Tipping the Balance between Concerted versus Sequential Proton

of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States. Inorg. Chem. , 2016, 55 (3),...
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Tipping the Balance between Concerted versus Sequential ProtonCoupled Electron Transfer Joshua S. Kretchmer and Thomas F. Miller III* Department of Chemistry and ChemicalEngineering, California Institute of Technology, Pasadena, California 91125, United States S Supporting Information *

ABSTRACT: We use quantized molecular dynamics simulations to investigate the competition between concerted and sequential proton-coupled electron-transfer (PCET) reaction mechanisms in inorganic catalysts. By analyzing reactive nonadiabatic PCET trajectories and computing both concerted and sequential rate constants, we characterize various molecular features that govern inorganic PCET reactions, including the solvent polarity, ligandmediated electron−proton interactions, and intrinsic protontransfer (PT) energy barrier. Using atomistic simulations with over 1200 atoms, we find that the symmetric iron biimidazoline system is extremely biased toward the concerted mechanism because of the strong ligand-mediated electron−proton interaction and the short PT distance. However, by investigating systembath models in which electron−proton interactions are shielded, which are representative of ruthenium terpyridylbenzoates and iron (tetraphenylporphyrin)benzoates, we predict that a crossover between the concerted and sequential PCET mechanisms may be possible either by increasing the polarity of the solvent or by increasing the intrinsic PT energy barrier. In addition, we predict the possibility of a crossover in the PCET mechanism by directly varying the strength of the ligand-mediated electron−proton interactions. The results presented here reveal new strategies for altering the competition between the competing PCET mechanisms and design principles for controlling PCET in catalytic systems.

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INTRODUCTION Proton-coupled electron transfer (PCET), in which both an electron and a proton undergo reactive transfer, is ubiquitous throughout chemistry and biology,1−4 including such examples as the tyrosine oxidation step of photosystem II5,6 and the proton pumping mechanism of cytochrome c oxidase.7,8 PCET reactions are typically categorized into two groups, concerted and sequential processes, based on the chronology of the electron- (ET) and proton-transfer (PT) events.9−12 Unraveling the factors that determine whether a given system follows a concerted versus sequential PCET mechanism is a significant experimental and theoretical challenge; in many cases, simply determining the dominant PCET mechanism is intractable.2,13 Consequently, it is only recently that there have been attempts to design chemical systems that follow a specific PCET mechanism.14,15 To inform this effort and to provide strategies for controlling charge-transfer reactions in inorganic catalysis, an improved understanding of the factors that govern the competition between PCET mechanisms is needed. Toward this goal, we utilize quantized molecular dynamics to explore various means by which the dominant mechanism may be altered in the symmetric PCET reaction of iron biimidazoline (Figure 1) and in other model systems for inorganic catalysis. The competition among sequential and concerted PCET mechanisms is typically illustrated in terms of a square scheme,9−12 as pictured in Figure 1b for the case of the PCET reaction in iron biimidazoline. The two sequential mechanisms, which exhibit either PT or ET as an initial step, © XXXX American Chemical Society

are indicated along the edges of the square scheme and involve the formation of a charge-separated intermediate; the concerted mechanism is indicated along the diagonal of the square scheme and bypasses the formation of the charge-separated intermediate. A factor that is commonly attributed to the relative favorability of the concerted PCET mechanism is that it does not incur the thermodynamic penalty associated with the formation of a charge-separated intermediate (Figure 1b).2,4,10,11 This has motivated the design of systems that exhibit a specific PCET reaction by controlling the driving force for the formation of the charge-separated intermediate through direct chemical modifications of the reactive species14 or by employing different combinations of oxidants and bases.15 However, other factors, such as interactions with the surrounding solvent and the strength of the nonadiabatic coupling, can contribute to the PCET mechanism.4,10,16−25 In this study, we thus aim to fully characterize the factors that govern the competing PCET mechanisms and, in doing so, identify strategies to alter the dominant mechanism. We begin by investigating the physical interactions that are found to contribute to the favorability of the concerted mechanism for the PCET self-exchange reaction involving iron biimidazoline complexes (Figure 1). Iron biimidazoline acts as a model for iron-containing enzymes that utilize PCET reactions Received: August 10, 2015

A

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principles for control of the PCET reactions in catalytic applications.

