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. 2020 Sep 8;6(9):1594–1601. doi: 10.1021/acscentsci.0c00756

Nonequilibrium Dynamics of Proton-Coupled Electron Transfer in Proton Wires: Concerted but Asynchronous Mechanisms

Joshua J Goings 1, Sharon Hammes-Schiffer 1,*
PMCID: PMC7517869  PMID: 32999935

Abstract

graphic file with name oc0c00756_0006.jpg

The coupling between electrons and protons and the long-range transport of protons play important roles throughout biology. Biomimetic systems derived from benzimidazole-phenol (BIP) constructs have been designed to undergo proton-coupled electron transfer (PCET) upon electrochemical or photochemical oxidation. Moreover, these systems can transport protons along hydrogen-bonded networks or proton wires through multiproton PCET. Herein, the nonequilibrium dynamics of both single and double proton transfer in BIP molecules initiated by oxidation are investigated with first-principles molecular dynamics simulations. Although these processes are concerted in that no thermodynamically stable intermediate is observed, the simulations predict that they are predominantly asynchronous on the ultrafast time scale. For both systems, the first proton transfer typically occurs ∼100 fs after electron transfer. For the double proton transfer system, typically the second proton transfer occurs hundreds of femtoseconds after the initial proton transfer. A machine learning algorithm was used to identify the key molecular vibrational modes essential for proton transfer: a slow, in-plane bending mode that dominates the overall inner-sphere reorganization, the proton donor–acceptor motion that leads to vibrational coherence, and the faster donor–hydrogen stretching mode. The asynchronous double proton transfer mechanism can be understood in terms of a significant mode corresponding to the two anticorrelated proton donor–acceptor motions, typically decreasing only one donor–acceptor distance at a time. Although these PCET processes appear concerted on the time scale of typical electrochemical experiments, attaching these BIP constructs to photosensitizers may enable the detection of the asynchronicity of the electron and multiple proton transfers with ultrafast two-dimensional spectroscopy. Understanding the fundamental PCET mechanisms at this level will guide the design of PCET systems for catalysis and energy conversion processes.

Short abstract

Neural networks help predict important motions in bioinspired molecules when they undergo proton-coupled electron transfer comprising one or two proton transfers along a proton wire.

Introduction

Inspired by the proton-coupled electron transfer (PCET) process between tyrosine and histidine in photosystem II,1−6 a variety of biomimetic molecules have been developed to explore the fundamental principles underlying PCET.3,7−14 For example, the tyrosine-histidine pair in photosystem II has been modeled by benzimidazole-phenol (BIP) constructs. These molecules have proven to be informative because of the ability to tune the PCET redox potential by substituent modification.14−17 Moreover, by increasing the number of benzimidazole linkers in the BIP constructs, molecular systems have been designed to transfer multiple protons upon oxidation in what is termed a one-electron with n proton transfers (EnPT) process, where systems with n values up to 4 have been synthesized and characterized.16 Although the thermodynamics of these multiproton PCET processes in the BIP constructs has been well-studied,14−18 the dynamics of these PCET processes is not yet well understood. In particular, the detailed mechanism in terms of the fundamental electron and proton transfer (PT) steps in these EnPT processes has not been resolved.

On electrochemical time scales, the PCET mechanisms in these BIP constructs appear to be concerted in that the protons are all on their donors in the neutral state and are all on their acceptors in the oxidized state, with no intermediates corresponding to partial proton transfer detected in infrared spectroelectrochemistry experiments.19−24 Density functional theory (DFT) calculations of proton-coupled redox potentials are also consistent with a concerted PCET mechanism in that the computed potentials corresponding to complete, concerted proton transfer upon oxidation agree with the experimental values.14−17 If a sequential PCET mechanism is defined to require a thermodynamically stable intermediate, these systems appear to correspond to concerted rather than sequential mechanisms. On the ultrafast time scale, however, the concerted PCET process may be asynchronous in that proton transfer may occur after electron transfer or one proton transfer may precede another proton transfer, even in the absence of a thermodynamically stable intermediate. This type of concerted but asynchronous mechanism could potentially be observed experimentally for a system composed of the BIP molecule attached to a photosensitizer such as a porphyrin, where the BIP molecule is oxidized photochemically rather than electrochemically.24−27 In the photochemical case, electron transfer occurs from the BIP molecule to the electronically excited photosensitizer. To better understand these fundamental mechanistic issues and generate predictions that could be experimentally detectable on the ultrafast time scale, such as tens or hundreds of femtoseconds, the nonequilibrium dynamics can be studied in model systems that mimic either electrochemical or photochemical oxidation of these BIP constructs.

