Tutorial on Computing Nonadiabatic Proton-Coupled Electron Transfer Rate Constants

Abstract

Proton-coupled electron transfer (PCET) is pervasive throughout chemistry, biology, and physics. Over the last few decades, we have developed a general theoretical formulation for PCET that includes the quantum mechanical effects of the electrons and transferring protons, including hydrogen tunneling, as well as the reorganization of the environment and the donor-acceptor fluctuations. Analytical rate constants have been derived in various well-defined regimes. This tutorial focuses on the vibronically nonadiabatic regime, in which a golden rule rate constant expression is applicable. The goal is to provide detailed instructions on how to compute the input quantities to this rate constant expression for PCET in molecules, proteins, and electrochemical systems. The required input quantities are the inner-sphere and outer-sphere reorganization energies, the diabatic proton potential energy profiles, the electronic coupling, the reaction free energy, and the proton donor-acceptor distance distribution function. Instructions on how to determine the degree of electron-proton nonadiabaticity, which is important for determining the form of the vibronic coupling, are also provided. Detailed examples are given for thermal enzymatic PCET, homogeneous molecular electrochemical PCET, photochemical molecular PCET, and heterogeneous electrochemical PCET. A Python-based package, pyPCET, for computing nonadiabatic PCET rate constants, along with example scripts, input data, output files, and detailed documentation are publicly available.

I. Introduction

Proton-coupled electron transfer (PCET) is ubiquitous in chemistry and biology.^1–10 This tutorial focuses on concerted PCET reactions, in which one electron and one proton transfer simultaneously without a thermodynamically stable intermediate. The extension to more complex PCET reactions involving multiple electrons or protons is straightforward and will be mentioned briefly. This tutorial describes the practical aspects of computing rate constants within the framework of our group’s PCET theory,^11–13 which treats the transferring proton(s) as well as all electrons quantum mechanically and describes the reaction in terms of nonadiabatic transitions between mixed electron-proton vibronic states. Instead of reviewing this theory,^{14, 15} we will focus on the calculation of the input quantities for the vibronically nonadiabatic PCET rate constant. As there are multiple ways to compute these rate constants, we will refer to applications to specific systems for the reader to find additional technical details. After explaining how to calculate the input quantities, we will explain how to analyze the electron-proton nonadiabaticity, which is important for determining the form of the vibronic coupling. In the final section of this tutorial, we will describe how the input quantities were computed for three different systems and will provide results and files for these three systems.

II. Vibronically nonadiabatic PCET rate constants

This tutorial focuses on the vibronically nonadiabatic rate constant. A PCET reaction is typically vibronically nonadiabatic when the vibronic coupling is much less than the thermal energy. The expression for PCET at proton donor-acceptor distance $R$ has the form¹²

$$k(R)=\sum _{\mu }{P}_{\mu }\sum _v\frac{{\left|V_{\mu v}\right|}^2}{\hslash }\sqrt{\frac{\pi }{\lambda k_{\text{B}}T}}\text{exp}\left[-\frac{{\left(\Delta G_{\mu v}^0+\lambda \right)}^2}{4\lambda k_{\text{B}}T}\right]$$ (1)

Here $k(R)$ is the PCET rate constant for proton donor-acceptor distance $R$, and the summations are over reactant and product vibronic states $\mu$ and $\nu$, respectively. $P_{\mu }$ is the Boltzmann population of vibronic state $\mu ,V_{\mu \nu }$ is the vibronic coupling between states $\mu$ and $\nu ,\hslash$ is the reduced Planck constant, $\lambda$ is the reorganization energy, $k_{\text{B}}$ is the Boltzmann constant, $T$ is the temperature, and $\Delta G_{\mu \nu }^0$ is the reaction free energy for vibronic states $\mu$ and $\nu$. Specifically, $\Delta G_{\mu \nu }^0=\Delta G^0+{\epsilon }_{\nu }-{\epsilon }_{\mu }$ contains contributions from the PCET reaction free energy for the reactant and product vibronic ground states, $\Delta G^0$, and the energies ${\epsilon }_{\mu }$ and ${\epsilon }_{\nu }$ of reactant $\mu$ and product $\nu$ vibronic states relative to their respective ground vibronic states. In general, $P_{\mu },V_{\mu \nu }$, and $\Delta G_{\mu \nu }^0$ depend on $R$. To obtain the overall PCET rate constant, typically the rate constant in Eq. (1) is thermally averaged over the range of relevant proton donor-acceptor distances $R$:¹⁶

$$k=\underset{0}{\overset{\infty }{\int }}k(R)P(R)dR$$ (2)

where $P(R)$ is the proton donor-acceptor distance distribution function. Note that $R$ is the distance between the proton donor and acceptor and is independent of the proton position, as discussed further in the section on generating the proton potential energy curves.

In Eq. (1), the form of the vibronic coupling is determined by the electron-proton nonadiabaticity. In the electronically nonadiabatic regime, $V_{\mu \nu }=V_{\text{el}}{S}_{\mu \nu }$, where $V_{\text{el}}$ is the electronic coupling between the reactant and product diabatic electronic states and $S_{\mu \nu }$ is the overlap between the reactant and product proton vibrational wavefunctions. A more general semiclassical (sc) form of the vibronic coupling is^{17, 18}

$$V_{\mu v}^{(\text{sc})}=\kappa \frac{{\Delta }_{\mu v}}{2}$$ (3)

where ${\Delta }_{\mu \nu }$ is the tunneling splitting on the ground adiabatic electronic state when diabatic states $\mu$ and $\nu$ are resonant, and $\kappa$ depends on the electronic coupling $V_{\text{el}}$ between the reactant and product diabatic electronic states as well as several other quantities related to the diabatic proton potential energy curves. In the electronically adiabatic limit, $\kappa =1$, and in the electronically nonadiabatic limit, $\kappa \ll 1$ and this expression becomes $V_{\mu \nu }=V_{\text{el}}{S}_{\mu \nu }$. Section IV will discuss how to compute $\kappa$ and determine the appropriate form of the vibronic coupling. For all other parts of this paper, we consider only the electronically nonadiabatic regime, where $V_{\mu \nu }=V_{\text{el}}{S}_{\mu \nu }$. Ref.¹⁹ provides an example of calculating the PCET rate constant using the vibronic coupling in its more general form. As mentioned above, the vibronic coupling can be computed to determine if a PCET reaction is vibronically adiabatic or vibronically nonadiabatic by comparing its magnitude to the thermal energy $k_{\text{B}}T$. Further details about computational diagnostics for determining vibronic and electron-proton nonadiabaticity are provided in Ref. 20.

The extension of Eq. (1) to electrochemical PCET²¹ involves integration over the one-electron states of the electrode, weighting by the Fermi distribution and the density of states. A general form of the cathodic electrochemical PCET rate constant at proton donor-acceptor distance $R$ and an applied electrode potential $E$ is

$$k(R,E)=\sum _{\mu }{P}_{\mu }\sum _v\frac{{\left|V_{\mu v}(R)\right|}^2}{\hslash }\sqrt{\frac{\pi }{\lambda k_{\text{B}}T}}\int f(\epsilon )\rho (\epsilon )\text{exp}\left[-\frac{{\left(\Delta G_{\mu v}(R,E,\epsilon )+\lambda \right)}^2}{4\lambda k_{\text{B}}T}\right]\text{d}\epsilon$$ (4)

where $\epsilon$ is the energy of the one-electron state of the electrode and $\Delta G_{\mu \nu }(R,E,\epsilon )=\Delta G(R,E)-\epsilon +{\epsilon }_{\nu }-{\epsilon }_{\mu }$. The $-\epsilon$ term indicates that the electrode state with energy $\epsilon$ transfers an electron to the molecule for the cathodic PCET reaction. As above, ${\epsilon }_{\mu }$ and ${\epsilon }_{\nu }$ are the reactant and product vibronic state energies, which in general depend on $R$. Here $f(\epsilon )$ is the Fermi distribution, and $\rho (\epsilon )$ is the density of states for the electrode, which is a property of the material and may be determined from first-principles calculations. The density of states $\rho (\epsilon )$ can be approximated as a constant ${\rho }_{\text{F}}$ when it is a slowly varying function of energy in the vicinity of the Fermi level, such as the sp-bands of noble metals. For an anodic electrochemical PCET reaction, the $-\epsilon$ term in the expression for $\Delta G_{\mu \nu }(R,E,\epsilon )$ is replaced by a $+\epsilon$ term, signifying that the electrode state with energy $\epsilon$ accepts an electron from the molecule for the anodic PCET reaction, and the Fermi distribution $f(\epsilon )$ in Eq. (4) is replaced by $1-f(\epsilon )$.

