Proton-coupled electron transfer (PCET) is key to the activation of the blue light using flavin (BLUF) domain photoreceptors. Here, to elucidate the photocycle of the central FMN-Gln-Tyr motif in the BLUF domain of OaPAC, we eliminated the intrinsic interfering W90 in the mutant design. We integrated the stretched exponential function into the target analysis to account for the dynamic heterogeneity arising from the active-site solvation relaxation and the flexible H-bonding network as shown in the molecular dynamics simulation results, facilitating a simplified expression of the kinetics model. We find that, in both the functional wild-type (WT) and the nonfunctional Q48E and Q48A, forward PCET happens in the range of 105 ps to 344 ps, with a kinetic isotope effect (KIE) measured to be ∼1.8 to 2.4, suggesting that the nature of the forward PCET is concerted. Remarkably, only WT proceeds with an ultrafast reverse PCET process (31 ps, KIE = 4.0), characterized by an inverted kinetics of the intermediate FMNH˙. Our results reveal that the reverse PCET is driven by proton transfer via an intervening imidic Gln.

Theoretical perspectives on non-Born-Oppenheimer effects in chemistry

The Born-Oppenheimer approximation, which assumes that the electrons respond instantaneously to the motion of the nuclei, breaks down for a wide range of chemical and biological processes. The rate constants of such nonadiabatic processes can be calculated using analytical theories, and the real-time nonequilibrium dynamics can be described using numerical atomistic simulations. The selection of an approach depends on the desired balance between accuracy and efficiency. The computational expense of generating potential energy surfaces on-the-fly often favours the use of approximate, robust and efficient methods such as trajectory surface hopping for large, complex systems. The development of formally exact non-Born-Oppenheimer methods and the exploration of well-defined approximations to such methods are critical for providing benchmarks and preparing for the next generation of faster computers. Thus, the parallel development of rigorous but computationally expensive methods and more approximate but computationally efficient methods is optimal. This Perspective briefly summarizes the available theoretical and computational non-Born-Oppenheimer methods and presents examples illustrating how analytical theories and nonadiabatic dynamics simulations can elucidate the fundamental principles of chemical and biological processes. These examples also highlight how theoretical calculations are able to guide the interpretation of experimental data and provide experimentally testable predictions for nonadiabatic processes. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Theoretical Modeling of Electrochemical Proton-Coupled Electron Transfer

Proton-coupled electron transfer (PCET) plays an essential role in a wide range of electrocatalytic processes. A vast array of theoretical and computational methods have been developed to study electrochemical PCET. These methods can be used to calculate redox potentials and pK_a values for molecular electrocatalysts, proton-coupled redox potentials and bond dissociation free energies for PCET at metal and semiconductor interfaces, and reorganization energies associated with electrochemical PCET. Periodic density functional theory can also be used to compute PCET activation energies and perform molecular dynamics simulations of electrochemical interfaces. Various approaches for maintaining a constant electrode potential in electronic structure calculations and modeling complex interactions in the electric double layer (EDL) have been developed. Theoretical formulations for both homogeneous and heterogeneous electrochemical PCET spanning the adiabatic, nonadiabatic, and solvent-controlled regimes have been developed and provide analytical expressions for the rate constants and current densities as functions of applied potential. The quantum mechanical treatment of the proton and inclusion of excited vibronic states have been shown to be critical for describing experimental data, such as Tafel slopes and potential-dependent kinetic isotope effects. The calculated rate constants can be used as input to microkinetic models and voltammogram simulations to elucidate complex electrocatalytic processes.

Proton-Coupled Electron Transfer: The Engine of Energy Conversion and Storage

Proton-coupled electron transfer (PCET) underpins energy conversion in chemistry and biology. Four energy systems are described whose discoveries are based on PCET: the water splitting chemistry of the Artificial Leaf, the carbon fixation chemistry of the Bionic Leaf-C, the nitrogen fixation chemistry of the Bionic Leaf-N and the Coordination Chemistry Flow Battery (CCFB). Whereas the Artificial Leaf, Bionic Leaf-C, and Bionic Leaf-N require strong coupling between electron and proton to reduce energetic barriers to enable high energy efficiencies, the CCFB requires complete decoupling of the electron and proton so as to avoid parasitic energy-wasting reactions. The proper design of PCET in these systems facilitates their implementation in the areas of (i) centralized large scale grid storage of electricity and (ii) decentralized energy storage/conversion using only sunlight, air and any water source to produce fuel and food within a sustainable cycle for the biogenic elements of C, N and P.