Herein, we used first-principles molecular dynamics methods to study the nonequilibrium dynamics of both single and double proton transfer BIP systems following oxidation (Figure 1). In these simulations, an equilibrium ensemble of neutral molecules was prepared, and electrochemical or photochemical oxidation was described by instantaneously removing an electron from the molecule, producing a nonequilibrium ensemble in the oxidized state. As this ensemble relaxes toward equilibrium, the protons transfer from their donors to their acceptors, which is the lowest free energy state, thereby elucidating the time scales and dynamical mechanisms of single or double proton transfer. Our simulations illustrate that the E1PT and E2PT processes are concerted but predominantly asynchronous, with the first proton transfer occurring slightly less than 100 fs after electron transfer, and the second proton transfer occurring hundreds of femtoseconds after the first proton transfer in the E2PT process for these particular systems.

Figure 1.

Figure 1

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Two benzimidazole-phenol (BIP) constructs explored in this study. Upon oxidation, the E1PT system exhibits single proton transfer, and the E2PT system exhibits double proton transfer.

To gain insight into the important modes governing the time scale of proton transfer, we trained an artificial neural network28 (ANN) to predict the proton transfer times as a function of the molecular coordinates.29 Although the ANN provides remarkably accurate predictions on the test data set, the main purpose of this study was to identify the most important molecular modes for predicting the proton transfer times using interpretable machine learning methods.30 The main advantage of the machine learning approach implemented herein is that it allows for large amounts of nonequilibrium data to be incorporated into the physical understanding of the mechanism. This approach has clear benefits over the more traditional approaches of calculating the differences between the optimized product and reactant geometries and associated free energies or computing free energy pathways that require quasi-equilibrium methods and a choice of reaction coordinate(s). These interpretable machine learning approaches also guide further inquiry by identifying significant molecular motions that can be followed up for additional study. In this context, the ANN identified a slow mode corresponding to an in-plane benzimidazole-phenol bend (∼450 fs period), a slightly faster mode corresponding to the proton donor–acceptor distance motion (∼95 fs), and a much faster mode corresponding to the stretch of the transferring proton (∼14 fs). Further inspection of these modes provided insight into the complex interactions of these molecular motions on the proton transfer process. Extensions of these concepts to the double proton transfer system provided additional insights into multiproton PCET processes.

Results and Discussion

Nonequilibrium First-Principles Molecular Dynamics Simulations

To simulate the nonequilibrium PCET dynamics for the E1PT and E2PT BIP molecules, we used a first-principles Born–Oppenheimer molecular dynamics (BOMD) approach. All of these simulations were performed in the gas phase at the B3LYP-D3(BJ)/6-31G** level of theory31−37 using the Q-Chem 5.2 electronic structure package.38 This level of theory was selected based on the good agreement between the computed and experimental redox potentials in previous works,15−17 indicating that the regions of the potential energy surface near equilibrium are accurate; however, the regions far from equilibrium are less certain. Additionally, because the experimental studies of these systems have been performed in aprotic, relatively low-dielectric environments, such as dichloromethane or acetonitrile, the gas phase simulations are a reasonable qualitative approximation to experimental conditions.

An ensemble of independent trajectories was generated by sampling initial positions and velocities along all normal modes with frequencies greater than 100 cm–1 from a ground-state Wigner distribution39−41 corresponding to the neutral molecule. Additional computational details are provided in the Supporting Information (SI). We propagated 240 independent trajectories for the E1PT system and 120 independent trajectories for the E2PT system. These initial configurations approximately represent the vibrational ground state of the neutral molecule. Given these initial conditions, an electron was removed, and the trajectories were propagated in the oxidized state. By using initial conditions from the neutral species and propagating in the oxidized state, each trajectory corresponded to a nonequilibrium dynamics simulation following either electrochemical or photochemical oxidation. The MD trajectories utilized a time step of 20 au (∼0.5 fs). The proton transfer time was defined as the time at which the distance between the proton and its acceptor becomes less than 1.05 Å, with the additional requirement that this distance remains less than 1.5 Å for at least 40 fs. Choosing a slightly longer or shorter time for the proton required to remain on its acceptor did not appreciably change any of the analysis (Figure S1).