The form of $\Delta G(R,E)$, which is the reaction free energy for the ground vibronic states at $R$ and $E$, differs for homogeneous^{13, 22, 23} and heterogeneous¹⁹ electrochemical PCET.²⁴ For the homogeneous case, $R$ is the proton donor-acceptor distance within the molecule, requiring thermal averaging of the rate constant over the proton donor-acceptor distance, weighting by $P(R)$, and another integral over the distance between the molecule and the electrode, weighted by the concentration of the molecular species. For the heterogeneous case of proton discharge to the electrode, $R$ is the distance from the proton donor to the electrode, and only a single integration over distance, weighting by the concentration of the proton donor molecular species, is required. In this expression, the applied potential $E$ impacts only $\Delta G_{\mu \nu }(R,E,\epsilon )$, although in principle the influence of $E$ on the proton potential energy curves and therefore the vibronic coupling could also be included. Moreover, as discussed below, the applied potential $E$ may also influence the proton donor-acceptor distribution function or local concentration of the proton donor or acceptor species. The rate constant expressions are given in Eq. 21–23 for homogeneous and Eqs. 28–30 for heterogeneous PCET in Ref.²⁴, along with a more extensive explanation of the various terms.

The first application of this nonadiabatic PCET theory to electrochemical PCET centered on an osmium aquo complex attached to a gold electrode via a self-assembled monolayer.²⁵ The methodology for computing the input quantities has evolved since this work, and therefore this initial application is not discussed below. Moreover, a more general rate constant expression that spans the vibronically adiabatic and nonadiabatic regimes for electrochemical PCET has been derived and applied to proton discharge on electrode surfaces^{26, 27} but will not be discussed herein.

We emphasize that these methods for computing input quantities are based on approximations that provide qualitative but not always quantitative results. Absolute rate constants are challenging to compute with a high level of quantitative accuracy, but relative rate constants and kinetic isotope effects are often comparable to experimental measurements.^{19, 23, 28–36}

III. Input quantities

A. Reorganization Energy

The reorganization energy is typically split into inner-sphere $({\lambda }_{\text{i}})$ and outer-sphere $({\lambda }_{\text{o}})$ contributions, which are assumed to be separable. The total reorganization energy is expressed as $\lambda ={\lambda }_{\text{i}}+{\lambda }_{\text{o}}$. The inner-sphere contributions involve rearrangement of the solute complex, whereas the outer-sphere contributions involve the rearrangement of the solvent and/or protein environment upon PCET. In early work, the inner-sphere reorganization energy for metal complexes was estimated based on changes in bond lengths and effective force constants of the ligands,^{28, 37} and the outer-sphere contributions were often calculated using the frequency-resolved cavity model (FRCM)³⁸ with CHELPG³⁹ charges.^{28, 30, 37, 40–44} More recent applications use the methods described below, which are typically more suitable.

Inner-Sphere Reorganization Energy

One of the most commonly used methods for calculating inner-sphere reorganization energy is the four-point scheme⁴⁵ extended to PCET.²² In this case, the inner-sphere reorganization energy is computed as

$${\lambda }_{\text{i}}=\frac{1}{2}\left[E_{\text{P}}\left(Q^{\text{R}}\right)-E_{\text{P}}\left(Q^{\text{P}}\right)+E_{\text{R}}\left(Q^{\text{P}}\right)-E_{\text{R}}\left(Q^{\text{R}}\right)\right]$$ (5)

Here $E_{\text{R}}$ indicates the PCET reactant state (i.e., the electron and proton are on their donors), and $E_{\text{P}}$ indicates the PCET product state (i.e., the electron and proton are on their acceptors). $Q^{\text{R}}$ is the optimized geometry for the PCET reactant state, and $Q^{\text{P}}$ is the optimized geometry for the PCET product state, although for the nonequilibrium terms, the position of the transferring proton is re-optimized with all other nuclei fixed, as denoted in Figure 1 below.

Figure 1.

Inner-sphere reorganization energy within nonadiabatic PCET theory can be computed using a four-point scheme involving two equilibrium geometry optimizations and two nonequillibrium geometry optimizations, for which only the transferring proton is optimized with all other nuclei fixed. The blue parabola denotes the reactant state, and the red parabola denotes the product state.

For this approach, the geometry of the solute is optimized in the gas phase for the reactant state to obtain $E_{\text{R}}\left(Q^{\text{R}}\right)$ (i.e., with the transferring proton and electron on their respective donors) and for the product to obtain $E_{\text{p}}\left(Q^{\text{P}}\right)$ (i.e., with the transferring proton and electron on their respective acceptors). To obtain the nonequilibrium energy $E_{\text{R}}\left(Q^{\text{P}}\right)$, the transferring proton is optimized on its donor for the reactant electronic state (i.e., the electron on its donor) with all other nuclei fixed at the optimized geometry for the product state. To obtain the nonequilibrium energy $E_{\text{P}}\left(Q^{\text{R}}\right)$, the transferring proton is optimized on its acceptor for the product electronic state (i.e., the electron on its acceptor) with all other nuclei fixed at the optimized geometry for the reactant state.

The energies used for the inner-sphere reorganization energy are computed in the gas phase. Moreover, these are electronic energies, not free energies, and therefore the geometries do not need to be stationary points in the gas phase, although often they are. Typically, the geometry optimizations of the solute are performed in the gas phase. However, if the reactant and/or product geometries optimized in the gas phase are significantly different from those optimized in solvent or protein, then it may be preferable to use the geometries optimized in solvent.^{31, 33} In this case, the energies are still computed in the gas phase, and the optimization of only the proton to obtain the nonequilibrium energies is still performed in the gas phase. For consistency, the transferring proton may also be optimized in the gas phase to compute the equilibrium energies, although this additional optimization is unlikely to alter the equilibrium energy significantly.

In some cases, the reactant and product diabatic electronic states can be obtained by simply changing the charge of the overall system. For example, in homogeneous electrochemical PCET or photoexcited PCET with an external photoreceptor, the charge of the molecule is different for the reactant and product.^{22, 23, 31, 33, 46, 47} In this case, the inner-sphere reorganization energy can be computed using the full molecular complex, as for the benzimidazole phenol (BIP) molecules;²³ the isolated molecules for intermolecular proton transfer, as for the tyrosine and tyrosyl radical in ribonucleotide reductase (RNR);⁴⁸ or separate fragments that have different charges for the reactant and product, as for the anthracene-phenol-pyridine triads.³⁴ In cases where the net charge of the system does not change and the fragment approach is not feasible, charge constraints could be used to maintain the correct electronic state by using a method such as constrained DFT (CDFT)^{49, 50} or another charge-constraining procedure. The optimal approach depends on the specific system studied.