Strategies for switching the mechanism of proton-coupled electron transfer reactions illustrated by mechanistic zone diagrams

The mechanism by which proton-coupled electron transfer (PCET) occurs is of fundamental importance and has great consequences for applications, e.g. in catalysis. However, determination and tuning of the PCET mechanism is often non-trivial. Here, we apply mechanistic zone diagrams to illustrate the competition between concerted and stepwise PCET-mechanisms in the oxidation of 4-methoxyphenol by Ru(bpy)₃³⁺-derivatives in the presence of substituted pyridine bases. These diagrams show the dominating mechanism as a function of driving force for electron and proton transfer (ΔG⁰_ET and ΔG⁰_PT) respectively [Tyburski et al., J. Am. Chem. Soc., 2021, 143, 560]. Within this framework, we demonstrate strategies for mechanistic tuning, namely balancing of ΔG⁰_ET and ΔG⁰_PT, steric hindrance of the proton-transfer coordinate, and isotope substitution. Sterically hindered pyridine bases gave larger reorganization energy for concerted PCET, resulting in a shift towards a step-wise electron first-mechanism in the zone diagrams. For cases when sufficiently strong oxidants are used, substitution of protons for deuterons leads to a switch from concerted electron–proton transfer (CEPT) to an electron transfer limited (ETPT_lim) mechanism. We thereby, for the first time, provide direct experimental evidence, that the vibronic coupling strength affects the switching point between CEPT and ETPT_lim, i.e. at what driving force one or the other mechanism starts dominating. Implications for solar fuel catalysis are discussed.

The mechanism by which proton-coupled electron transfer (PCET) occurs is of fundamental importance and has great consequences for applications, e.g. in catalysis.

Electron-Induced Proton Transfer

The pathway of activationless proton transfer induced by an electron-transfer reaction is studied theoretically. Long-range electron transfer produces highly nonequilibrium medium polarization that can drive proton transfer through an activationless transition during the process of thermalization, dynamically altering the screening of the electron-proton Coulomb interaction by the medium. The cross electron-proton reorganization energy is the main energy parameter of the theory, which exceeds in magnitude the proton-transfer reorganization energy roughly by the ratio of the electron-transfer to proton-transfer distance. This parameter, which can be either positive or negative, is related to the difference in pK_a values in two electron-transfer states. The relaxation time of the medium is on the (sub)picosecond time scale, which establishes the characteristic time for activationless proton transfer. Microscopic calculations predict substantial retardation of the collective relaxation dynamics compared to the continuum estimates due to the phenomenology analogous to de Gennes narrowing. Nonequilibrium medium configuration promoting proton transfer can be induced by either thermal or photoinduced charge transfer.

C-H oxidation in fluorenyl benzoates does not proceed through a stepwise pathway: revisiting asynchronous proton-coupled electron transfer

2-Fluorenyl benzoates were recently shown to undergo C–H bond oxidation through intramolecular proton transfer coupled with electron transfer to an external oxidant. Kinetic analysis revealed unusual rate-driving force relationships. Our analysis indicated a mechanism of multi-site concerted proton–electron transfer (MS-CPET) for all of these reactions. More recently, an alternative interpretation of the kinetic data was proposed to explain the unusual rate-driving force relationships, invoking a crossover from CPET to a stepwise mechanism with an initial intramolecular proton transfer (PT) (Costentin, Savéant, Chem. Sci., 2020, 11, 1006). Here, we show that this proposed alternative pathway is untenable based on prior and new experimental assessments of the intramolecular PT equilibrium constant and rates. Measurement of the fluorenyl 9-C–H pK_a, H/D exchange experiments, and kinetic modelling with COPASI eliminate the possibility of a stepwise mechanism for C–H oxidation in the fluorenyl benzoate series. Implications for asynchronous (imbalanced) MS-CPET mechanisms are discussed with respect to classical Marcus theory and the quantum-mechanical treatment of concerted proton–electron transfer.

2-Fluorenyl benzoates were recently shown to undergo C–H bond oxidation through intramolecular proton transfer coupled with electron transfer to an external oxidant.