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METHODOLOGY In this section, we provide the RPMD equations of motion and summarize the key features of the systems that are used to investigate PCET reaction mechanisms. A full description of the RPMD methodology and the system details can be found in sections 1 and 2 in the Supporting Information (SI). The RPMD equations of motion used to simulate the PCET dynamics involving the quantized transferring electron, the quantized transferring proton, and the N classical particles associated with the remaining atoms in the system are36,45 1 ∇q(α) v̇ (eα) = ωne 2(q (eα + 1) + q (eα − 1) − 2q (eα)) − me e α − k)(1/ nep) + 1) U (q (eα), q(( , Q) p

(1)

v̇ (pγ ) = ωnp 2(q(pγ + 1) + q(pγ − 1) − 2q(pγ )) 1 − mp

nep

∑ ∇q l=1

(γ ) p

γ − 1)nep + l) , q(pγ ), Q) U (q (( e

(2)

and 1 V̇j = − neMj

np

nep

∑ ∑ ∇Q U (q((e γ− 1)n

ep + l)

j

γ=1 l=1

, q(pγ ), Q)

(3)

where ne is the number of ring-polymer beads for the transferring electron, me is the mass for the electron, and q(α) e and v(α) e are the respective position and velocity vectors for the αth ring-polymer bead of the electron; the corresponding quantities for the transferring proton are indicated using subscript “p”. Because of its more strongly quantized nature, the transferring electron is described using a larger number of ring-polymer beads than the transferring proton; thus, in eqs 1−3, it is assumed that nep = ne/np is an integer number, and

Figure 1. (a) Symmetric PCET exchange reaction in iron biimidazoline. (b) PCET scheme depicting the sequential and concerted PCET mechanisms in iron biimidazoline. The sequential mechanism proceeds through the charge-separated intermediate along the horizontal and vertical edges of the schematic; the concerted mechanism proceeds along the diagonal, bypassing formation of the charge-separated intermediate. (c) Structures of the reactant, product, and intermediate species associated with the PCET reaction.

⎢ α − 1⎥ ⎥ k = α − nep⎢ ⎢⎣ nep ⎥⎦

(4)

where ⌊...⌋ denotes the floor function. The cyclic constraint of (ne) (np) the ring polymer is satisfied via q(0) and q(0) e = qe p = qp , and the intrabead harmonic frequencies are ωne = ne/βℏ and ωnp = np/βℏ, where β = 1/kBT is the inverse temperature. The position, velocity, and mass for the jth classical degree of freedom are given by Qj, Vj, and Mj, respectively, and Q = {Q1, ..., QN}. The potential energy function of the system is given by U(qe,qp,Q), such that the interaction of the transferring electron with its environment is described using a one-electron pseudopotential.46 Such a pseudopotential can be employed for the description of the PCET reaction because the iron centers exhibit the high-spin configuration and the combined spin state of the iron centers does not change throughout the PCET reaction; the transferred down-spin electron is treated as a distinguishable electron in the mean-field pseudopotential of the remaining up-spin electrons in the metal d orbitals, as has been previously done in the treatment of high-spin iron aquo complexes.47 The competing PCET mechanisms for the self-exchange reaction in iron biimidazoline (Figure 1) are studied using a fully atomistic representation consisting of over 1200 atoms

during the catalytic cycle,26−28 such as the C−H bond activation in lipoxygenase enzymes.29−32 By then varying these physical interactions in both iron biimidazoline and other model inorganic catalysts with a weaker ligand-mediated electron−proton interaction, such as ruthenium terpyridylbenzoates33,34 and iron (tetraphenylporphyrin)benzoates,13 we find that the dominant mechanism for PCET switches between the sequential and concerted pathways. We investigate the competing PCET mechanisms using the ring-polymer molecular dynamics (RPMD) method.35,36 RPMD is a Feynman path-integral method that enables inclusion of electronic and nuclear quantum effects in classical molecular dynamics simulations and has been shown to accurately describe PCET kinetics and to allow for the direct comparison of concerted and sequential PCET mechanisms.16 The accuracy of RPMD follows from the method’s exact description of statistical fluctuations35,37−39 and its formal connection to semiclassical instanton theory.40−44 The current work investigates the various means by which the dominant PCET mechanism can be altered in inorganic reactions, and it suggests new design B

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reproduce the driving force for the first step in the sequential mechanism, (ii) the strength of the Coulombic interaction between the electron and the solvent is modified to reproduce the reorganization energy for the isolated ET reaction in iron biimidazoline,27 and (iii) the point charges assigned to the iron atoms in both complexes are chosen such that the RPMD rate for the concerted self-exchange PCET mechanism in iron biimidazoline is consistent with experiment,27 while still maintaining the stability of the electron ring polymer on the iron atoms. It is important to note that none of the parameters in the atomistic representation are chosen to manipulate or fit the relative favorability between the concerted and sequential mechanisms. In addition to the fully atomistic representation, we also employ a previously developed system-bath model for condensed-phase PCET,16 which allows for examination of a broader class of systems beyond iron biimidazoline. The system-bath model captures the essential physics of PCET reactions and includes a quantized electron and a quantized proton, both of which interact with a single classical coordinate that models the collective motion of the solvent. This solvent mode is further coupled to a dissipative classical harmonic bath.