Outer-Sphere Reorganization Energy

The outer-sphere reorganization energy can often be computed based on an electrostatic model. For homogeneous PCET in solution, a two-sphere model is often used. In this case, the analytical expression is⁵¹

$${\lambda }_{\text{o}}=(\Delta q{)}^2\left(\frac{1}{{\epsilon }_{\infty }}-\frac{1}{{\epsilon }_0}\right)\left(\frac{1}{2R_{\text{A}}}+\frac{1}{2R_{\text{B}}}-\frac{1}{R_{\text{AB}}}\right)$$ (6)

Here $\Delta q$ is the change in charge of the solute complex, ${\epsilon }_{\infty }$ is the optical dielectric constant of the solvent, and ${\epsilon }_0$ is the static dielectric constant of the solvent. The two spheres have radii $R_{\text{A}}$ and $R_{\text{B}}$, and the distance between their centers is $R_{\text{AB}}$, as shown in Figure 2a. When the two spheres are assumed to be touching, $R_{\text{AB}}=R_{\text{A}}+R_{\text{B}}$. In the case of homogeneous electrochemical PCET, the outer-sphere reorganization energy ${\lambda }_{\text{o}}$ of a solute complex in a spherical cavity of radius $a$ at distance $d$ from the electrode surface (Figure 2b) can be calculated according to

$${\lambda }_{\text{o}}=\frac{(\Delta q{)}^2}{2}\left(\frac{1}{{\epsilon }_{\infty }}-\frac{1}{{\epsilon }_0}\right)\left(\frac{1}{a}-\frac{1}{2d}\right)$$ (7)

Figure 2.

Methods to compute outer-sphere reorganization energy. (a) The outer-sphere reorganization energy for homogeneous PCET in solution can be calculated with a two-sphere model, where the primary parameters are the radii of the two spheres, $R_{\text{A}}$ and $R_{\text{B}}$, and the distance between their centers, $R_{\text{AB}}$. (b) The outer-sphere reorganization energy for homogeneous electrochemical PCET can be calculated by using a model with a sphere of radius $a$ at a distance $d$ from the electrode. Often it is assumed that $d=a$.

For both cases, the radius of the sphere can be estimated by equating the volume of the sphere to the volume of the cavity obtained with a dielectric continuum model such as the polarizable continuum model (PCM) in conjunction with a DFT calculation of the solute. Note that the spherical approximation may not be applicable to highly non-spherical molecules. For the homogeneous case, the distance between the centers is typically chosen to be the distance between the electron donor and acceptor. For the electrochemical case, often the sphere is assumed to be tangent to the electrode surface to determine the distance $d$. In some cases, such as hydrogen transfer between two tyrosine residues⁴⁸ or electron transfer between an electrode and an adsorbed molecule,³⁶ the outer-sphere reorganization energy is approximated to be zero because the change in charge distribution of the solute is very small. Often the outer-sphere reorganization energy for PCET is approximated as that for electron transfer if the electron is transferring over a much longer distance than the proton.^{35, 47} A range of applications have used Eq. (6)^{31, 33–35} and Eq. (7)^{22, 23, 46} to calculate the outer-sphere reorganization energy.

More sophisticated approaches have also been developed for computing the outer-sphere reorganization energy for electrochemical PCET. These approaches are based on the integral equation formalism with the polarizable continuum model (IEF-PCM) framework.^52–54 The effects of the electric double layer have been included in a multilayer IEF-PCM approach and have been found to be very small, only slightly reducing the solvent reorganization energy.⁵³ However, implicit solvent models do not capture the effects of explicit solvent and ions that comprise the electric double layer. We have found that the simple models give very similar results as the more sophisticated approaches for the systems studied and typically agree well with the available experimentally measured reorganization energies.^52–54 Thus, the simple analytical models are often sufficient.

Total Reorganization Energy with Molecular Dynamics

The total reorganization energy can also be estimated using molecular dynamics (MD) simulations. In this case, the total reorganization energy of the system, which includes both inner- and outer-sphere contributions, can be calculated as⁵⁵

$$\lambda =\frac{{\sigma }^2}{2k_{\text{B}}T}$$ (8)

where ${\sigma }^2$ is the variance of the energy gap between the reactant and product diabatic electronic states corresponding to electron transfer. Again, the reorganization energy for PCET may be approximated as that for electron transfer if the electron is transferring over a much longer distance than the proton.

In practice, the energy gap is sampled for both the reactant and product diabatic electronic states by propagating MD trajectories on the corresponding potential energy surfaces, typically using an appropriately parameterized molecular mechanical force field. For each conformation sampled on the reactant (product) surface, the energy gap can be computed as the difference between the product and reactant diabatic state energies by altering the molecular mechanical force field parameters to obtain the product (reactant) state energy. Alternatively, the energy gap can be computed for each conformation using a QM/MM approach with a method such as CDFT. The reorganization energy is calculated from the variance obtained by sampling on the reactant and product states using Eq. (8), and the resulting two values are averaged to obtain the total reorganization energy. If the linear response approximation and other approximations underlying Marcus theory for electron transfer are valid, the sampling on each state will produce Gaussian distributions, the reorganization energies computed on the two states will be similar, and the separation between the average energy gap values obtained by sampling on the reactant and product states will be twice the reorganization energy. However, nonergodic effects in heterogeneous systems such as proteins could lead to significant deviations from this ideal behavior.⁵⁶

This method was used to study PCET in soybean lipoxygenase²⁹ and in the ${\alpha }_3\text{Y}$ de novo protein.⁴⁷ The use of MD to predict total reorganization energies at electrochemical interfaces is challenging due to the necessity of including explicit solvent, ions, and often an atomistic description of the electrode with applied potential.^57–59 This approach is applicable to PCET reactions simulated in explicit solvent and/or protein environments, but it is significantly more computationally expensive than the dielectric continuum approaches described above due to the necessity of conformational sampling.

B. Proton Potential Energy Profiles

The proton potential energy profiles are a critical part of the PCET rate constant calculation. These proton potential energy profiles are generated as a function of the proton coordinate $r_{\text{p}}$, which is typically defined along an axis corresponding to the motion of the proton from its donor to its acceptor. Note that this coordinate is distinct from the proton donor-acceptor distance $R$, which is the distance between the proton donor and acceptor and is independent of the proton position. The vibronically nonadiabatic rate constant is derived using the golden rule formalism to describe the PCET reaction in terms of nonadiabatic transitions between reactant and product diabatic vibronic states. In this case, the proton potential energy profiles are obtained for the reactant and product diabatic electronic states (i.e., the electron is on its donor or acceptor, respectively). As discussed above, in some cases, the reactant and product diabatic electronic states can be obtained by simply changing the charge of the overall system. This strategy is applicable to systems where the charge of the molecule is different for the reactant and product, such as in homogeneous electrochemical PCET or photoexcited PCET with an external photoreceptor.^{22, 23, 31, 33, 46, 47} In some cases, the proton potential energy profiles can be computed for a fragment that has a different charge for the reactant and product, such as the phenol-pyridine fragment in the anthracene-phenol-pyridine triads.³⁴ In other cases, charge constraints must be used to maintain the correct electronic state for the full molecular system by using a method such as CDFT, as for the tyrosine/tyrosyl radical complex in RNR.⁴⁸

In early work, the proton potential energy profiles were represented as Morse potentials with the parameters such as the equilibrium distance, bond dissociation energy, and vibrational frequency obtained from quantum chemistry calculations. In this case, the dependence on the proton donor-acceptor distance $R$ can be approximated by altering the distance between the minima of the reactant and product Morse potentials, which are mirrored relative to each other.^{19, 29, 35} Morse potentials do not always provide reliable results, however, and should be used only if a more reliable approach is not possible. Morse potentials are especially problematic for cases in which the proton potential energy profiles should be asymmetric double-well potentials that allow significant contributions from vibronic states with the proton localized in the higher-energy well.