Solvent and Temperature Effects on Photoinduced Proton-Coupled Electron Transfer in the Marcus Inverted Region

Concerted proton-coupled electron transfer (PCET) in the Marcus inverted region was recently demonstrated (Science 2019, 364, 471-475). Understanding the requirements for such reactivity is fundamentally important and holds promise as a design principle for solar energy conversion systems. Herein, we investigate the solvent polarity and temperature dependence of photoinduced proton-coupled charge separation (CS) and charge recombination (CR) in anthracene-phenol-pyridine triads: 1 (10-(4-hydroxy-3-(4-methylpyridin-2-yl)benzyl)anthracene-9-carbonitrile) and 2 (10-(4-hydroxy-3-(4-methoxypyridin-2-yl)benzyl)anthracene-9-carbonitrile). Both the CS and CR rate constants increased with increasing polarity in acetonitrile:n-butyronitrile mixtures. The kinetics were semi-quantitatively analyzed where changes in dielectric and refractive index, and thus consequently changes in driving force (-ΔG°) and reorganization energy (λ), were accounted for. The results were further validated by fitting the temperature dependence, from 180 to 298 K, in n-butyronitrile. The analyses support previous computational work where transitions to proton vibrational excited states dominate the CR reaction with a distinct activation free energy (ΔG*CR ∼ 140 meV). However, the solvent continuum model fails to accurately describe the changes in ΔG° and λ with temperature via changes in dielectric constant and refractive index. Satisfactory modeling was obtained using the results of a molecular solvent model [J. Phys. Chem. B 1999, 103, 9130-9140], which predicts that λ decreases with temperature, opposite to that of the continuum model. To further assess the solvent polarity control in the inverted region, the reactions were studied in toluene. Nonpolar solvents decrease both ΔG°CR and λ, slowing CR into the nanosecond time regime for 2 in toluene at 298 K. This demonstrates how PCET in the inverted region may be controlled to potentially use proton-coupled CS states for efficient solar fuel production and photoredox catalysis.

Proton-Coupled Electron Transfer Guidelines, Fair and Square

Proton-coupled electron transfer (PCET) reactions are fundamental to energy transformation reactions in natural and artificial systems and are increasingly recognized in areas such as catalysis and synthetic chemistry. The interdependence of proton and electron transfer brings a mechanistic richness of reactivity, including various sequential and concerted mechanisms. Delineating between different PCET mechanisms and understanding why a particular mechanism dominates are crucial for the design and optimization of reactions that use PCET. This Perspective provides practical guidelines for how to discern between sequential and concerted mechanisms based on interpretations of thermodynamic data with temperature-, pressure-, and isotope-dependent kinetics. We present new PCET-zone diagrams that show how a mechanism can switch or even be eliminated by varying the thermodynamic (ΔGPT° and ΔGET°) and coupling strengths for a PCET system. We discuss the appropriateness of asynchronous concerted PCET to rationalize observations in organic reactions, and the distinction between hydrogen atom transfer and other concerted PCET reactions. Contemporary issues and future prospects in PCET research are discussed.

Theoretical Description of the Primary Proton-Coupled Electron Transfer Reaction in the Cytochrome bc1 Complex

The cytochrome bc₁ complex is a transmembrane enzymatic protein complex that plays a central role in cellular energy production and is present in both photosynthetic and respiratory chain organelles. Its reaction mechanism is initiated by the binding of a quinol molecule to an active site, followed by a series of charge transfer reactions between the quinol and protein subunits. Previous work hypothesized that the primary reaction was a concerted proton-coupled electron transfer (PCET) reaction because of the apparent absence of intermediate states associated with single proton or electron transfer reactions. In the present study, the kinetics of the primary bc₁ complex PCET reaction is investigated with a vibronically nonadiabatic PCET theory in conjunction with all-atom molecular dynamics simulations and electronic structure calculations. The computed rate constants and relatively high kinetic isotope effects are consistent with experimental measurements on related biomimetic systems. The analysis implicates a concerted PCET mechanism with significant hydrogen tunneling and nonadiabatic effects in the bc₁ complex. Moreover, the employed theoretical framework is shown to serve as a general strategy for describing PCET reactions in bioenergetic systems.