(Figure 2). The dynamics of the transferring proton and electron are quantized using RPMD, while the dynamics of the

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RESULTS AND DISCUSSION In presenting the results, we first analyze the competing sequential and concerted PCET mechanisms in the selfexchange reaction in iron biimidazoline (Figure 1) to identify the physical interactions that govern the favorability of the concerted mechanism. We then investigate the effects of modulating the strength of these interactions to probe strategies for altering the dominant PCET mechanism. Concerted versus Sequential PCET in Iron Biimidazoline. We begin by examining the competing PCET reaction mechanisms for the self-exchange reaction in iron biimidazoline solvated in acetonitrile. To illustrate the energetic landscape associated with the concerted and sequential mechanisms for PCET in this system, Figure 3 presents the 2D free-energy (FE) profile along with representative samples from the ensemble of reactive trajectories for this system along a collective variable that tracks the motion of the electron, θe, and a collective variable that tracks the motion of the proton,

Figure 2. Snapshots of the atomistic representation in the (a) transition state, (b) reactant well, and (c) product well for the PCET reaction in iron biimidazoline. Two of the biimidizoline complexes are shown in gray, with the iron centers in orange and the nitrogen atoms that act as proton donors/acceptors in green; the transferring electron is shown in blue, and the transferring proton is shown in red. Both the transferring proton and electron are illustrated in the representation of the path-integral beads, which are used in the RPMD method to quantize the description of the transferring electron and proton. The explicit acetonitrile molecules are shown in white. The transition state (a) is characterized by a configuration in which the electron resides on both the donor and acceptor iron atoms, and the proton is equidistant from both nitrogen atoms; for the reactant (b) and product (c), the electron and proton are associated with the donor and acceptor complexes, respectively.

remaining heavy atoms are described in the classical limit (eqs 1−3); quantization of all other degrees of freedom is straightforward using RPMD,48−50 although the current work focuses only on the effect of quantizing the transferring particles. We neglect the explicit inclusion of the perchlorate counterions to model the dilute solution of iron biimidazoline present in the experimental setup. The potential energy function, U(qe,qp,Q), of the iron biimidazoline system includes pairwise Coulombic and Lennard-Jones interactions, for which the atomic charges of the biimidazoline ligands depend on the position of the proton, and an additional interaction that models the hydrogen bond between the proton and the two iron biimidazoline complexes. The majority of the parameters used to define these interactions are obtained from previously developed force fields.51,52 However, a few parameters in the interaction potentials are fit to experimental data in order to provide a realistic description of iron biimidazoline. Specifically, (i) the strength of the Coulombic interaction between the electron and the iron biimidazoline ligands is modified to

Figure 3. Reactive RPMD trajectories revealing distinct concerted (red) and sequential ET−PT (orange) and PT−ET (purple) reaction mechanisms for the PCET reaction in the atomistic representation of iron biimidazoline. The trajectories are projected onto the FE surface in the electron bead-count coordinate, θe, and the proton collective variable, θp, with contour lines indicating FE increments of 1 kcal/mol. Snapshots of the RPMD simulations from each of the four PCET basins of stability are also shown. C