A more sophisticated analytical approach is to use a two-state empirical valence bond (EVB) model for each diabatic proton potential, allowing asymmetric double-well potentials. In this case, the diagonal elements are represented as Morse potentials and sometimes non-bonding and electrostatic terms that depend on $R$. The off-diagonal elements can be constant or depend on $R$. The dependence of the proton potential energy profiles on $R$ is incorporated by diagonalizing the reactant and product 2 × 2 EVB Hamiltonians to obtain the corresponding ground diabatic electronic states for each $R$ value.²⁸ Several applications have used this EVB model approach^{28, 30, 37, 40–44}

The most reliable approach is the diabatic grid-based approach. The first step of this procedure is to generate a series of average reactant/product structures for different proton donor-acceptor distances. These average structures approximately correspond to the crossing point along an inner-sphere solute coordinate connecting the reactant and product equilibrium structures, although in some cases a weighted average rather than an equal average is used.⁶⁰ This procedure is ideal for cases where the reactant and product have different charges because the reaction is electrochemical or photochemical with an external photosensitizer. This strategy involves the following steps for each $R$ value:

• Optimize the reactant and product structures with $R$ constrained.

• Overlay the reactant and product structures so the proton donor and acceptor are superimposed and minimize the RMSD of the remaining nuclear positions while maintaining this constraint. Then generate the average structure by averaging all the Cartesian coordinates of the individual nuclei.

• Optimize the proton on its donor for the reactant state and then optimize the proton on its acceptor for the product state, while all other nuclei remain fixed to the average structure.

• Generate the proton coordinate axis that passes through the two optimized proton positions.

• Move the proton along a grid spanning this axis, performing single-point energy calculations in implicit solvent for the reactant and product states at each grid point to generate the reactant and product diabatic proton potential energy profiles.

• Fit the data to a polynomial or spline to interpolate/extrapolate and obtain a smooth curve that spans a sufficient region such that the potential is high enough on both sides.

The steps for generating the average reactant/product structure, proton coordinate grid, and reactant and product proton potential energy profiles are schematically depicted in Figure 3. The geometry optimizations in this procedure are usually performed in implicit solvent but can also be performed in the gas phase, depending on the system being studied. The result of this diabatic grid-based approach is a series of reactant and product proton potential energy profiles for a grid of $R$ values.

Figure 3.

Schematic depiction of the process used to generate proton potential energy profiles for an average reactant/product geometry at a fixed proton donor-acceptor distance $R$. In the diagram, a blue background indicates that all of the nuclei are at the optimized geometry for the reactant state at a fixed $R$, and a red background indicates that all of the nuclei are at the optimized geometry for the product state at a fixed $R$. A purple background indicates that all of the nuclei except the transferring proton are at the average reactant/product geometry for a fixed $R$. The terms reactant and product in this figure refer to the electronic state. An example of the resulting proton potential energy profiles is shown in Figure 4.

This diabatic grid-based approach has been applied to a wide range of electrochemical or photochemical PCET reactions for which the reactant and product have different charges.^{23, 31, 33, 46, 47, 61} This approach can also be implemented using a fragment of the molecule rather than the full molecule to generate the proton potentials.³⁴ In some cases, the reactant and product proton potential energy profiles can be obtained with CDFT for a series of average reactant/product geometries with different $R$ values obtained using the procedure depicted in Figure 3.^{36, 48} The proton potential energy profiles can also be obtained by using CDFT or another charge-constraining approach for a single geometry and including the dependence of the potentials on $R$ using an EVB model approach.⁶⁰

After obtaining the series of proton potential energy profiles for a set of $R$ values, the one-dimensional time-independent Schrödinger equation is solved numerically for the proton moving in these potentials to obtain the proton vibrational wavefunctions and associated energy levels using a method such as the Fourier Grid Hamiltonian (FGH) method.^{62, 63} The wavefunctions are used to calculate the overlap term $S_{\mu \nu }$ (middle panel of Figure 4), and the energy levels are used to calculate the Boltzmann population $P_{\mu }$ and $\Delta G_{\mu \nu }^0(R)=\Delta G^0+{\epsilon }_{\nu }-{\epsilon }_{\mu }$, where ${\epsilon }_{\mu }$ and ${\epsilon }_{\nu }$ (left and right panels of Figure 4) are the energies of the reactant and product proton vibrational energy levels relative to their respective ground state vibrational energy levels. The proton potential energy profiles may also be calculated in multiple dimensions for PCET involving multiple proton transfer reactions.^{22, 23}

Figure 4.

Proton potential energy profiles and proton vibrational wavefunctions for the reactant (left, blue) and product (right, red). The ground vibrational state for each potential energy profile is set to have zero energy. The proton vibrational energy levels are labeled, and the overlap $S_{\mu \nu }$ between the reactant and product ground vibrational wavefunctions is shown in purple and labeled by $S_{00}$.

C. Electronic Coupling

In the electronically nonadiabatic limit, the vibronic coupling is $V_{\mu \nu }=V_{\text{el}}{S}_{\mu \nu }$. In this section, we discuss the calculation of the electronic coupling, $V_{\text{el}}$, which is the coupling between the reactant and product diabatic electronic states, where the electron is localized on its donor or acceptor, respectively. Note that in the calculation of kinetic isotope effects (KIEs) or other quantities that depend on the ratio of calculated rate constants, such as Tafel slopes or relative rate constants for species with different substituents, the electronic coupling cancels and therefore is not needed.

The electronic coupling depends on the electron donor-acceptor distance $R_{\text{ET}}$ and is often assumed to decay exponentially with this distance:

$$V_{\text{el}}\left(R_{\text{ET}}\right)=V_{\text{el}}\left({\bar{R}}_{\text{ET}}\right)\text{exp}\left[-\left({\beta }^{\prime }/2\right)\left(R_{\text{ET}}-{\bar{R}}_{\text{ET}}\right)\right]$$ (9)

where $V_{\text{el}}\left({\bar{R}}_{\text{ET}}\right)$ is the electronic coupling at the equilibrium electron donor-acceptor distance ${\bar{R}}_{\text{ET}},{\beta }^{\prime }$ is a decay constant typically ~1–2 Å⁻¹, and $R_{\text{ET}}$ is the electron donor-acceptor distance. This distance dependence of the electronic coupling has been used in several applications.^{19, 35, 60}

When the magnitude of the electronic coupling is required, it is often computed with the constrained DFT configuration interaction (CDFT-CI) approach.^{49, 50, 64} The geometry at which to compute the electronic coupling can be generated in a variety of ways. One option is to generate an average reactant/product geometry by superimposing the proton donor-acceptor axes of the equilibrium reactant and product geometries with the midpoints between the proton donor and acceptor at the same position, producing a single geometry with an average proton donor-acceptor distance. Another option is to generate an average reactant/product geometry by superimposing the proton donor and acceptor atoms for reactant and product geometries at a fixed proton donor-acceptor distance $R$ for a series of $R$ values, leading to the same geometries as those used to generate the proton potential energy profiles. For either option, the root-mean square deviation (RMSD) of the positions of the remaining nuclei is minimized while maintaining these constraints on the proton donor-acceptor axis. The average structure is obtained by averaging all the Cartesian coordinates of the nuclei. The electronic coupling can be computed for this average structure with the CDFT-CI approach along a grid of proton coordinates with all other nuclei fixed. The electronic coupling usually depends on the proton coordinate, and the value used in the calculation of the PCET rate constant can be obtained by averaging over the relevant range of proton coordinates or calculating it at the crossing point of the aligned proton potential energy profiles (see section IVB).