Theoretical Study of Shallow Distance Dependence of Proton-Coupled Electron Transfer in Oligoproline Peptides

Long-range electron transfer is coupled to proton transfer in a wide range of chemically and biologically important processes. Recently the proton-coupled electron transfer (PCET) rate constants for a series of biomimetic oligoproline peptides linking Ru(bpy)₃²⁺ to tyrosine were shown to exhibit a substantially shallower dependence on the number of proline spacers compared to the analogous electron transfer (ET) systems. The experiments implicated a concerted PCET mechanism involving intramolecular electron transfer from tyrosine to Ru(bpy)₃³⁺ and proton transfer from tyrosine to a hydrogen phosphate dianion. Herein these PCET systems, as well as the analogous ET systems, are studied with microsecond molecular dynamics, and the ET and PCET rate constants are calculated with the corresponding nonadiabatic theories. The molecular dynamics simulations illustrate that smaller ET donor-acceptor distances are sampled by the PCET systems than by the analogous ET systems. The shallower dependence of the PCET rate constant on the ET donor-acceptor distance is explained in terms of an additional positive, distance-dependent electrostatic term in the PCET driving force, which attenuates the rate constant at smaller distances. This electrostatic term depends on the change in the electrostatic interaction between the charges on each end of the bridge and can be modified by altering these charges. On the basis of these insights, this theory predicted a less shallow distance dependence of the PCET rate constant when imidazole rather than hydrogen phosphate serves as the proton acceptor, even though their pK_a values are similar. This theoretical prediction was subsequently validated experimentally, illustrating that long-range electron transfer processes can be tuned by modifying the nature of the proton acceptor in concerted PCET processes. This level of control has broad implications for the design of more effective charge-transfer systems.

Modeling voltammetry curves for proton coupled electron transfer: The importance of nuclear quantum effects

We investigate rates of proton-coupled electron transfer (PCET) in potential sweep experiments for a generalized Anderson-Holstein model with the inclusion of a quantized proton coordinate. To model this system, we utilize a quantum classical Liouville equation embedded inside of a classical master equation, which can be solved approximately with a recently developed algorithm combining diffusional effects and surface hopping between electronic states. We find that the addition of nuclear quantum effects through the proton coordinate can yield quantitatively (but not qualitatively) different IV curves under a potential sweep compared to electron transfer (ET). Additionally, we find that kinetic isotope effects give rise to a shift in the peak potential, but not the peak current, which would allow for quantification of whether an electrochemical ET event is proton-coupled or not. These findings suggest that it will be very difficult to completely understand coupled nuclear-electronic effects in electrochemical voltammetry experiments using only IV curves, and new experimental techniques will be needed to draw inferences about the nature of electrochemical PCET.

Quantum electrocatalysts: theoretical picture, electrochemical kinetic isotope effect analysis, and conjecture to understand microscopic mechanisms

The fundamental aspects of quantum electrocatalysts are discussed together with the newly developed electrochemical kinetic isotope effect (EC-KIE) approach.

Advances and challenges for experiment and theory for multi-electron multi-proton transfer at electrified solid–liquid interfaces

Understanding microscopic mechanism of multi-electron multi-proton transfer reactions at complexed systems is important for advancing electrochemistry-oriented science in the 21st century.

Quantum proton tunneling in multi-electron/-proton transfer electrode processes

Quantum proton tunneling in multi-electron/-proton transfer electrode processes were investigated in order to understand their possible microscopic mechanisms.

Theoretical analysis of the inverted region in photoinduced proton-coupled electron transfer

Photoinduced proton-coupled electron transfer (PCET) plays a key role in a wide range of energy conversion processes, and understanding how to design systems to control the PCET rate constant is a significant challenge. Herein a theoretical formulation of PCET is utilized to identify the conditions under which photoinduced PCET may exhibit inverted region behavior. In the inverted region, the rate constant decreases as the driving force increases even though the reaction becomes more thermodynamically favorable. Photoinduced PCET will exhibit inverted region behavior when the following criteria are satisfied: (1) the overlap integrals corresponding to the ground reactant and the excited product proton vibrational wavefunctions become negligible for a low enough product vibronic state and (2) the reaction free energies associated with the lower excited product proton vibrational wavefunctions contributing significantly to the rate constant are negative with magnitudes greater than the reorganization energy. These criteria are typically not satisfied by harmonic or Morse potentials but are satisfied by more realistic asymmetric double well potentials because the proton vibrational states above the barrier correspond to more delocalized proton vibrational wavefunctions with nodal structures leading to destructive interference effects. Thus, this theoretical analysis predicts that inverted region behavior could be observed for systems with asymmetric double well potentials characteristic of hydrogen-bonded systems and that the hydrogen/deuterium kinetic isotope effect will approach unity and could even become inverse in this region due to the oscillatory nature of the highly excited vibrational wavefunctions. These insights may help guide the design of more effective energy conversion devices.