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Inorganic Chemistry θp.16,47 The electron collective variable θe is termed an electron “bead-count” coordinate and reports on the fraction of ringpolymer beads located on the iron atom associated with either the donor or acceptor complex (eq S43). The electron beadcount coordinate takes a value of θe = −1 when the electron is localized on the donor iron atom and a value of θe = 1 when the electron is localized on the acceptor iron atom, as can be seen in the snapshots in Figure 3. The proton collective variable θp is defined as the difference between the distances that separate the ring-polymer centroid and the two nitrogen atoms participating in the hydrogen bond (eq S46). The proton collective variable takes a value of θp = −0.5 Å when the proton is bonded to the donor nitrogen atom and a value of θp = 0.5 Å when the proton is bonded to the acceptor nitrogen atom. The FE profile in Figure 3 exhibits four distinct minima corresponding to the reactant, product, and charge-separated intermediate states depicted in the PCET square scheme (Figure 1b). Snapshots of the RPMD simulations of the atomistic representation in each minimum are also illustrated in Figure 3. Distinct channels on the FE surface connect the various basins of stability. Overlaid on the FE surface are representative samples from the ensemble of reactive RPMD trajectories for PCET. The trajectories cluster within the channels, illustrating the competing sequential (purple and orange) and concerted (red) reaction mechanisms. These trajectories can be used for the direct comparison and calculation of the rates for the competing PCET mechanisms, illustrating an important advantage of the RPMD method for the investigation of PCET reactions. Similar results have been observed for the investigation of PCET reactions in model systems.16 We now show that the RPMD simulations predict that the concerted PCET mechanism is dominant for the self-exchange reaction in iron biimidazoline, in agreement with previous experimental results.27 To determine the dominant mechanism, we compute the RPMD reaction rates for both the concerted and sequential processes. The bimolecular thermal reaction rate is obtained through the product of the RPMD unimolecular rate associated with the dimerized iron biimidazoline precursor complex and the equilibrium constant for the formation of this precursor complex.13,28,53,54 The unimolecular thermal reaction rate associated with the precursor complex is calculated using RPMD from the product of the Boltzmann-weighted activation FE and the reaction transmission coefficient.35,55,56 Full details of the rate calculations can be found in section 3.1 in the SI. The FE profile used for calculation of the concerted PCET reaction rate, which corresponds to the channel along the diagonal in the 2D FE profile in Figure 3, is presented in Figure 4a. As was shown previously for concerted PCET in systembath models,16 the FE profile exhibits a sharp rise as a function of θ e associated with the formation of ring-polymer configurations in which the electron spans the two iron sites (Figure 4a, inset), followed by a more gradual change associated with reorientation of the solvent. The RPMD rate for the bimolecular concerted PCET reaction is calculated to be k = (5 ± 2) × 102 M−1 s−1; as expected, the RPMD rate is in reasonable agreement with the experimentally measured PCET rate,27 k = (5.8 ± 0.6) × 103 M−1 s−1, particularly given the sensitivity of the calculated absolute rates to the details of the potential energy surface. Although we have chosen parameters to recover the concerted PCET rate, as described in the Methodology section, no parameters were chosen to fit the

Figure 4. 1D FE profile in the electron bead-count coordinate, F(θe), associated with (a) the concerted PCET reaction and (b) the ET reaction in the sequential ET followed by the PT mechanism. (c) 1D FE profile in the proton collective variable, F(θp), associated with the PT reaction in the sequential ET followed by the PT mechanism.

sequential PCET rate and thus the relative favorability of the two mechanisms as discussed below. For a general PCET reaction, the two sequential PCET mechanisms corresponding to either ET followed by PT (orange trajectories, Figure 3) or PT followed by ET (purple trajectories, Figure 3) will exhibit different rates. However, because of the symmetry of the PCET reaction considered here, the rates for these two sequential PCET mechanisms are necessarily identical. Consequently, we focus solely on the calculation of the rate for the mechanism corresponding to sequential ET followed by PT. Parts b and c of Figure 4 present the 1D FE profiles along θe and θp associated with the ET Fe II(H 2bim) + Fe III(Hbim) ET

HooI Fe III(H 2bim) + Fe II(Hbim)

(5)

and PT Fe III(H 2bim) + Fe II(Hbim) PT

⇌ Fe III(Hbim) + Fe II(H 2bim)

(6)

reactions in the sequential ET followed by the PT mechanism, respectively. The FE barrier along θp for the PT reaction is on the order of the thermal energy kBT at the experimental D

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gap takes a value of ΔU = ±5 kcal/mol when the proton and electron are associated with the donor or acceptor, respectively. The FE profile in Figure 5 exhibits two basins of stability corresponding to the concerted PCET reactant and product species. Overlaid on the FE surface are representative samples from the ensemble of reactive RPMD trajectories (red) for the concerted PCET reaction. The RPMD trajectories exhibit a Marcus-type solvent-gating mechanism, which has been seen previously for PCET16 reactions in system-bath models and which is assumed in the derivation of many PCET rate theories.4,57−59 The solvent-gating mechanism is illustrated, from left to right, by the black arrows and the snapshots of the atomistic representation along the dynamical trajectories in Figure 5. The mechanism is characterized by an initial solvent fluctuation that brings the system to configurations at which the vibronic diabats for the transfer of electron and proton are nearly degenerate, ΔU ≈ 0, followed by the rapid transfer of both the electron and proton. Finally, the solvent relaxes, trapping the electron and proton on the acceptor molecule. Figure 6 illustrates how the solvent fluctuations observed in Figure 5 couple to the reactive system and provide an