This strategy assumes that electron transfer occurs fast enough to avoid perturbing the equilibrium ensemble of neutral molecules and that the electrons within the BIP molecule respond adiabatically to the nuclear motions. This approach also neglects the nuclear quantum effects of the transferring protons. Alternative strategies using real-time time-dependent density functional theory42 and its extension within the nuclear-electronic orbital (NEO) framework43 to treat the transferring protons quantum mechanically, possibly on systems including the photosensitizer, would address these issues but are not yet applicable to these types of systems. Moreover, the times computed for proton transfer are not directly comparable to putative solution-phase experiments because the calculations are performed in the gas phase. However, the current strategy provides insights about the qualitative mechanisms and identifies the critical vibrational modes for proton transfer.

Of the 240 nonequilibrium trajectories for the E1PT system, all exhibited proton transfer within 350 fs, with a median time to proton transfer of 79 fs. A representative trajectory showing proton transfer is given in Figure 2A. The distribution of proton transfer times is approximately bimodal, with peaks around 50 and 150 fs (Figure 3A). A detailed analysis of this bimodal distribution will be given below. In this system, the PCET mechanism is typically concerted but asynchronous, with the proton transfer occurring ∼100 fs after electron transfer; however, the mechanism is synchronous within ∼15 fs for a small fraction of the trajectories. This type of asynchronicity would not be observable on the time scales of electrochemical experiments, in which the system is oxidized in an essentially quasi-equilibrium manner, but could potentially be observable in ultrafast photochemical experiments with very short time scale resolutions.

Figure 2.

Figure 2

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(A) The proton transfer process for a representative trajectory for the E1PT system, with proton transfer occurring at 58 fs. (B) The proton transfer process for a representative trajectory for the E2PT system, with the phenolic proton transfer (blue) occurring at 47 fs followed by the other proton transfer (orange) occurring at 512 fs. Movies of these two trajectories are provided as Web-enhanced objects (Movies 1 and 2, respectively).

Figure 3.

Figure 3

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(A) Histogram and estimated probability density (solid blue line) of proton transfer times after oxidation, as obtained from 240 BOMD trajectories of the E1PT system. The median proton transfer time is 79 fs and is indicated by the vertical dashed blue line. The distribution is approximately bimodal and shows peaks in the probability density at ∼50 and ∼150 fs. The proton transfer has occurred in all trajectories by 350 fs. (B) E2PT joint distribution plot for the trajectories of the E2PT system that exhibited double proton transfer. The green circles indicate the phenolic proton transfer (PT1) and the other proton transfer (PT2) times after oxidation for each trajectory. The dashed black line indicates perfect synchronicity between the two proton transfers. The first proton transfer occurs relatively quickly (median 116 fs), whereas the second proton transfer typically occurs much later (median 712 fs).

Of the 120 nonequilibrium trajectories for the E2PT system, the median time for the first proton transfer was 116 fs, and the median time for the second proton transfer when it occurred was 712 fs. The median first proton transfer time for the E2PT system is fairly similar to the median time of 79 fs observed for the E1PT system. Typically, the double proton transfer reaction is asynchronous, with the second proton transfer occurring quite a bit later (Figure 2B). Defining synchronous proton transfer as the two proton transfers occurring within 20 fs of each other, the asynchronous mechanism with the phenolic proton transferring first was observed in 83% of the trajectories; however, 13% of the trajectories exhibited the reverse order, and 3% of the trajectories exhibited nearly synchronous double proton transfer. Note that these percentages pertain only to the 89 trajectories exhibiting double proton transfer within the maximum trajectory length of 1450 fs. As discussed above for the E1PT system, this asynchronicity of double proton transfer would not be observable on the time scale of electrochemical experiments but could potentially be detected photochemically.44,45