From Fundamental Theories to Quantum Coherences in Electron Transfer

Photoinduced electron transfer (ET) is a cornerstone of energy transduction from light to chemistry. The past decade has seen tremendous advances in the possible role of quantum coherent effects in the light-initiated energy and ET processes in chemical, biological, and materials systems. The prevalence of such coherence effects holds a promise to increase the efficiency and robustness of transport even in the face of energetic or structural disorder. A primary motive of this Perspective is to work out how to think about "coherence" in ET reactions. We will discuss how the interplay of basic parameters governing ET reactions-like electronic coupling, interactions with the environment, and intramolecular high-frequency quantum vibrations-impact coherences. This includes revisiting the insights from the seminal work on the theory of ET and time-resolved measurements on coherent dynamics to explore the role of coherences in ET reactions. We conclude by suggesting that in addition to optical spectroscopies, validating the functional role of coherences would require simultaneous mapping of correlated electron motion and atomically resolved nuclear structure.

Proton Discharge on a Gold Electrode from Triethylammonium in Acetonitrile: Theoretical Modeling of Potential-Dependent Kinetic Isotope Effects

The discharge of protons on electrode surfaces, known as the Volmer reaction, is a ubiquitous reaction in heterogeneous electrocatalysis and plays an important role in renewable energy technologies. Recent experiments with triethylammonium (TEAH⁺) donating the proton to a gold electrode in acetonitrile demonstrate significantly different Tafel slopes for TEAH⁺ and its deuterated counterpart, TEAD⁺. As a result, the kinetic isotope effect (KIE) for the hydrogen evolution reaction changes considerably as a function of applied potential. Herein a vibronically nonadiabatic approach for proton-coupled electron transfer (PCET) at an electrode interface is extended to heterogeneous electrochemical processes and is applied to this system. This approach accounts for the key effects of the electrical double layer and spans the electronically adiabatic and nonadiabatic regimes, as found to be necessary for this reaction. The experimental Tafel plots for TEAH⁺ and TEAD⁺ are reproduced using physically reasonable parameters within this model. The potential-dependent KIE or, equivalently, isotope-dependent Tafel slope is found to be a consequence of contributions from excited electron-proton vibronic states that depend on both isotope and applied potential. Specifically, the contributions from excited reactant vibronic states are greater for TEAD⁺ than for TEAH⁺. Thus, the two reactions proceed by the same fundamental mechanism yet exhibit significantly different Tafel slopes. This theoretical approach may be applicable to a wide range of other heterogeneous electrochemical PCET reactions.

Promoting proton coupled electron transfer in redox catalysts through molecular design

Mini-review on using the secondary coordination sphere to facilitate multi-electron, multi-proton catalysis.

Controlling Electrons and Protons through Theory: Molecular Electrocatalysts to Nanoparticles