temperature of 300 K (Figure 4c). It follows that the chargeseparated intermediate is only short-lived, and the sequential ET followed by the PT mechanism proceeds via rate-limiting ET, such that the rate constant for the sequential PCET mechanism is given by the forward rate for the sequential ET reaction. The RPMD rate for the bimolecular sequential PCET is thus calculated to be k = (1.2 ± 0.5) × 10−1 M−1 s−1. A comparison of the RPMD reaction rates for the concerted and sequential PCET mechanisms reveals that the concerted mechanism proceeds with a significantly faster rate and is the dominant mechanism for the self-exchange reaction in iron biimidazoline. These simulation results are consistent with previous experimental analysis27 validating the employed simulation model; none of the parameters of the atomistic representation are chosen to fit the relative favorability between the concerted and sequential mechanisms. Having confirmed that the concerted mechanism is favored in iron biimidazoline, we now focus on the physical interactions that determine the PCET mechanism in iron biimidazoline. We begin by investigating the thermodynamic penalty associated with the formation of the charge-separated intermediate. By comparing the FE profiles in Figure 4a and Figure 4b, it is clear that the driving force for the initial ET step in the sequential mechanism is significantly higher in energy (11 kcal/mol) than the thermal energy, whereas the driving force for the concerted mechanism is zero because of the symmetry of the reaction. The energetic cost of forming the charge-separated intermediate thus biases the PCET mechanism in iron biimidazoline toward the concerted process. We now investigate the role of solvent interactions in determining the PCET mechanism. Figure 5 presents the 2D

Figure 6. Total dipole of the reactive complex in the direction connecting the two iron atoms, dtot(t), (black), as well as the individual components arising from the transferring electron, de(t) (blue), transferring proton, dp(t) (red), and charge distribution of the biimidazoline ligand, dL(t) (green), for the concerted PCET reaction in the atomistic representation of iron biimidazoline.

interesting additional mechanistic basis for the favorability of the concerted mechanism in iron biimidazoline. Figure 6 presents the total dipole of the reactive complex in the direction connecting the two iron atoms, dtot(t), as a function of time. The total dipole can be decomposed into contributions arising from the position of the transferring electron, de(t), the position of the transferring proton, dp(t), and the charge distribution of the biimidazoline ligands, dL(t), also pictured in Figure 6. It is important to note that the charge distribution of the biimidazoline ligands depends on the position of the proton. The dipoles are calculated as a nonequilibrium average over the full ensemble of time-evolved reactive RPMD trajectories; see the SI, sections 3.1.1 and 3.1.9.60−62 The trajectories are averaged such that time zero corresponds to the moment when each trajectory crosses the dividing surface defined by when the electron and proton are evenly shared between the donor and acceptor species; negative and positive time then correspond to configurations on the reactant or product side of the dividing surface, respectively. Figure 6 shows that the contribution to the total dipole arising from the proton, dp(t), is small at all times because of the small distance over which the proton transfers. In

Figure 5. Reactive RPMD trajectories (red) for the concerted PCET reaction in the atomistic representation of iron biimidazoline revealing a Marcus-type solvent-gating mechanism indicated by the black arrows. The trajectories are projected onto the FE surface in the electron bead-count coordinate θe and the concerted PCET energy gap coordinate ΔU, with contour lines indicating FE increments of 2 kcal/mol. Snapshots of the RPMD simulations from the reactant, transition state, and product regions of the concerted PCET reaction are also shown.

FE profile along the electron bead-count coordinate θe and the concerted PCET energy gap coordinate ΔU computed for the concerted pathway, section 3.1.5 in the SI. The concerted PCET energy gap coordinate (eq S.48) describes the energy difference between the vibronic state corresponding to the electron and proton associated with the donor complex and the vibronic state corresponding to the electron and proton associated with the acceptor complex.4,57−59 The PCET energy E

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Figure 7. (a) Unimolecular rates, (b) solvent reorganization energy, (c) driving force for the sequential ET prior to the PT reaction, and (d) Marcus barrier associated with the concerted (black) and sequential (red) PCET mechanisms in iron biimidazoline as a function of the polarity of the solvent.

comparison, the contributions arising from the electron, de(t), and charge distribution of the ligands, dL(t), are large at all times. Furthermore, the contributions arising from the electron and charge distribution of the ligands are opposite in sign and switch on similar time scales. This cancellation of charge leads to the magnitude of the total dipole being smaller than the contribution arising from the electron at all times. Consequently, the degree to which the polar solvent couples to the reactive species during the concerted PCET reaction is reduced by a cancellation of charge between the transferring electron and the time-dependent charge distribution of the biimidazoline ligands. A similar reduction in the solvent reorganization energy for the concerted PCET mechanism was seen in previous theoretical work utilizing Fermi Golden-Rule rate expressions for analysis of the PCET, isolated ET, and isolated PT reactions in iron biimidazoline.28 The presence of the solvent thus provides a second physical interaction that leads to the favorability of the concerted mechanism for the selfexchange reaction in iron biimidazoline. In this section, we have utilized the RPMD method to investigate the competing sequential and concerted PCET mechanisms in the self-exchange reaction in iron biimidazoline. We have illustrated two physical interactions that lead to the favorability of the concerted mechanism: (i) the concerted mechanism avoids the high-energy charge-separated intermediate formed during the sequential mechanism and (ii) the concerted mechanism mitigates the high-energy barrier associated with solvent reorganization. With an understanding of some of the physical interactions that contribute to the favorability of the concerted mechanism, we can now begin to investigate how modulating these interactions alters the relative favorability between the two PCET mechanisms, as discussed in the following section. Controlling the Dominant PCET Mechanism. Having demonstrated the physical interactions that lead to the favorability of the concerted mechanism for the self-exchange reaction in iron biimidazoline, we now modulate these interactions to identify new strategies to alter the dominant PCET mechanism, first in iron biimidazoline and then in other