D. Reaction Free Energy

The PCET reaction free energy, also referred to as the negative of the PCET driving force, is an important parameter in the rate constant expression. The reaction free energy can often be calculated directly from DFT calculations in solution.^{33, 34} Calculation of the PCET reaction free energies with DFT entails geometry optimizations of the reactant and product and inclusion of zero-point energy (ZPE) and entropic contributions based on calculated vibrational frequencies within the harmonic approximation. Solvation free energy contributions are typically included through a dielectric continuum model such as C-PCM. For PCET in enzymes, the reaction free energy may be calculated directly from quantum mechanical/molecular mechanical free energy simulations.⁴⁸

In some cases, the reaction free energy can be estimated based on experimental measurements.^{28, 32, 35, 47} The experimental measurements used to compute PCET reaction free energies are typically redox potentials and $\text{p}{K}_{\text{a}}$ values. The PCET reaction free energy can be written as a sum of reaction free energies for electron transfer followed by proton transfer or vice versa, as clearly shown by the thermodynamic cycles in a square diagram. The electron transfer reaction free energy can be obtained from the difference between the redox potentials of the electron donor and acceptor, and the proton transfer reaction free energy can be obtained from the difference between the $\text{p}{K}_{\text{a}}$ values of the proton donor and acceptor. In many cases, the experimental values for these thermodynamic quantities are more reliable than the absolute computational values, although relative redox potentials and $\text{p}{K}_{\text{a}}$ values can often be computed reliably.⁶⁵

In the case of electrochemical PCET, the reaction free energy will depend on the applied potential $E$. Additionally, the reaction free energy for heterogeneous electrochemical PCET reactions should include work contributions associated with the transport of the proton donor/acceptor and its conjugate base/acid in and out of the electric double layer (EDL). The electrostatic part of these work terms also depends on the applied potential at point $R$ in the EDL. For example, the reaction free energy for the Volmer reaction with the proton donor at a distance $R$ from the electrode surface can be expressed as a sum of the reaction free energy for dissociative adsorption of H₂ on the electrode surface, contributions from the applied electrode potential, and work terms that contain electrostatic and non-electrostatic contributions:¹⁹

$$\Delta G(R,E)=\Delta G_{\text{ads}}+e\left(E-E_{\text{RHE}}\right)-W_{{\text{HA}}^{+}}(R,E)+W_{\text{A}}(R,E)$$ (10)

Here $\Delta G_{\text{ads}}$ is the free energy change for adding a proton-electron pair to the uncharged electrode surface, and $e\left(E-E_{\text{RHE}}\right)$ is the applied potential on the RHE scale. $W_{{\text{HA}}^{+}}$ and $W_{\text{A}}$ are work terms associated with bringing the proton donor ${\text{HA}}^{+}$ and its conjugate base into or out of the double layer from bulk solution. The work term for charged species depends on the electrostatic potential at a distance $R$ from the electrode. The dependence of the work terms on the potential $E$ can be included through an extended Guoy-Chapman-Stern (GCS) model^66–69 of the EDL to compute the electrostatic potential at a distance $R$ from the electrode. Eq. (10) has been used in several applications to heterogeneous PCET.^{19, 36}

When the standard rate constant is computed for an electrochemical reaction, the reaction free energy is not needed. In this case, the cathodic and anodic rate constants are expressed as functions of the overpotential $\eta$, which in turn is defined relative to the equilibrium potential. At the equilibrium potential, the anodic and cathodic rate constants are equal. In practice, however, the thermally averaged cathodic and anodic rate constants are not necessarily equal at $\eta =0$ because the proton donor-acceptor distance probability distribution functions may be different for the anodic and cathodic processes. In this case, we can calculate the anodic and cathodic rate constants and choose the value of $\eta$ such that these two rate constants are equal.^{22, 23, 46} The resulting rate constant is considered to be the standard rate constant.

E. Proton Donor-Acceptor Distance Distribution Function

Inclusion of the proton donor-acceptor motion is important and strongly impacts the vibronic coupling, as well as the reaction free energies of the excited vibronic states. Some of the very early calculations assumed fixed proton donor-acceptor distances, which is not advisable. In other early work, the dependence of the rate constant on the proton donor-acceptor distance $R$ was included using analytical rate constant expressions assuming that the vibronic coupling decreases exponentially with $R$:⁷⁰

$$V_{\mu \nu }(R)\approx V_{\text{el}}{S}_{\mu \nu }^0\text{exp}\left[-{\alpha }_{\mu \nu }\left(R-{\bar{R}}_{\mu }\right)\right]$$ (11)

Here ${\bar{R}}_{\mu }$ is the proton donor-acceptor distance for the equilibrium reactant state, and $S_{\mu v}^0$ is the overlap computed at this distance. In this case, the proton potential energy profiles were only computed at the equilibrium $R$ distance for the reactant state, and the shapes of the proton potentials were assumed to be independent of $R$, which is not always a good assumption. Moreover, the assumption of the exponential decay of the overlap breaks down in some cases, especially for highly excited vibrational states. This analytical rate constant expressions was used to study PCET in soybean lipoxygenase.²⁹ Due to the assumptions mentioned above, however, this approach is not considered to be as reliable as the thermal averaging approach given by Eq. (2).

The probability distribution function for the proton donor-acceptor distance, $P(R)$, is required for thermal averaging of the PCET rate constant according to Eq. (2). The distribution function $P(R)$ can be expressed in the general form

$$P(R)=C_{\text{N}}\text{exp}[-\beta U(R)]$$ (12)

where $C_{\text{N}}$ is a normalization constant, $\beta =1/k_{\text{B}}T$, and $U(R)$ is the potential energy profile describing the interaction of the proton donor and acceptor species. If this distribution function becomes zero at large proton donor-acceptor distances, it can be normalized, and Eq. (12) is rewritten as:

$$P(R)=\frac{\text{exp}[-\beta U(R)]}{{\int }_0^{\infty }\text{exp}[-\beta U(R)]dR}$$ (13)

When $U(R)$ is harmonic and $\text{exp}[-\beta U(R)]$ is Gaussian, it is convenient to change the lower limit of the integral in the denominator to $-\infty$ because the integral can be computed analytically. However, for a general form of $U(R)$, this integral is computed numerically. The potential energy profile $U(R)$ can often be obtained from DFT geometry optimizations with constrained $R$ values,³¹ possibly fit to an analytical functional form.³⁶ In some cases, classical MD simulations can be used to obtain the probability distribution function $P(R)$ using umbrella sampling with the weighted histogram analysis method (WHAM)⁷¹ or by generating a histogram from unrestrained MD.^{47, 48} Another method for obtaining $P(R)$ is through QM/MM finite temperature string simulations with umbrella sampling.³²

Many of our studies have used the following procedure to obtain the proton donor-acceptor interaction potential $U(R)$ for molecular systems.^{22, 23, 46} The proton donor-acceptor interaction potential is assumed to be a harmonic potential, $U(R)=-k_{\text{eff}}(R-\bar{R}{)}^2/2$. Here $\bar{R}$ is the equilibrium proton donor-acceptor distance, and $k_{\text{eff}}$ is an effective force constant. The effective force constant can be obtained by projecting all the normal modes for an optimized structure onto the proton donor-acceptor axis and summing up the appropriately weighted force constants using methodology described elsewhere.^{22, 46} For most thermal or photochemical systems,³⁴ the proton donor-acceptor distance $\bar{R}$ and the effective force constant $k_{\text{eff}}$ are obtained for the reactant equilibrium structure because $P(R)$ is defined as the probability distribution function for the reactant. For some electrochemical systems,^{22, 23, 46} $\bar{R}$ is computed as the average of the proton donor-acceptor distance in the reactant and product equilibrium structures, and $k_{\text{eff}}$ is computed as the average effective force constant obtained for the reactant and product equilibrium structures because the reaction is reversible, both anodic and cathodic rate constants are computed, and detailed balance must be ensured.

