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

Conformational control over proton-coupled electron transfer in metalloenzymes

From the reduction of dinitrogen to the oxidation of water, the chemical transformations catalysed by metalloenzymes underlie global geochemical and biochemical cycles. These reactions represent some of the most kinetically and thermodynamically challenging processes known and require the complex choreography of the fundamental building blocks of nature, electrons and protons, to be carried out with utmost precision and accuracy. The rate-determining step of catalysis in many metalloenzymes consists of a protein structural rearrangement, suggesting that nature has evolved to leverage macroscopic changes in protein molecular structure to control subatomic changes in metallocofactor electronic structure. The proton-coupled electron transfer mechanisms operative in nitrogenase, photosystem II and ribonucleotide reductase exemplify this interplay between molecular and electronic structural control. We present the culmination of decades of study on each of these systems and clarify what is known regarding the interplay between structural changes and functional outcomes in these metalloenzyme linchpins.

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.

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.

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.

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.

Dissecting Reaction Mechanisms and Catalytic Contributions in Flavoprotein Fumarate Reductases

The interconversion between fumarate and succinate is fundamental to the energy metabolism of nearly all organisms. This redox reaction is catalyzed by a large family of enzymes, fumarate reductases and succinate dehydrogenases, using hydride and proton transfers from a flavin cofactor and a conserved Arg side-chain. These flavoenzymes also have substantial biomedical and biotechnological importance. Therefore, a detailed understanding of their catalytic mechanisms is valuable. Here, calibrated electronic structure calculations in a cluster model of the active site of the Fcc₃ fumarate reductase were employed to investigate various reaction pathways and possible intermediates in the enzymatic environment and to dissect interactions that contribute to catalysis of fumarate reduction. Carbanion, covalent adduct, carbocation, and radical intermediates were examined. Significantly lower barriers were obtained for mechanisms via carbanion intermediates, with similar activation energies for hydride and proton transfers. Interestingly, the carbanion bound to the active site is best described as an enolate. Hydride transfer is stabilized by a preorganized charge dipole in the active site and by the restriction of the C1-C2 bond in a twisted conformation of the otherwise planar fumarate dianion. But, protonation of a fumarate carboxylate and quantum tunneling effects are not critical for catalysis of the hydride transfer. Calculations also suggest that the driving force for enzyme turnover is provided by regeneration of the catalytic Arg, either coupled with flavin reduction and decomposition of a proposed transient state or directly from the solvent. The detailed mechanistic description of enzymatic reduction of fumarate provided here clarifies previous contradictory views and provides new insights into catalysis by essential flavoenzyme reductases and dehydrogenases.

Electronic coupling and electron transfer in hydrogen-bonded mixed-valence compounds

Electron transfer provided by hydrogen bonds represents a unique and highly significant area of research, as it has a crucial role to play in a wide variety of chemical and biological systems. The hydrogen-bonded mixed-valence system, in the form of donor-hydrogen bond-acceptor, provides an ideal platform for exploring thermally-induced electron transfer across this non-covalent unit. Over the past decades, ongoing progress has been made in this field. Here we critically assess some studies on the qualitative and quantitative evaluation of electronic coupling and thermal electron transfer across hydrogen bond interface. Additionally, selected experimental examples are discussed in terms of intervalence charge transfer, with particular attention paid to the proton-coupled and often overlooked proton-uncoupled electron transfer pathway in hydrogen-bonded mixed-valence systems. We further highlight the major limitations of this research area and suggest potential directions for future exploration.

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.