model inorganic systems, such as ruthenium terpyridylbenzoates and iron (tetraphenylporphyrin)benzoates. Competing PCET Mechanisms in Iron Biimidazoline. We begin by investigating the effect of solvent polarity on the relative favorability between the competing PCET mechanisms during the self-exchange reaction in iron biimidazoline. Figure 7a presents the rates for the concerted (black) and sequential (red) mechanisms in iron biimidazoline as a function of the polarity of the solvent. The polarity of the solvent in the fully atomistic representation of iron biimidazoline is quantified through the permanent dipole moment associated with the atomic charges on the acetonitrile molecule obtained from the molecular mechanics force field. Although increasing the polarity of the solvent does not lead to a crossover in the dominant PCET mechanism for iron biimidazoline, the relative favorability between the concerted and sequential PCET mechanisms is clearly affected by the solvent polarity; as the solvent polarity is increased, the sequential mechanism becomes more favorable in comparison to the concerted mechanism. The dependence of the PCET rates on the solvent polarity in Figure 7a can be understood in terms of the contributions to the Marcus barrier associated with the solvent reorganization for both the sequential and concerted PCET mechanisms4,59,63,64 ΔG⧧ =

(λ 0 + ΔG°)2 4λ

(7)

where λ0 is the solvent reorganization energy associated with either the concerted PCET reaction or the sequential ET reaction prior to PT; as before, it is sufficient to analyze only the sequential ET reaction because the sequential PCET mechanism occurs via rate-limiting ET. We thus investigate the solvent dependence of both the solvent reorganization energy and driving force associated with the concerted PCET and sequential ET reaction prior to PT. The reorganization energies associated with the concerted PCET (black) and sequential ET (red) reactions are presented in Figure 7b. The reorganization energy associated with both mechanisms increases with increasing solvent polarity, but F

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and iron (tetraphenylporphyrin)benzoates,13 in which the ligand-mediated interaction is weakened because of the spatial separation of the redox and basic sites. In these systems, the ligand-mediated electron−proton interaction is characterized by shifts in the redox potential of the molecules upon protonation. We utilize two basic models to simulate the behavior of such systems. In the first, we decrease the strength of the Coulombic interaction between the electron and biimidazoline ligands in the fully atomistic representation to mimic a reduced ligandmediated electron−proton interaction, as described in section 2.1 of the SI; recall that the charge distribution of the biimidazoline ligands depends on the position of the proton. In the second, we develop a system-bath model for condensedphase PCET that allows for direct variation of the physical interactions that impact the PCET reaction, as described in section 2.2 of the SI. In the system-bath model, we specifically modulate (i) the solvent polarity, (ii) the strength of the ligandmediated electron−proton interaction, and (iii) the intrinsic PT energy barrier. As discussed below, we do not present results that involve varying the intrinsic ET energy barrier because this quantity is found to have little effect on the competition between the concerted and sequential mechanisms for systems in which the sequential mechanism involves rate-limiting ET, such as those investigated in this work. Experimentally, the ligand-mediated electron−proton interaction can be varied through spatial separation of the redox and basic sites13,33,34 or through direct chemical modifications of the reactive species.2,14 It is worth noting that in many cases varying the ET distance leads to a spatial separation of the redox and basic sites. The intrinsic PT energy barrier can be varied by changing the PT distance or the chemical species participating in the hydrogen bond.65,66 Figure 8 presents the rates for the concerted (black) and sequential (red) PCET mechanisms as a function of the solvent