For heterogeneous electrochemical PCET reactions involving proton transfer to an electrode surface or to a molecule adsorbed on the surface, the total current density $j(E)$ depends on the applied potential $E$, and $P(R)$ is replaced by the local concentration of the proton donor, $c_{{\text{HA}}^{+}}(R,E)$. In this case, the current density is given by

$$j(E)=F\underset{0}{\overset{\infty }{\int }}k(R,E)c_{{\text{HA}}^{+}}(R,E)\text{d}R$$ (14)

and

$$c_{{\text{HA}}^{+}}(R,E)=c_{{\text{HA}}^{+}}^0\text{exp}\left[-\beta W_{{\text{HA}}^{+}}(R,E)\right]$$ (15)

where $c_{{\text{HA}}^{+}}^0$ is the bulk concentration of the proton donor ${\text{HA}}^{+}$ and $W_{{\text{HA}}^{+}}(R,E)$ is the work term associated with bringing the proton donor ${\text{HA}}^{+}$ into the EDL from bulk solution. As discussed above in Sec. IIID, this work term can be computed using an electrostatic model for the EDL. Eq. (14) has been used in applications to electrochemical PCET,^{19, 36} where either Eq. (15) or another form of the potential-dependent proton donor concentration is used.

IV. Electron-proton nonadiabaticity

A. Theoretical background

The general expression for the vibronic coupling is given in Eq. (3). In this section, we explain how to compute the prefactor $\kappa$, which characterizes the PCET reaction as electronically adiabatic, where the electrons respond instantaneously to the proton motion, or electronically nonadiabatic, where the electrons do not respond instantaneously to the proton motion. The prefactor $\kappa$ is given by¹⁷

$$\kappa =\sqrt{2\pi p}\frac{e^{p\text{ln}p-p}}{\Gamma (p+1)}$$ (16)

where $\Gamma (x)$ is the gamma function, and the adiabaticity parameter is defined as $p={\tau }_{\text{p}}/{\tau }_{\text{e}}$. The effective proton tunneling time is ${\tau }_{\text{p}}=V_{\text{el}}/\left(|\Delta F|v_{\text{t}}\right)$, and the electronic transition time is ${\tau }_{\text{e}}=\hslash /V_{\text{el}}$. Here, $|\Delta F|$ is the difference between the slopes of the diabatic proton potential energy curves at the crossing point, and the tunneling velocity is $v_t=\sqrt{2\left(V_{\text{c}}-E\right)/m_{\text{p}}}$, where $V_{\text{c}}$ is the energy at which the diabatic potential energy curves cross, $E$ is the tunneling energy, defined as the energy of the aligned diabatic reactant and product vibrational levels, and $m_{\text{p}}$ is the mass of a proton. When $p\gg 1$, the PCET reaction is electronically adiabatic (i.e., $\kappa \approx 1$ and $V^{(\text{sc})}\approx V^{(\text{ad})}$), whereas when $p\ll 1$, the reaction is electronically nonadiabatic (i.e., $\kappa <1$ and $V^{(\text{sc})}\approx V^{(\text{nad})}$). The vibronic coupling in the adiabatic limit is $V^{(\text{ad})}={\Delta }_{\mu \nu }/2$, where ${\Delta }_{\mu \nu }$ is the tunneling splitting on the ground electronic state when diabatic states $\mu$ and $\nu$ are resonant, and the vibronic coupling in the electronically nonadiabatic limit is $V^{(\text{nad})}=V_{\text{el}}{S}_{\mu \nu }$.

These quantities can be computed from the diabatic proton potential energy profiles described above.^{18, 72} Typically, they are computed for the ground reactant and product vibronic states, although the extension to excited vibronic states has been implemented.¹⁹ After generating the reactant and product proton potential energy profiles and computing the proton vibrational ground states, the potentials are shifted in energy so that the reactant and product proton vibrational ground states are aligned. This shift ensures that the vibronic coupling is computed when the reactant and product vibronic states are degenerate, presumably due to fluctuations of the solvent or protein environment, according to the golden rule formalism. Then $\left(V_c-E\right)$ is the energy difference between the crossing point of the diabatic proton potential energy curves and the aligned proton vibrational ground states. The difference in slopes, $|\Delta F|$, is computed numerically from these curves using the finite difference method. The electronic coupling $V_{\text{el}}$ is computed along the proton coordinate grid as described in Sec. IIIC, and the overlap $S_{00}$ between the reactant and product proton vibrational wavefunctions is computed numerically. The tunneling splitting ${\Delta }_{00}$ is computed as the energy difference between the lowest two proton vibrational states for the proton potential energy profile corresponding to the ground adiabatic electronic state derived from the aligned diabatic proton potential energy curves (see section IVB). This adiabatic proton potential energy profile is typically a symmetric or nearly symmetric double-well potential. In this case, the lowest two proton vibrational states are associated with antisymmetric and symmetric proton vibrational wavefunctions. Thus, we have all the quantities necessary to compute the vibronic coupling in any regime and to determine the degree of electron-proton nonadabaticity. Moreover, the magnitude of the vibronic coupling determines the vibronic nonadiabaticity: the PCET reaction is vibronically nonadiabatic (adiabatic) if the vibronic coupling is much less than (greater than) the thermal energy $k_{\text{B}}T$.

B. Example for PCET reaction in ribonucleotide reductase

To illustrate the procedure for computing the general form of the vibronic coupling and determining the degree of electron-proton nonadiabaticity, we provide an example for the PCET reaction between the Y356 and Y731 residues in RNR. Specifically, this PCET reaction occurs between tyrosine and a tyrosyl radical in the protein environment. Additional computational details are provided in Ref.⁷³, and additional examples can be found in Refs.¹⁸ and⁷².

The proton potential energy profiles and electronic coupling were obtained for this PCET reaction using the procedure described in Sec. VA below. The initial configuration for this PCET reaction was obtained from QM/MM free energy simulations. The proton potential energy profiles were obtained for the reactant and product diabatic electronic states by performing CDFT calculations at each grid point along the proton coordinate. These profiles were computed for the same configuration of the tyrosine/tyrosyl pair modeling Y356/Y731 both in the gas phase and including the protein/solvent environment through electrostatic embedding. The electronic coupling, $V_{\text{el}}$, between the diabatic states was determined using CDFT-CI at each grid point along the proton coordinate. Both the proton potential energy profiles and the electronic coupling were interpolated to construct smooth curves. The interpolated proton potential energy profiles were then shifted to ensure that their ground vibrational energy levels are equal. The proton coordinate at which the aligned potential energy curves intersect was then identified. The value of $V_{\text{el}}$ at this crossing point is an input quantity for the electronic transition time ${\tau }_{\text{e}}$ and the effective tunneling time ${\tau }_{\text{p}}$. The slopes of the diabatic proton potential energy curves at the crossing point were computed numerically using the finite difference method to determine $|\Delta F|$, which is an input quantity for ${\tau }_{\text{p}}$ and the adiabaticity parameter $p$. The aligned proton potential energy profiles for the reactant and product diabatic electronic states are shown in blue and red, respectively, with the slopes at the crossing point indicated as black dashed lines, in Figure 5a.

Figure 5.