Kinetic model for reversible radical transfer in ribonucleotide reductase

The enzyme ribonucleotide reductase (RNR), which catalyzes the reduction of ribonucleotides to deoxynucleotides, is vital for DNA synthesis, replication, and repair in all living organisms. Its mechanism requires long-range radical translocation over ∼32 Å through two protein subunits and the intervening aqueous interface. Herein, a kinetic model is designed to describe reversible radical transfer in Escherichia coli RNR. This model is based on experimentally studied photoRNR systems that allow the photochemical injection of a radical at a specific tyrosine residue, Y356, using a photosensitizer. The radical then transfers across the interface to another tyrosine residue, Y731, and continues until it reaches a cysteine residue, C439, which is primed for catalysis. This kinetic model includes radical injection, an off-pathway sink, radical transfer between pairs of residues along the pathway, and the conformational flipping motion of Y731 at the interface. Most of the input rate constants for this kinetic model are obtained from previous experimental measurements and quantum mechanical/molecular mechanical free-energy simulations. Ranges for the rate constants corresponding to radical transfer across the interface are determined by fitting to the experimentally measured Y356 radical decay times in photoRNR systems. This kinetic model illuminates the time evolution of radical transport along the tyrosine and cysteine residues following radical injection. Further analysis identifies the individual rate constants that may be tuned to alter the timescale and probability of the injected radical reaching C439. The insights gained from this kinetic model are relevant to biochemical understanding and protein-engineering efforts with potential pharmacological implications.

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.

Electrocatalysis in Alkaline Media and Alkaline Membrane-Based Energy Technologies

Hydrogen energy-based electrochemical energy conversion technologies offer the promise of enabling a transition of the global energy landscape from fossil fuels to renewable energy. Here, we present a comprehensive review of the fundamentals of electrocatalysis in alkaline media and applications in alkaline-based energy technologies, particularly alkaline fuel cells and water electrolyzers. Anion exchange (alkaline) membrane fuel cells (AEMFCs) enable the use of nonprecious electrocatalysts for the sluggish oxygen reduction reaction (ORR), relative to proton exchange membrane fuel cells (PEMFCs), which require Pt-based electrocatalysts. However, the hydrogen oxidation reaction (HOR) kinetics is significantly slower in alkaline media than in acidic media. Understanding these phenomena requires applying theoretical and experimental methods to unravel molecular-level thermodynamics and kinetics of hydrogen and oxygen electrocatalysis and, particularly, the proton-coupled electron transfer (PCET) process that takes place in a proton-deficient alkaline media. Extensive electrochemical and spectroscopic studies, on single-crystal Pt and metal oxides, have contributed to the development of activity descriptors, as well as the identification of the nature of active sites, and the rate-determining steps of the HOR and ORR. Among these, the structure and reactivity of interfacial water serve as key potential and pH-dependent kinetic factors that are helping elucidate the origins of the HOR and ORR activity differences in acids and bases. Additionally, deliberately modulating and controlling catalyst-support interactions have provided valuable insights for enhancing catalyst accessibility and durability during operation. The design and synthesis of highly conductive and durable alkaline membranes/ionomers have enabled AEMFCs to reach initial performance metrics equal to or higher than those of PEMFCs. We emphasize the importance of using membrane electrode assemblies (MEAs) to integrate the often separately pursued/optimized electrocatalyst/support and membranes/ionomer components. Operando/in situ methods, at multiscales, and ab initio simulations provide a mechanistic understanding of electron, ion, and mass transport at catalyst/ionomer/membrane interfaces and the necessary guidance to achieve fuel cell operation in air over thousands of hours. We hope that this Review will serve as a roadmap for advancing the scientific understanding of the fundamental factors governing electrochemical energy conversion in alkaline media with the ultimate goal of achieving ultralow Pt or precious-metal-free high-performance and durable alkaline fuel cells and related technologies.