because of the cancellation of charge during the concerted PCET reaction, the solvent reorganization energy for concerted PCET is always lower than that for the sequential PCET mechanism and increases more slowly. In principle, this behavior would lead to a favorability of the concerted mechanism at high solvent polarity. Figure 7c presents the driving force for the concerted PCET (black) and sequential ET (red) reactions as a function of the solvent polarity. The driving force for the sequential ET reaction is observed to decrease with increasing solvent polarity because of an increased stabilization of the charge-separated intermediate formed following the sequential ET reaction; the driving force associated with the concerted PCET reaction is always zero because the concerted reaction is symmetric. This behavior would lead to a favorability of the sequential mechanism at high solvent polarity, compensating for the increased reorganization energy. Figure 7d presents the total Marcus barrier associated with the concerted (black) and sequential (red) PCET mechanisms, which accounts for both the solvent reorganization energy and the driving force. The Marcus barrier associated with the sequential mechanism is observed to decrease with increasing solvent polarity because of the decrease in the driving force; the Marcus barrier associated with the concerted mechanism is observed to increase with increasing solvent polarity because of the increase in the solvent reorganization energy. The trend in the Marcus barrier explains the increase in the favorability of the sequential mechanism in comparison to the concerted mechanism in iron biimidazoline with increasing solvent polarity. The lack of a crossover between the PCET mechanisms, as shown in Figure 7, illustrates the robustness of the concerted PCET mechanism in iron biimidazoline with respect to changes in the solvent polarity. A major contribution to the robustness of the concerted PCET mechanism in iron biimidazoline is the strength of the interaction between the electron and proton, which leads to the large driving force for the sequential ET reaction, as observed in Figures 4b and 7c. In addition, the PT distance in iron biimidazoline is short; the distance between the nitrogen atoms participating in the hydrogen bond is 2.67 Å in the crystal structure,27 and the total distance the proton transfers in the simulations is ∼0.5 Å. Consequently, the concerted mechanism does not incur a large energetic penalty associated with the simultaneous transfer of both the electron and proton.4,10,28 An increase in the solvent polarity is insufficient to overcome these simultaneous factors in iron biimidazoline and cause a crossover between the PCET mechanisms. Competing PCET Mechanisms beyond Iron Biimidazoline. Utilizing the insights from the self-exchange reaction in iron biimidazoline, we now turn to systems with a weakened interaction between the electron and proton to potentially observe a transition between the concerted and sequential mechanisms. The electron−proton interaction is governed mainly by the coupled electronic structure between the redox and proton donor/acceptor regions of the molecule and not by the direct Coulomb interaction between the electron and proton. In this way, the presence of the ligand mediates the interaction between the electron and proton, and we thus refer to this interaction as the ligand-mediated electron−proton interaction in the remainder of the manuscript. Examples of such systems with a weakened ligand-mediated electron− proton interaction include ruthenium terpyridylbenzoates33,34

Figure 8. Unimolecular rates calculated using RPMD for the concerted (black) and sequential (red) PCET reactions for systems with weakened ligand-mediated electron−proton interactions as a function of the polarity of the solvent.

polarity for the modified iron biimidazoline system. The same qualitative trend as that in Figure 7a is observed, in which the relative favorability of the sequential mechanism increases with increasing solvent polarity. However, for this system we observe that the rates for the concerted and sequential PCET mechanisms become equivalent in highly polar solvents, illustrating that variations in the solvation environment provide a viable strategy for controlling the dominant PCET mechanism. G

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Inorganic Chemistry We now extend our analysis of the competition between concerted and sequential PCET by examining a system-bath model for condensed-phase PCET in such systems as the benzoates discussed above. Figure 9a presents the rates for the

corresponds to about an 11.5 kcal/mol change in the driving force for the initial ET reaction in the sequential mechanism. The figure illustrates that a crossover between the sequential and concerted PCET mechanisms can be obtained through a direct change in the strength of the ligand-mediated electron− proton interaction. The sequential mechanism dominates in systems with a weak interaction, such as the benzoates; the concerted mechanism dominates in systems with a strong interaction, such as iron biimidazoline. Such a result is in agreement with previous experimental work, which designed systems with a specific PCET mechanism by controlling the driving force for the formation of the charge-separated intermediate.14,15 Last, Figure 9c presents rates for the concerted (black) and sequential (red) PCET mechanisms as a function of the height of the PT energy barrier, V0; the presented range of V0 corresponds to about a 0.2 Å change in the PT distance. A clear crossover between PCET mechanisms is observed, with the concerted mechanism favored for small values of the PT barrier and the sequential mechanism favored for large values, as has been suggested by previous theoretical work.17 The rate for the sequential mechanism is largely independent of the height of the PT energy barrier because the sequential mechanism proceeds through a rate-limiting ET mechanism; deviations from such behavior in Figure 9c are due to changes in the equilibrium electron−proton distance arising from the varying PT barrier. On the other hand, the concerted mechanism depends strongly on the barrier height because of the simultaneous transfer of both the electron and proton. Consequently, systems that exhibit short hydrogen bonds are biased toward the concerted mechanism, as was found for the iron biimidazoline system. The observed trend holds for systems in which the sequential mechanism involves a ratelimiting ET step, such as those investigated in this work. If, on the other hand, the sequential mechanism involves a ratelimiting PT step, such as that for systems with a large PT distance or a short ET distance, a crossover between PCET mechanisms may be observed upon variation of the intrinsic ET energy barrier. The results presented in this section illustrate novel strategies for altering the dominant PCET mechanism through modifications to the reactive species or surrounding environment. In particular, we have provided evidence for a clear crossover between concerted and sequential PCET mechanisms in systems with a weakened ligand-mediated electron− proton interaction through variation of either the solvent polarity, the strength of the ligand-mediated electron−proton interaction, or the height of the intrinsic PT energy barrier.