The development of renewable energy sources that are environmentally friendly and economical is of critical importance. The effective utilization of such energy sources relies on catalysts to facilitate the interconversion between electrical and chemical energy through multielectron, multiproton reactions. The design of effective catalysts for these types of energy conversion processes requires the ability to control the localization and movement of electrons and protons, as well as the coupling between them. Theoretical calculations, in conjunction with experimental validation and feedback, are playing a key role in these catalyst design efforts. A general theory has been developed for describing proton-coupled electron transfer (PCET) reactions, which encompass all reactions involving the coupled transfer of electrons and protons, including sequential and concerted mechanisms for multielectron, multiproton processes. In addition, computational methods have been devised to compute the input quantities for the PCET rate constant expressions and to generate free energy pathways for molecular electrocatalysts. These methods have been extended to heterogeneous PCET reactions to enable the modeling of PCET processes at electrode and nanoparticle surfaces. Three distinct theoretical studies of PCET reactions relevant to catalyst design for energy conversion processes are discussed. In the first application, theoretical calculations of hydrogen production catalyzed by hangman metalloporphyrins predicted that the porphyrin ligand is reduced, leading to dearomatization and proton transfer from the carboxylic acid hanging group to the meso carbon of the porphyrin rather than the metal center, producing a phlorin intermediate. Subsequent experiments isolated and characterized the phlorin intermediate, validating this theoretical prediction. These molecular electrocatalysts exemplify the potential use of noninnocent ligands to localize electrons and protons on different parts of the catalyst and to direct their motions accordingly. In the second application, theoretical calculations on substituted benzimidazole phenol molecules predicted that certain substituents would lead to multiple intramolecular proton transfer reactions upon oxidation. Subsequent experiments verified these multiproton reactions, as well as the predicted shifts in the redox potentials and kinetic isotope effects. These bioinspired molecular systems demonstrate the potential use of multiproton relays to enable the transport of protons over longer distances along specified pathways, as well as the tuning of redox potentials through this movement of positive charge. In the third application, theoretical studies of heterogeneous PCET in photoreduced ZnO nanoparticles illustrated the significance of proton diffusion through the bulk of the nanoparticle as well as interfacial PCET to an organic radical in solution at its surface. These theoretical calculations were consistent with prior experimental studies of this system, although theoretical methods for heterogeneous PCET have not yet attained the level of predictive capability highlighted for the molecular electrocatalysts. These examples suggest that theory will play a significant role in the design of both molecular and heterogeneous catalysts to control the movement and coupling of electrons and protons. The resulting catalysts will be essential for the development of renewable energy sources to address current energy challenges.

Diabatic model for electrochemical hydrogen evolution based on constrained DFT configuration interaction

The accuracy of density functional theory (DFT) based kinetic models for electrocatalysis is diminished by spurious electron delocalization effects, which manifest as uncertainties in the predicted values of reaction and activation energies. In this work, we present a constrained DFT (CDFT) approach to alleviate overdelocalization effects in the Volmer-Heyrovsky mechanism of the hydrogen evolution reaction (HER). This method is applied a posteriori to configurations sampled along a reaction path to correct their relative stabilities. Concretely, the first step of this approach involves describing the reaction in terms of a set of diabatic states that are constructed by imposing suitable density constraints on the system. Refined reaction energy profiles are then recovered by performing a configuration interaction (CDFT-CI) calculation within the basis spanned by the diabatic states. After a careful validation of the proposed method, we examined HER catalysis on open-ended carbon nanotubes and discovered that CDFT-CI increased activation energies and decreased reaction energies relative to DFT predictions. We believe that a similar approach could also be adopted to treat overdelocalization effects in other electrocatalytic proton-coupled electron transfer reactions, e.g., in the oxygen reduction reaction.

Coupled-perturbed DFTB-QM/MM metadynamics: Application to proton-coupled electron transfer

We present a new concept of free energy calculations of chemical reactions by means of extended sampling molecular dynamics simulations. Biasing potentials are applied on partial atomic charges, which may be combined with atomic coordinates either in a single collective variable or in multi-dimensional biasing simulations. The necessary additional gradients are obtained by solving coupled-perturbed equations within the approximative density-functional tight-binding method. The new computational scheme was implemented in a combination of Gromacs and Plumed. As a prospective application, proton-coupled electron transfer in a model molecular system is studied. Two collective variables are introduced naturally, one for the proton transfer and the other for the electron transfer. The results are in qualitative agreement with the extended free simulations performed for reference. Free energy minima as well as the mechanism of the process are identified correctly, while the topology of the transition region and the height of the energy barrier are only reproduced qualitatively. The application also illustrates possible difficulties with the new methodology. These may be inefficient sampling of spatial coordinates when atomic charges are biased exclusively and a decreased stability of the simulations. Still, the new approach represents a viable alternative for free energy calculations of a certain class of chemical reactions, for instance a proton-coupled electron transfer in proteins.