Structural and Thermodynamic Effects on the Kinetics of C-H Oxidation by Multisite Proton-Coupled Electron Transfer in Fluorenyl Benzoates

Our recent experimental and theoretical investigations have shown that fluorene C-H bonds can be activated through a mechanism in which the proton and electron are transferred from the C-H bond to a separate base and oxidant in a concerted, elementary step. This multisite proton-coupled electron transfer (MS-PCET) mechanism for C-H bond activation was shown to be facilitated by shorter proton donor-acceptor distances. With the goal of intentionally modulating this donor-acceptor distance, we have now studied C-H MS-PCET in the 3-methyl-substituted fluorenyl benzoate (2-Flr-3-Me-BzO^-). This derivative was readily oxidized by ferrocenium oxidants by initial C-H MS-PCET, with rate constants that were 6-21 times larger than those for 2-Flr-BzO^- with the same oxidants. Structural comparisons by X-ray crystallography and by computations showed that addition of the 3-methyl group caused the expected steric compression; however, the relevant C···O^- proton donor-acceptor distance was longer, due to a twist of the carboxylate group. The structural changes induced by the 3-Me group increased the basicity of the carboxylate, weakened the C-H bond, and reduced the reorganization energy for C-H bond cleavage. Thus, the rate enhancement for 2-Flr-3-Me-BzO^- was due to effects on the thermochemistry and kinetic barrier, rather than from compression of the C···O^- proton donor-acceptor distance. These results highlight both the challenges of controlling molecules on the 0.1 Å length scale and the variety of parameters that affect PCET rate constants.

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.

Nuclear-electronic orbital methods: Foundations and prospects

The incorporation of nuclear quantum effects and non-Born-Oppenheimer behavior into quantum chemistry calculations and molecular dynamics simulations is a longstanding challenge. The nuclear-electronic orbital (NEO) approach treats specified nuclei, typically protons, quantum mechanically on the same level as the electrons with wave function and density functional theory methods. This approach inherently includes nuclear delocalization and zero-point energy in molecular energy calculations, geometry optimizations, reaction paths, and dynamics. It can also provide accurate descriptions of excited electronic, vibrational, and vibronic states as well as nuclear tunneling and nonadiabatic dynamics. Nonequilibrium nuclear-electronic dynamics simulations beyond the Born-Oppenheimer approximation can be used to investigate a wide range of excited state processes. This Perspective provides an overview of the foundational NEO methods and enumerates the prospects for using these methods as building blocks for future developments. The conceptual simplicity and computational efficiency of the NEO approach will enhance its accessibility and applicability to diverse chemical and biological systems.

Tunneling and Nonadiabatic Effects on a Proton-Coupled Electron Transfer Model for the Qo Site in Cytochrome bc1

Cytochrome bc₁ is a fundamental enzyme for cellular respiration and photosynthesis. This dimeric protein complex catalyzes a proton-coupled electron transfer (PCET) from the reduced coenzyme-Q substrate (Q) to a bimetallic iron-sulfur cluster in the Q_o active site. Herein, we combine molecular dynamics simulations of the complete cytochrome bc₁ protein with electronic-structure calculations of truncated models and a semiclassical tunneling theory to investigate the electron-proton adiabaticity of the initial reaction catalyzed in the Q_o site. After sampling possible orientations between the Q substrate and a histidine side chain that functions as hydrogen acceptor, we find that a truncated model composed by ubiquinol-methyl and imidazole-iron(III)-sulfide captures the expected changes in oxidation and spin states of the electron donor and acceptor. Diabatic electronic surfaces obtained for this model with multiconfigurational wave function calculations demonstrate that this reaction is electronic nonadiabatic, and proton tunneling is faster than mixing of electronic configurations. These results indicate the formalism that should be used to calculate vibronic couplings and kinetic parameters for the initial reaction in the Q_o site of cytochrome bc₁. This framework for molecular simulation may also be applied to investigate other PCET reactions in the Q-cycle or in various metalloproteins that catalyze proton translocation coupled to redox processes.