Figure 9. Rates for a system-bath model of PCET calculated using RPMD for the concerted (black) and sequential (red) mechanisms as a function of (a) the solvent polarity as defined for the system-bath model, (b) the strength of the electron−proton interaction, μep, and (c) the height of the intrinsic barrier for PT, V0. The solvent polarity is defined in terms of the interaction potential of the system-bath model, and the parameters μep and V0 appear in the interaction potential for the system-bath model, as defined in sections 2.2 and 3.3 of the SI; all three of these parameters are given in atomic units. The range of parameters is chosen to correspond to physically reasonable ranges of interactions as described in the main text.

concerted (black) and sequential (red) PCET mechanisms as a function of the solvent polarity as defined for the system-bath models, section 3.2 in the SI. We observe the same qualitative trend as that in Figures 7 and 8, validating the ability for the system-bath model to capture the essential physics governing PCET reactions. However, Figure 9a exhibits a distinct crossover between the PCET mechanisms; the concerted mechanism is the dominant PCET mechanism at low solvent polarity, while the sequential mechanism is the dominant PCET mechanism at high solvent polarity. By comparing Figures 7a and 8, we clearly observe that the strength of the ligand-mediated electron−proton interaction has a strong effect on the PCET mechanism. To further elucidate this point, Figure 9b presents the rates for the concerted (black) and sequential (red) PCET mechanisms as a function of the strength of the ligand-mediated interaction, μep, as defined in the system-bath model; the presented range of μep

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CONCLUSION In this work, we utilize quantized molecular dynamics simulations to examine the competition between concerted and sequential PCET reaction mechanisms in inorganic complexes. The results presented here provide a range of strategies for altering the competition between the PCET mechanisms and design principles for inorganic catalysts. Through analysis of the full ensemble of reactive PCET trajectories and calculation of both the sequential and concerted rate constants, we first characterize the various molecular features that contribute to the favorability of the concerted mechanism in the symmetric iron biimidazoline system. We find that the concerted mechanism during the selfexchange reaction in iron biimidazoline is extremely robust H

DOI: 10.1021/acs.inorgchem.5b01821 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry

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because of the strong ligand-mediated electron−proton interaction and the short PT distance, such that no investigated change in the solvent environment is sufficient to induce a crossover from the concerted to the sequential mechanism. However, for systems in which the ligand-mediated interaction is shielded, such as ruthenium terpyridylbenzoates33,34 and iron (tetraphenylporphyrin)benzoates,13 we predict a crossover between the sequential and concerted PCET mechanisms through variations of either the solvent polarity or the intrinsic PT energy barrier. Experimentally, the PT barrier can be varied through modulations of the PT distance or modifications of the chemical species involved in the hydrogen bond.65,66 In addition, we illustrate that a crossover between PCET mechanisms can be induced by varying the strength of the ligand-mediated electron−proton interaction, which may be experimentally achieved through variation of the spatial separation of the proton and electron donor/acceptor pairs13,33,34 or through direct chemical modification of the donor/acceptor species.2,14 Taken together, these results suggest a wider scope of the possible methods for the future design of novel PCET systems.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.5b01821. Detailed descriptions of the RPMD methodology, the simulation models employed, and the calculation details (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation (NSF) CAREER Award under Grant CHE-1057112 and the U.S. Department of Energy under Grant DE-SC0006598. Additionally, J.S.K. acknowledges support from an NSF Graduate Research Fellowship under Grant DGE-1144469, and T.F.M. acknowledges support from a Camille and Henry Dreyfus Foundation New Faculty Award and an Alfred P. Sloan Foundation Research Fellowship. Computing resources were provided by the National Energy Research Scientific Computing Center and the ASCR Leadership Computing Challenge Program.

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DOI: 10.1021/acs.inorgchem.5b01821 Inorg. Chem. XXXX, XXX, XXX−XXX