Machine Learning to Identify Important Modes

In order to aid in the interpretation of the PCET mechanism, a feedforward artificial neural network (ANN) was trained on the nonequilibrium dynamics. Following the methods given in ref (29), the data set was constructed by extracting geometries and corresponding proton transfer times every 2 fs along the trajectories until the time of proton transfer and representing these geometries in the normal mode coordinates associated with the oxidized reference molecule. This procedure yielded a total data set of 12381 molecular geometries, each labeled with the corresponding time until proton transfer from that geometry. The data set was randomly divided into a training set (80%, 9905 frames), a validation set (10%, 1238 frames), and a test set (10%, 1238 frames). The test set was not used in the training or selection of model hyperparameters. A grid search was used to optimize the network hyperparameters, and the details are given in Table S1. Additional computational details are given in the SI. The final selected model gave the lowest prediction error on the validation set and consisted of 3 hidden layers with 768 neurons per layer, using a dropout rate of 25% to prevent overfitting. The predicted versus actual proton transfer times for the test set are depicted in Figure 4, illustrating that the model yields a robust fit to the test data. Although this accuracy provides validation for the model, our main goal is to determine the correlations and features used by the model to achieve this high accuracy. Such an analysis will provide physical insight into the modes most relevant to proton transfer in the BIP molecule.

Figure 4.

Figure 4

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Predictions of the E1PT proton transfer time from the ANN compared to the actual proton transfer time for the test set (1238 frames). Perfect agreement between the predicted and actual proton transfer times is depicted by the dashed black line. The root-mean-square deviation (RMSD), the mean absolute error (MAE), and the coefficient of determination (R2) are also given.

To understand the correlations that the neural network has learned, we used permutation importance46 and Shapley additive explanations (SHAP) to interrogate the model.47 Permutation importance determines what features (i.e., modes in this case) are most important to the accuracy of the already trained model by randomly shuffling the test data and measuring the mean increase in the model error. If a mode is particularly important to the model then randomly shuffling its values should lead to a greater increase in prediction error, as the new inputs are random noise. Conversely, if a mode is unimportant then randomly shuffling its values will have a smaller impact on the model error. We performed permutation importance testing on the trained model for 200 iterations (Figure S2A). This analysis indicates that the model relies primarily on mode 5, followed by modes 24 and 119 to a lesser extent. In contrast to permutation importance testing, Shapley additive explanations do not assign significance to a feature based on how much error is introduced when a feature is removed. Rather, the SHAP method computes the contribution that each feature made to a predicted value. Computing Shapley values for an ensemble of predictions provides a sense of the global behavior of a model, as illustrated by a plot of the average magnitude of the Shapley values for the six most significant modes (Figure S2B). Consistent with the permutation importance testing, the model clearly relies on mode 5 to make its predictions, followed by modes 24 and 119.

Identification and Analysis of Important Modes for PT

The ANN identified the vibrational modes that contribute most strongly to the proton transfer reaction in the E1PT system. The physical nature of these modes is illustrated in Figure 5. The most important mode was identified to be mode 5, which corresponds to a predominantly in-plane bending motion that brings the phenol and benzimidazole portions together. Projection of the difference between the aligned optimized structures of the oxidized and neutral species onto the normal modes for the oxidized species (Figure S3) indicates that mode 5 dominates the inner-sphere reorganization for the overall PCET reaction. This finding is consistent with the identification of this mode as the most important mode for proton transfer by the ANN. Figure 5A illustrates how the ensemble of trajectories evolves along mode 5. The sudden oxidation of the molecules equilibrated in the neutral state places the system out of equilibrium with respect to this mode, resulting in forces that push the molecules toward the equilibrium geometry of the oxidized state.

Figure 5.

Figure 5

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Three most significant modes of the E1PT system identified by the ANN. The panels on the left depict the evolution of the ensemble of trajectories with the proton transfer time identified by a closed circle. The middle of the figure depicts the motions along each mode from blue to red. The right portion of the figure provides information about each mode. These collective motions dominate the proton transfer process after oxidation. (A) The most significant mode at 72 cm–1 corresponds to the predominantly in-plane bending motion that brings the phenol and benzimidazole portions together after oxidation. (B) The second most significant mode at 356 cm–1 corresponds to proton donor–acceptor vibrational motion, and proton transfer is more probable when the donor–acceptor distance is sufficiently short. (C) The third most significant mode at 2420 cm–1 corresponds to the proton donor–hydrogen stretching mode, and proton transfer is more probable when this distance is sufficiently long.