Recent progress in approximate quantum dynamics methods for the study of proton-coupled electron transfer reactions

Proton-coupled electron transfer (PCET) reactions are ubiquitous natural processes at the heart of energy conversion reactions in photosynthesis and respiration, DNA repair, and diverse enzymatic reactions. Theoretical formulation and computational method developments have eyed modeling of thermal and photoinduced PCET for the last three decades. The accumulation of these studies, collected in dozens of reviews, accounts, and perspectives, has firmly established the influence of quantum effects, including non-adiabatic electronic transitions, vibrational relaxation, zero-point energy, and proton tunneling, on the rate and mechanism of PCET reactions. Here, we focus on some recently-developed methods, spanning the last eight years, that can quantitatively capture these effects in the PCET context and provide efficient means for their qualitative description in complex systems. The theoretical background of each method and their accuracy with respect to exact results are discussed and the results of relevant PCET simulations based on each method are presented.

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.

Polariton-Mediated Electron Transfer via Cavity Quantum Electrodynamics

We investigate the polariton-mediated electron transfer reaction in a model system with analytic rate constant theory and direct quantum dynamical simulations. We demonstrate that the photoinduced charge transfer reaction between a bright donor state and dark acceptor state can be significantly enhanced or suppressed by coupling the molecular system to the quantized radiation field inside an optical cavity. This is because the quantum light-matter interaction can influence the effective driving force and electronic couplings between the donor state, which is the hybrid light-matter excitation, and the molecular acceptor state. Under the resonance condition between the photonic and electronic excitations, the effective driving force can be tuned by changing the light-matter coupling strength; for an off-resonant condition, the same effect can be accomplished by changing the molecule-cavity detuning. We further demonstrate that using both the electronic coupling and light-matter coupling helps to extend the effective couplings across the entire system, even for the dark state that carries a zero transition dipole. Theoretically, we find that both the counter-rotating terms and the dipole self-energy in the quantum electrodynamics Hamiltonian are important for obtaining an accurate polariton eigenspectrum as well as the polariton-mediated charge transfer rate constant, especially in the ultrastrong coupling regime.

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.

Highly Polarized Coumarin Derivatives Revisited: Solvent-Controlled Competition Between Proton-Coupled Electron Transfer and Twisted Intramolecular Charge Transfer

Linking a polarized coumarin unit with an aromatic substituent via an amide bridge results in weak electronic coupling that affects the intramolecular electron-transfer (ET) process. As a result of this, interesting solvent-dependent photophysical properties can be observed. In polar solvents, electron transfer in coumarin derivatives of this type induces a mutual twist of the electron-donating and -accepting molecular units (TICT process) that facilitates radiationless decay processes (internal conversion). In the dyad with the strongest intramolecular hydrogen bond, the planar form is stabilized, such that twisting can only occur in highly polar solvents, whereas a fast proton-coupled electron-transfer (PCET process) occurs in nonpolar n-alkanes. The k_PCET rate constant decreases linearly with the energy of the fluorescence maximum in different solvents. This observation can be explained in terms of competition between electron- and proton-transfer from a highly polarized (ca. 15 D) and fluorescent locally excited (¹ LE) state to a much less polarized (ca. 4 D) charge-transfer (¹ CT) state, a unique occurrence. Photophysical measurements performed for a family of related coumarin dyads, together with results of quantum-chemical computations, give insight into the mechanism of the ET process, which is followed by either a TICT or a PCET process. Our results reveal that dielectric solvation of the excited state slows down the PCET process, even in nonpolar solvents.

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.

Quantum effects in complex systems: summarizing remarks

Quantum mechanical phenomena such as coherence, spin dynamics, and tunneling have been observed in biological, electrochemical, polymeric, and many other condensed phase processes. This paper summarizes the diverse contributions to the Faraday Discussion on quantum effects in complex systems. Various processes exhibiting quantum mechanical behavior were examined using advanced spectroscopic and theoretical methods. An emerging theme was the critical importance of feedback between experiment and theory, particularly in the form of experimental testing of theoretical predictions.