Nonadiabaticity analysis for the PCET reaction between Y356 and Y731 with the proton potential energy profiles computed in the gas phase. The corresponding plots including the protein/solvent environment are very similar. (a) Aligned diabatic proton potential energy curves illustrating the reactant (blue) and product (red) diabatic electronic states. The energy at the crossing point $V_{\text{c}}$ and the tunneling energy $E$ are indicated. The black dashed lines in (a) show the tangent lines of the diabatic proton potential energy curves at the crossing point. $|\Delta F|$ is calculated as the difference between the slopes of these dashed lines. (b) Adiabatic (black dashed lines) and diabatic (blue and red solid lines) proton potential energy curves for this PCET reaction. (c) The lowest two proton vibrational states with symmetric (orange) and antisymmetric (green) wave functions associated with the ground adiabatic electronic state. The slight asymmetry in these wave functions arises because this system is not symmetric. The antisymmetric wave function is shifted upward for visual aid, and the tunneling splitting ${\Delta }_{00}$ is indicated.

The proton potential energy profiles associated with the ground and excited adiabatic electronic states were obtained by diagonalizing the 2 × 2 Hamiltonian in the diabatic representation. In this case, the diagonal elements were the diabatic proton potential energies for the reactant and product states, and the off-diagonal elements were the proton coordinate-dependent $V_{\text{el}}$. The adiabatic proton potential energy profiles are shown as black dashed lines in Figure 5b. The tunneling splitting, ${\Delta }_{00}$, was then computed as the energy difference between the lowest two proton vibrational states for the proton potential energy profile corresponding to the ground adiabatic electronic state (Figure 5c). The values of the relevant parameters and the vibronic coupling for this PCET reaction with the proton potential energy profiles computed both in the gas phase and including the protein/solvent environment are given in Table 1.

Table 1.

Electronic Coupling, Tunneling Splitting, Effective Tunneling Time, Electronic Transition Time, Adiabaticity Parameter, Prefactor, and Vibronic Couplings Calculated for the PCET Reaction between Y356 and Y731 with Proton Potentials Computed in the Gas Phase and Including the Protein/Solvent Environment. Data originally found in Ref.⁷³.

Parameters	Gas Phase	Including Environment
$V_{\text{el}}(\text{kcal}/\text{mol})$	2.53	2.93
${\Delta }_{00}(\text{kcal}/\text{mol})$	7.76 × 10−2	1.02 × 10−1
${\tau }_{\text{p}}(\text{fs})$	0.19	0.24
${\tau }_{\text{e}}(\text{fs})$	6.00	5.18
$p$	3.16 × 10−2	4.66 × 10−2
$\kappa$	0.394	0.459
$V^{(\text{sc})}(\text{kcal}/\text{mol})$	1.53 × 10–2	2.35 × 10−2
$V^{(\text{nad})}(\text{kcal}/\text{mol})$	1.05 × 10–2	1.52 × 10−2
$V^{(\text{ad})}(\text{kcal}/\text{mol})$	3.88 × 10–2	5.12 × 10−2

This analysis shows that the PCET reaction between Y356 and Y731 is vibronically nonadiabatic, given that the vibronic coupling is smaller than the thermal energy, and is electronically nonadiabatic, given that the adiabaticity parameter $p\ll 1$, the prefactor $\kappa <1$, and the semiclassical vibronic coupling is similar to the nonadiabatic vibronic coupling, i.e., $V^{(sc)}\approx V^{\text{(nad)}}$. The nonadiabatic nature is also evident from the relatively small splitting between the ground and first excited adiabatic proton potential energy profiles and the similarity between the adiabatic and diabatic curves for all proton coordinates except small deviations in the avoided crossing region (Figure 5b). Note that the results including the protein/solvent environment are less clearly electronically nonadiabatic PCET because there is a small difference between $V^{(\text{sc})}$ and $V^{(\text{nad})}$ despite the finding that $p\ll 1$. Clearer examples of electronically adiabatic and nonadiabatic PCET can be found for the gas phase benzyl-toluene and phenoxyl-phenol systems, respectively.^{18, 72}

V. Examples for computing nonadiabatic PCET rate constant

In this section, we present examples for how the input quantities for nonadiabatic PCET rate constants were calculated for three different systems: biological PCET between two tyrosine residues in RNR,⁴⁸ homogeneous electrochemical PCET or photochemical PCET in a benzimidazole phenol,²³ and heterogeneous electrochemical PCET at a surface-immobilized cobalt porphyrin.³⁶ For more complete details, we refer the reader to the original publications.

A. PCET Between Tyrosine and Tyrosyl in an Open Configuration

In this example, PCET occurs between a tyrosine and tyrosyl radical (Figure 6a), resulting in a net hydrogen atom transfer. This is an example of a biological PCET process, as this reaction occurs between Y356 and Y731 in RNR. As described in Sec. IVB, this PCET reaction has been shown to be vibronically and electronically nonadiabatic. In this regime, the PCET rate constant is given by Eqs. (1) and (2) with $V_{\mu \nu }=V_{\text{el}}{S}_{\mu \nu }$.

Figure 6.

(a) PCET occurring between a tyrosine (right) and tyrosyl radical (left), involving a net hydrogen-atom transfer with the transferring proton labeled in blue. (b) Proton donor-acceptor distance distribution function $P(R)$ (green), PCET rate constant $k(R)$ (orange), and the product of these two functions for the PCET reaction between Y356 and Y731 in RNR as a function of proton donor acceptor distance $R$. The maximum of the product $P(R)k(R)$ is at 2.54 Å, whereas the maximum of the probability distribution function $P(R)$ is at 2.80 Å.

Reorganization Energy

For this system, we consider only the inner-sphere reorganization energy because the outer-sphere contribution is negligible given that the reaction corresponds to the net hydrogen atom transfer between Y356 and Y731 with minimal charge redistribution. The four-point method given in Eq. (5) is used to compute the inner-sphere reorganization energy, treating the tyrosine and tyrosyl radicals as independent fragments. Thus, the inner-sphere reorganization energy can be expressed as ${\lambda }_i=E_{\text{Tyr}·}\left(Q_{\text{Tyr}}\right)-E_{\text{Tyr}}\left(Q_{\text{Tyr}}\right)+E_{\text{Tyr}}\left(Q_{\text{Tyr}·}\right)-E_{\text{Tyr}·}\left(Q_{\text{Tyr}·}\right)$, where $E_{\text{Tyr}}\left(Q_{\text{Tyr}}\right)$ and $E_{\text{Tyr}·}\left(Q_{\text{Tyr}·}\right)$ are the energies of the optimized geometries of the tyrosine and tyrosyl radical, respectively, while $E_{\text{Tyr}}.\left(Q_{\text{Tyr}}\right)$ and $E_{\text{Tyr}}\left(Q_{\text{Tyr}·}\right)$ correspond to the energies of the radical and neutral tyrosine at the optimized geometry of the other state. Note that the factor of two in the denominator of Eq. (5) is cancelled by a factor of two that arises because the reactant and product contain both a tyrosine and a tyrosyl radical. The resulting inner-sphere reorganization energy is ${\lambda }_{\text{i}}=18.86\text{kcal}/\text{mol}$.

Proton Potential Energy Profiles

We generated average structures and proton coordinate axes for a series of proton donor-acceptor distances $R$ using the diabatic grid-based approach described in Section IIIB. For this system, $R$ is defined as the distance between the hydroxyl oxygen atoms, and the initial geometry used for the constrained geometry optimizations was selected from QM/MM free energy simulations. To generate the proton potential energy profiles, the hydrogen atom was placed at each grid point along the proton coordinate axis, and single-point energy CDFT-CI calculations were performed with spin constraints imposed to localize the unpaired spin on Y356 for the reactant and Y731 for the product. These calculations produced the reactant and product diabatic proton potential energy curves for a series of $R$ values.