Mode 24 corresponds predominantly to the proton donor–acceptor vibrational mode, which is known to be important for proton transfer.48,49Figure 5B illustrates how the ensemble evolves along mode 24. Proton transfer occurs when the molecule is displaced in the positive direction along mode 24 because this direction corresponds to the proton donor and acceptor atoms moving closer together. If the proton can transfer during the turning point associated with a short proton donor–acceptor distance because the OH distance is sufficiently long, it will do so (Figure S4), but otherwise, the proton transfer will be delayed until the donor and acceptor come closer together again during a subsequent vibration of mode 24. Because the proton donor–acceptor distance must be short enough to allow proton transfer to occur, a coherence along this vibrational coordinate occurs, as shown in the left panel of Figure 5B. This ensemble coherence results in the multimodal distribution of proton transfer times shown in Figure 3A. Mode 24 corresponds to an energy of 356 cm–1 or a vibrational period of ∼100 fs. Interestingly, the first “wave” of proton transfers is observed at ∼50 fs, followed by another resurgence at ∼150 fs, and an even smaller shoulder at ∼250 fs, consistent with the vibrational period of this mode.

Mode 119 corresponds to the proton donor–hydrogen stretch coordinate, as shown in Figure 5C. Large displacements along this mode correspond to an increase of the proton–donor distance as the proton transfers from the donor to the acceptor. The ANN detects this mode as significant because the overall proton transfer reaction involves large displacement along this mode; however, vibrations along this mode alone are not necessarily sufficient without movement along additional modes. In particular, a combination of motion along this mode and the proton donor–acceptor mode is required for proton transfer, explaining why the proton does not always transfer for short proton donor–acceptor distances (Figure S4).

We applied the lessons learned from the E1PT application of the ANN to the E2PT system. During the double proton transfer process, the E2PT molecule rearranges enough that representing the structural changes during double PT in terms of the reactant normal modes is not physically meaningful. Thus, the application of the ANN used for the E1PT system is not as straightforward for the E2PT system. However, the other techniques used to understand the E1PT system still apply to the E2PT system, and the results can be interpreted in the context of the knowledge provided by the ANN for the E1PT system. Therefore, we projected the difference between the aligned optimized structures of the oxidized and neutral E2PT species onto the normal modes for the oxidized species (Figure S3B). This analysis identified significant displacement along a single mode (mode 4, depicted in Figure S5A) corresponding to in-plane bending of the phenol, benzimidazole, and pyridine moieties upon oxidation. Analogous to the E1PT system, this mode dominates the inner-sphere reorganization for the overall multiproton PCET reaction. By further analogy with the E1PT system, we expect that displacement along the proton donor–acceptor modes and proton donor–hydrogen stretch modes will also be significant for the proton transfer reactions.

We identified two donor–acceptor modes for the phenolic proton transfer, alongside the phenolic proton stretch mode (Figures S5B, S5C, and S5D), that appear to play significant roles. The analysis suggests that the proton donor–acceptor displacements associated with the two proton transfers are anticorrelated for the more important mode of this type and correlated for the less important mode of this type (Figures S5 and S6). The anticorrelation of the two proton donor–acceptor motions for the more important mode may be one of the causes for the observed asynchronicity of the two proton transfers. A related observation regarding this asynchronicity is that the proton donor–acceptor distance is typically shorter at the interface exhibiting proton transfer than at the other interface during the first proton transfer reaction (Figure S7). Furthermore, limiting our analysis to the trajectories that exhibit phenolic proton transfer first, we observe that, analogous to the E1PT system, the phenolic proton transfer is associated with simultaneous displacements along the proton donor–acceptor mode(s) and the phenolic proton stretch mode (Figure S6). In the general case, however, the single and double proton transfers in the E2PT system exhibit a complex coupling that is not as straightforward to disentangle, but the same fundamental principles apply.