Elucidating Proton-Coupled Electron Transfer Mechanisms of Metal Hydrides with Free Energy- and Pressure-Dependent Kinetics

Proton-coupled electron transfer (PCET) was studied in a series of tungsten hydride complexes with pendant pyridyl arms ([(PyCH₂Cp)WH(CO)₃], PyCH₂Cp = pyridylmethylcyclopentadienyl), triggered by laser flash-generated Ru^III-tris-bipyridine oxidants, in acetonitrile solution. The free energy dependence of the rate constant and the kinetic isotope effects (KIEs) showed that the PCET mechanism could be switched between concerted and the two stepwise PCET mechanisms (electron-first or proton-first) in a predictable fashion. Straightforward and general guidelines for how the relative rates of the different mechanisms depend on oxidant and base are presented. The rate of the concerted reaction should depend symmetrically on changes in oxidant and base strength, that is on the overall ΔG⁰_PCET, and we argue that an "asynchronous" behavior would not be consistent with a model where the electron and proton tunnel from a common transition state. The observed rate constants and KIEs were examined as a function of hydrostatic pressure (1-2000 bar) and were found to exhibit qualitatively different dependence on pressure for different PCET mechanisms. This is discussed in terms of different volume profiles of the PCET mechanisms as well as enhanced proton tunneling for the concerted mechanism. The results allowed for assignment of the main mechanism operating in the different cases, which is one of the critical questions in PCET research. They also show how the rate of a PCET reaction will be affected very differently by changes of oxidant and base strength, depending on which mechanism dominates. This is of fundamental interest as well as of practical importance for rational design of, for example, catalysts for fuel cells and solar fuel formation, which operate in steps of PCET reactions. The mechanistic richness shown by this system illustrates that the specific mechanism is not intrinsic to a specific synthetic catalyst or enzyme active site but depends on the reaction conditions.

Strategies for Enhancing the Rate Constant of C-H Bond Cleavage by Concerted Proton-Coupled Electron Transfer

Recently selective C-H bond cleavage under mild conditions with weak oxidants was reported for fluorenyl-benzoates. This mechanism is based on multi-site concerted proton-coupled electron transfer (PCET) involving intermolecular electron transfer to an outer-sphere oxidant coupled to intramolecular proton transfer to a well-positioned proton acceptor. The electron transfer driving force depends predominantly on the oxidant, and the proton transfer driving force depends mainly on the basicity of the carboxylate, which is influenced by the substituent on the benzoate fragment. Experiments showed that the rate constants are much more sensitive to the carboxylate basicity than to the redox potential of the oxidant. Herein a vibronically nonadiabatic PCET theory is used to explain how changing the driving force for the electron and proton transfer components of the reaction through varying the oxidant and the substituent, respectively, impacts the PCET rate constant. In addition to increasing the driving force for proton transfer, enhancing the basicity of the carboxylate also decreases the equilibrium proton donor-acceptor distance, thereby facilitating the sampling of shorter proton donor-acceptor distances. This additional effect arising from the strong dependence of proton transfer on the proton donor-acceptor distance provides an explanation for the greater sensitivity of the rate constant to the carboxylate basicity than to the redox potential of the oxidant. These fundamental insights have broad implications for developing new strategies to activate C-H bonds, specifically by designing systems with shorter equilibrium proton donor-acceptor distances.

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.

Concerted proton-electron transfer reactions in the Marcus inverted region

Electron transfer reactions slow down when they become very thermodynamically favorable, a counterintuitive interplay of kinetics and thermodynamics termed the inverted region in Marcus theory. Here we report inverted region behavior for proton-coupled electron transfer (PCET). Photochemical studies of anthracene-phenol-pyridine triads give rate constants for PCET charge recombination that are slower for the more thermodynamically favorable reactions. Photoexcitation forms an anthracene excited state that undergoes PCET to create a charge-separated state. The rate constants for return charge recombination show an inverted dependence on the driving force upon changing pyridine substituents and the solvent. Calculations using vibronically nonadiabatic PCET theory yield rate constants for simultaneous tunneling of the electron and proton that account for the results.