Proton-coupled electron transfer in the electrocatalysis of CO2 reduction: prediction of sequential vs. concerted pathways using DFT

Herein we investigate computationally in detail the mechanism of the formation of the carboxylate adduct during the electroreduction of CO₂ in water catalysed by cobalt porphyrin complexes. Specifically, we address qualitatively the competition between the concerted and sequential pathways for the proton-coupled electron transfer. We use a simple methodology for accurate computation of the pK_a of the neutral and anionic carboxylate intermediates, [CoP-COOH] and [CoP-COOH]^- (where CoP is a cobalt porphine complex), based on the isodesmic proton-exchange reaction scheme. The predicted values are used as in input for a theoretical model that describes the transition between the sequential and concerted pathways. The activation of the sequential pathway (ET-PT) that leads to the formation of the neutral [CoP-COOH] intermediate at pH ≈ 3.5 (pK_a[CoP-COOH] = 3.5 ± 0.4), as predicted by the calculations, is in good agreement with the drastic increase in the faradaic efficiency of the CO₂ reduction reaction towards CO at pH = 3 compared to pH = 1, as experimentally observed. This confirms the existence of the CO₂ anionic adduct [CoP-CO₂]^- as a viable intermediate at pH = 3 and its crucial role for the pH dependence of the faradaic efficiency for the CO₂ reduction. The analysis also shows that when the pH is significantly higher than the pK_a of the neutral carboxylate adduct, the CO₂ reduction has to go through an alternative pathway with the formation of the anionic carboxylate intermediate [CoP-COOH]^-. It is formed through a concerted proton-electron transfer step from the anionic CO₂ adduct [CoP-CO₂]^- when the pH is below ∼8.6 (pK_a[CoP-COOH]^- = 8.6 ± 0.4). At pH ≈ 8.6 and above, another decoupled ET-PT is predicted to take place, leading to the formation of a dianionic CO₂ adduct [CoP-CO₂]^2-.

Proton-Coupled Electron Transfer: Moving Together and Charging Forward

Proton-coupled electron transfer (PCET) is ubiquitous throughout chemistry and biology. This Perspective discusses recent advances and current challenges in the field of PCET, with an emphasis on the role of theory and computation. The fundamental theoretical concepts are summarized, and expressions for rate constants and kinetic isotope effects are provided. Computational methods for calculating reduction potentials and pKa's for molecular electrocatalysts, as well as insights into linear correlations and non-innocent ligands, are also described. In addition, computational methods for simulating the nonadiabatic dynamics of photoexcited PCET are discussed. Representative applications to PCET in solution, proteins, electrochemistry, and photoinduced processes are presented, highlighting the interplay between theoretical and experimental studies. The current challenges and suggested future directions are outlined for each type of application, concluding with an overall view to the future.

Concerted proton-electron transfers: fundamentals and recent developments

Proton-coupled electron transfers (PCET) are ubiquitous in natural and synthetic processes. This review focuses on reactions where the two events are concerted. Semiclassical models of such reactions allow their kinetic characterization through activation versus driving force relationships, estimates of reorganization energies, effects of the nature of the proton acceptor, and H/D kinetic isotope effect as well as their discrimination from stepwise pathways. Several homogeneous reactions (through stopped-flow and laser flash-quench techniques) and electrochemical processes are discussed in this framework. Once the way has been rid of the improper notion of pH-dependent driving force, water appears as a remarkable proton acceptor in terms of reorganization energy and pre-exponential factor, thanks to its H-bonded and H-bonding properties, similarly to purposely synthesized "H-bond train" molecules. The most recent developments are in modeling and description of emblematic concerted proton-electron transfer (CPET) reactions associated with the breaking of a heavy-atom bond in an all-concerted process.

Theory of the transition from sequential to concerted electrochemical proton-electron transfer

A theory for the calculation of potential energy surfaces of electrochemical proton-coupled electron transfer is considered and parameterized on the basis of thermodynamic relations. The paper discusses the qualitatively different potential energy surfaces predicted by the theory, and their relation to the existence of sequential and concerted proton-electron transfer pathways. The concomitant activation energies for sequential and concerted PET are calculated. The applied overpotential may change the qualitative shape of the PES and therefore the mechanism of the proton-coupled electron transfer reaction.

pH-dependent reduction potentials and proton-coupled electron transfer mechanisms in hydrogen-producing nickel molecular electrocatalysts