Conclusion

Herein, we report a first-principles nonequilibrium MD study of single and double proton transfer in biomimetic BIP molecules initiated by electrochemical or photochemical oxidation. For the E1PT system, an ANN was used to identify important correlations between molecular vibrational modes and the time to proton transfer. The ANN identified three vibrational modes as significant. The most important mode is a slow in-plane bending mode that dominates the overall inner-sphere reorganization associated with PCET. Another important mode is the proton donor–acceptor motion, corresponding to a period of ∼100 fs, which is responsible for the multimodality and apparent vibrational coherence observed in the proton transfer times because the proton donor–acceptor distance must be short enough to allow proton transfer. The third important mode is the significantly higher-frequency proton donor–hydrogen stretching mode. The donor–hydrogen distance must be sufficiently long simultaneously with a sufficiently short donor–acceptor distance to allow proton transfer. An analysis of the ensemble of trajectories confirms the physical significance of all three of these modes identified by the ANN. Analogous modes were found to be important for the phenolic proton transfer in the E2PT system. Moreover, the significance of anticorrelated proton donor-acceptor motions was implicated as a cause of the asynchronicity of the two proton transfers.

These simulations predict that both the E1PT and E2PT systems proceed predominantly by a concerted but asynchronous mechanism. In both cases, the first proton transfer occurs ∼100 fs after the electron transfer that would be induced electrochemically or photochemically. For the E2PT system, typically the second proton transfer occurs nearly 600 fs after the initial phenolic proton transfer; however, a small number of trajectories exhibited the reverse order or a synchronous process. The ultrafast time scale of these processes suggests that there is no thermodynamically stable intermediate corresponding to only electron transfer in either system or only single proton transfer in the E2PT system. Thus, on the time scale of electrochemical experiments, which are performed at quasi-equilibrium, the process will be considered to be concerted, consistent with prior spectroelectrochemical experiments and thermodynamic calculations of proton-coupled redox potentials. If these BIP systems are attached to photosensitizers, however, ultrafast two-dimensional spectroscopy may be able to detect the lag between electron transfer and proton transfer or between the two proton transfers in multiproton PCET reactions. In these cases, the electron and proton transfers would need to be detected on the sub-100 fs time scale. Moreover, E3PT and E4PT systems are expected to show similar concerted but asynchronous mechanisms, albeit with greater complexity in the coupling among the molecular vibrational modes.

The specific delays between electron transfer and the first proton transfer and between the multiple proton transfers can be tuned by altering the terminal proton acceptor and adding electron donating and withdrawing substituents along the proton transfer pathway. From a practical perspective, changing the fundamental mechanism and time scales of PCET reactions may be important in systems where multiple potential proton transfer pathways are involved, such as in photosystem II.50 In these cases, changing the mechanism to be more or less asynchronous and varying the delay times between proton transfer steps could help favor one pathway over another. Further theoretical and experimental studies of these systems, particularly on the ultrafast time scale, will be of great interest to disentangle the complex coupling between electron and proton transfer steps in multiproton PCET processes.

Acknowledgments

We thank Tom and Ana Moore for useful discussions about these systems. This material is based upon work supported by the Air Force Office of Scientific Research (AFOSR) under AFOSR Award No. FA9550-18-1-0134. This work used the Extreme Science and Engineering Discovery Environment (XSEDE),51 which is supported by National Science Foundation under grant number ACI-1548562. Specifically, this work used Comet at the San Diego Supercomputer Center through allocation TG-MCB120097.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acscentsci.0c00756.

  • Additional computational details, impact of the definition of proton transfer on proton transfer times for the E1PT system, hyperparameter search for the artificial neural network, permutation importance and Shapley additive explanations, geometric differences between oxidized and neutral species, relationship between proton stretch and donor–acceptor distance for E1PT and E2PT molecules at the time of proton transfer, important modes for phenolic proton transfer in the E2PT system, relationship between proton donor–acceptor distances for the E2PT system, and optimized geometries for the neutral and oxidized species in the E1PT and E2PT systems (PDF)

Animations of the proton transfer in the E1PT and E2PT systems, described in Figure 2, are available as video files in the HTML version of the manuscript.

The authors declare no competing financial interest.

Supplementary Material

oc0c00756_si_001.pdf (1.3MB, pdf)

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