Probing photodissociation dynamics using ring polymer molecular dynamics

The performance of the ring polymer molecular dynamics (RPMD) approach to simulate typical photodissociation processes is assessed. The correct description of photodissociation requires the calculation of correlation functions or expectation values associated with non-equilibrium initial conditions, which was shown to be possible with RPMD very recently [J. Chem. Phys. 145, 204118 (2016)]. This approach is combined with treatment of the nonadiabatic dynamics employing the ring polymer surface hopping approach (RPSH), which is based on Tully's fewest switches surface hopping (FSSH) approach. The performance is tested using one-dimensional photodissociation models. It is found that RPSH with non-equilibrium initial conditions can well reproduce the time-dependent dissociation probability, and adiabatic and diabatic populations for cases where the crossing point is below and above the Franck-Condon point, respectively, while standard FSSH fails to reproduce the exact quantum dynamics in the latter case. Thus, it is shown that RPSH is an efficient and accurate alternative to standard FSSH, which is one of the most widely employed approaches to study photochemistry.

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.

Solvent-dependent transition from concerted electron-proton to proton transfer in photoinduced reactions between phenols and polypyridine Ru complexes with proton-accepting sites

The bimolecular rate coefficients (kobsq) for quenching the metal-to-ligand charge transfer excited states of two Ru polypyridine complexes containing H-bond accepting sites by six p-substituted phenols exhibit abrupt deviations from the expected linear correlations of log kobsq with phenol's Hammett σp constant. This pattern is attributed to a transition of the quenching mechanism from a concerted electron-proton transfer (EPT) to a proton transfer (PT); the latter becomes predominant for the most acidic phenols in acetonitrile, but not in dichloromethane. This assertion is supported by a detailed thermochemical analysis, which also excludes the quenching pathways involving electron transfer from phenols with or without deprotonation of phenols to the solvent, either concerted or sequential. The transition from EPT to PT upon the σp increase is consistent/supported by the magnitudes of the measured and computed PhOH/OD kinetic isotope effects and by the observed reduction of the EPT product yields upon replacing the low σp methoxyphenol by the high σp nitrophenol. In addition to modulating the relative contribution of the EPT and PT quenching pathways, the solvent strongly affects the bimolecular rate coefficients for the EPT quenching proper. Unlike with H-atom transfer reactions, this kinetic solvent effect could not be quantitatively accounted for by the phenol-solvent H-bonding alone, which suggests a solvent effect on the H-bonding constants in the phenol-Ru complex precursor exciplexes and/or on the unimolecular EPT rate coefficients within these exciplexes.

Theoretical Study of C-H Bond Cleavage via Concerted Proton-Coupled Electron Transfer in Fluorenyl-Benzoates

Developing new strategies to activate and cleave C-H bonds is important for a broad range of applications. Recently a new approach for C-H bond activation using multi-site concerted proton-coupled electron transfer (PCET) involving intermolecular electron transfer to an oxidant coupled to intramolecular proton transfer was reported. For a series of oxidants reacting with 2-(9 H-fluoren-9-yl)benzoate, experimental studies revealed an atypical Brønsted α, defined as the slope of the logarithm of the PCET rate constant versus the logarithm of the equilibrium constant or the scaled driving force. Herein this reaction is modeled with a vibronically nonadiabatic PCET theory. Hydrogen tunneling, thermal sampling of the proton donor-acceptor mode, solute and solvent reorganization, and contributions from excited vibronic states are found to play important roles. The calculations qualitatively reproduce the experimental observation of a Brønsted α significantly less than 0.5 and explain this shallow slope in terms of exoergic processes between pairs of electron-proton vibronic states. These fundamental mechanistic insights may guide the design of more effective strategies for C-H bond activation and cleavage.