The nickel-based P2(Ph)N2(Bn) electrocatalysts comprised of a nickel atom and two 1,5-dibenzyl-3,7-diphenyl-1,5-diaza-3,7-diphosphacyclooctane ligands catalyze H2 production in acetonitrile. Recent electrochemical experiments revealed a linear dependence of the Ni(II/I) reduction potential on pH with a slope of 57 mV/pH unit, implicating a proton-coupled electron transfer (PCET) process with the same number of electrons and protons transferred. The combined theoretical and experimental studies herein provide an explanation for this pH dependence in the context of the overall proposed catalytic mechanism. In the proposed mechanisms, the catalytic cycle begins with a series of intermolecular proton transfers from an acid to the pendant amine ligand and electrochemical electron transfers to the nickel center to produce the doubly protonated Ni(0) species, a precursor to H2 evolution. The calculated Ni(II/I) reduction potentials of the doubly protonated species are in excellent agreement with the experimentally observed reduction potential in the presence of strong acid, suggesting that the catalytically active species leading to the peak observed in these cyclic voltammetry (CV) experiments is doubly protonated. The Ni(I/0) reduction potential was found to be slightly more positive than the Ni(II/I) reduction potential, indicating that the Ni(I/0) reduction occurs spontaneously after the Ni(II/I) reduction, as implied by the experimental observation of a single CV peak. These results suggest that the PCET process observed in the CV experiments is a two-electron/two-proton process corresponding to an initial double protonation followed by two reductions. On the basis of the experimental and theoretical data, the complete thermodynamic scheme and the Pourbaix diagram were generated for this catalyst. The Pourbaix diagram, which identifies the most thermodynamically stable species at each reduction potential and pH value, illustrates that this catalyst undergoes different types of PCET processes for various pH ranges. These thermodynamic insights will aid in the design of more effective molecular catalysts for H2 production.

Gaining insight into the chemistry of lipoxygenases: a computational investigation into the catalytic mechanism of (8R)-lipoxygenase

Lipoxygenases (LOXs) are ubiquitous in nature and catalyze a range of life-essential reactions within organisms. In particular they are critical to the formation of eicosanoids, which are critical for normal cell function. However, a number of important questions about the reactivity and mechanism of these enzymes still remain. Specifically, although the initial step in the mechanism of LOXs has been well studied, little is known of subsequent steps. Thus, with use of a quantum mechanical/molecular mechanical approach, the complete catalytic mechanism of (8R)-LOX was investigated. The results have provided a better understanding of the general chemistry of LOXs as a whole. In particular, from comparisons with soybean LOX-1, it appears that the initial proton-coupled electron transfer may be very similar among all LOXs. Furthermore, LOXs appear to undergo multistate reactivity where potential spin inversion of an electron may occur either in the attack of O(2) or in the regeneration of the active site. Lastly, it is shown that with the explicit modeling of the environment, the regeneration of the active center likely occurs via the rotation of the intermediate followed by an outer-sphere [Formula: see text] transfer as opposed to the formation of a "purple intermediate" complex.

Nonadiabatic dynamics of photoinduced proton-coupled electron transfer: comparison of explicit and implicit solvent simulations

Theoretical approaches for simulating the ultrafast dynamics of photoinduced proton-coupled electron transfer (PCET) reactions in solution are developed and applied to a series of model systems. These processes are simulated by propagating nonadiabatic surface hopping trajectories on electron-proton vibronic surfaces that depend on the solute and solvent nuclear coordinates. The PCET system is represented by a four-state empirical valence bond model, and the solvent is treated either as explicit solvent molecules or as a dielectric continuum, in which case the solvent dynamics is described in terms of two collective solvent coordinates corresponding to the energy gaps associated with electron and proton transfer. The explicit solvent simulations reveal two distinct solvent relaxation time scales, where the faster time scale relaxation corresponds to librational motions of solvent molecules in the first solvation shell, and the slower time scale relaxation corresponds to the bulk solvent dielectric response. The charge transfer dynamics is strongly coupled to both the fast and slow time scale solvent dynamics. The dynamical multistate continuum theory is extended to include the effects of two solvent relaxation time scales, and the resulting coupled generalized Langevin equations depend on parameters that can be extracted from equilibrium molecular dynamics simulations. The implicit and explicit solvent approaches lead to qualitatively similar charge transfer and solvent dynamics for model PCET systems, suggesting that the implicit solvent treatment captures the essential elements of the nonequilibrium solvent dynamics for many systems. A combination of implicit and explicit solvent approaches will enable the investigation of photoinduced PCET processes in a variety of condensed phase systems.