We used the FGH method to compute the proton vibrational wavefunctions and energy levels, which enabled the calculation of the Boltzmann populations $P_{\mu }$, reaction free energies $\Delta G_{\mu \nu }^0$, and overlap integrals $S_{\mu \nu }$. Our calculations included up to the 7^th excited vibronic state for the reactant and product for each $R$ value to ensure convergence.

Electronic coupling

The electronic coupling was calculated with the CDFT-CI method for several different proton donor acceptor distances $R$. For each $R$ value, the diabatic proton potential energy profiles described above were shifted to align the reactant and product ground vibrational energy levels, and the electronic coupling was computed at the crossing point using the CDFT-CI method. The electronic coupling did not change significantly for different proton coordinates near the crossing point for the relevant $R$ values. Therefore, a constant electronic coupling of $V_{\text{el}}=0.8\text{kcal}/\text{mol}$ was used for the rate constant calculations. This value should be viewed as only qualitatively meaningful given the underlying approximations and limitations of the CDFT-CI approach.

Reaction free energy

The reaction free energy was obtained from the QM/MM free energy string simulations of this reaction,⁷⁴ including conformational sampling of the solvated protein. The value used for these calculations is $\Delta G_{00}^0=0$.

Proton Donor-Acceptor Distribution Function

To compute the probability distribution function $P(R)$, we used both unrestrained classical MD simulations and umbrella sampling simulations to ensure adequate sampling. The resulting distribution functions were similar for both unrestrained and restrained simulations. For the calculations described herein, we used $P(R)$ obtained from umbrella sampling simulations and fit the data to the exponential of a fourth-order polynomial. The resulting curve is shown in Fig. 6b.

PCET rate constant

We applied a quadratic fit to the natural logarithm of $k(R)$ and then exponentiated it to obtain the orange curve in Fig. 6b. The overall PCET rate constant was obtained by numerical integration of the product $k(R)P(R)$, as given in Eq. (2).

B. Homogeneous Electrochemical PCET in Benzimiadazole Phenol

In this example, the benzimidazole phenol (BIP) molecule is oxidized electrochemically, as shown in Figure 7. In this homogeneous electrochemical PCET process, an electron is transferred from the phenol to the electrode, coupled with an intramolecular proton transfer from phenol to benzimidazole. Thus, the reactant is neutral, and the product is positively charged due to electrochemical oxidation. This PCET reaction is vibronically and electronically nonadiabatic, allowing the use of Eq. (1) with $V_{\mu \nu }=V_{\text{el}}{S}_{\mu \nu }$. The corresponding homogeneous electrochemical rate constant expression is given in Eq. (1) of Ref.²³. The goal was to compute the standard rate constant, defined to be the rate constant at the overpotential for which the anodic and cathodic rate constants are the same. The kinetic isotope effect, defined as the ratio of the standard rate constant for hydrogen transfer and deuterium transfer, was also calculated.

Figure 7.

Homogeneous electrochemical PCET in a BIP molecule. When the BIP molecule is oxidized by the electrode, the proton transfer occurs intramolecularly from the donor oxygen to the acceptor nitrogen.

Reorganization Energy

The inner-sphere reorganization energy was calculated with the four-point scheme using Eq. (5) with geometry optimizations and energy calculations in the gas phase. For this electrochemical PCET, the electronic state was controlled by changing the charge of the molecule: the molecule was neutral for the reactant and positively charged for the product. The outer-sphere reorganization energy was calculated using Eq. (7) with the radius obtained from the C-PCM cavity volume for the optimized structure and assuming the sphere is tangent to the electrode (i.e., $d=a$). The inner-sphere and outer-sphere reorganization energies were computed to be ${\lambda }_{\text{i}}=11.6\text{kcal}/\text{mol}$ and ${\lambda }_{\text{o}}=9.8\text{kcal}/\text{mol}$.

Proton Potential Energy Profiles

For the proton potential energy profiles, we generated average reactant/product structures and proton coordinate axes at a range of proton donor-acceptor distances $R$ using the diabatic grid-based approach described in Section IIIC. For this system, $R$ is the distance between the oxygen and nitrogen atoms. The electronic state was controlled by changing the charge of the molecule. The single-point energies were computed at the DFT level with implicit acetonitrile solvent.

Electronic Coupling

The electronic coupling was not needed for the calculation of relative rate constants and KIEs and thus was not calculated.

Reaction Free Energy

The reaction free energy is not needed to compute the standard rate constant for this system. As discussed in Sec. IIID, in this case the cathodic and anodic rate constants are expressed as functions of the overpotential $\eta$, which in turn is defined relative to the equilibrium potential. The standard rate constant was computed by calculating the anodic and cathodic rate constants and choosing the value of $\eta$ such that these two rate constants are equal, which corresponds to $\eta =0$ for this system. The work terms in Eq. (10) were not included in the reaction free energy because the reaction was assumed to occur outside the EDL.

Proton Donor-Acceptor Distribution Function

The potential $U(R)$ was assumed to be harmonic and was used in conjunction with Eq. (13) to compute the proton donor-acceptor distribution function $P(R)$. The effective force constant was obtained for the reactant and product by projecting normal modes onto the proton donor-acceptor axis and summing up the appropriately weighted force constants. The equilibrium distance and effective force constant were calculated separately for the reactant and product structures optimized in the gas phase. Then the average equilibrium distance and average effective force constant were used to ensure detailed balance for this reversible reaction. (Note that the equilibrium distance and average effective force constant for the reactant equilibrium structure is used for most thermal and photochemical PCET reactions.³⁴)

PCET rate constant

The total rate constant for this homogeneous electrochemical PCET reaction was computed by numerically integrating Eq. (2), where the integrand is the product of the homogeneous electrochemical rate constant given in Eq. (1) of Ref.²³ and $P(R)$. This procedure was performed for both H and D to calculate the kinetic isotope effect. $P(R)$ and the reorganization energy were assumed to be the same for H and D in this system. Thus, changing the transferring nucleus from H to D only influenced the PCET rate constant via $P_{\mu },\Delta G_{\mu \nu }^0$, and $S_{\mu \nu }$.

To illustrate a photochemical PCET reaction, we also consider the case where an external photosensitizer oxidizes the BIP molecule.⁷⁵ The reaction free energy will depend on the oxidation potential of the photosensitizer and the proton-coupled oxidation potential of the BIP molecule. These quantities can be obtained experimentally or computationally. Here we assume that the reaction free energy is $\Delta G_{00}^0=-5.0\text{kcal}/\text{mol}$. The total rate constant for this photochemical PCET react was computed by numerically integrating Eq. (2) using Eq. (1) as the rate constant. Again, the kinetic isotope effect can be obtained by performing this procedure for both H and D.

C. Heterogeneous Electrochemical PCET at a Surface-Immobilized Cobalt Porphyrin

In this example, the proton is transferred from a hydronium ion in aqueous solution to the cobalt of cobalt tetraphenylporphyrin (CoTPP), which is adsorbed on a graphitic surface, while the electron is transferred from the graphitic surface to the CoTPP (Figure 8a).³⁶ The system has a net positive charge with a single unpaired electron localized on the CoTPP for the reactant and on the graphene for the product. The majority of the input quantities for the PCET rate constant were obtained with a cluster-based model with the CoTPP immobilized on a C₉₆H₂₄ graphitic flake, although the geometry optimizations at constrained $R$ values were performed with periodic DFT.

Figure 8.

(a) Proton transfer from hydronium in aqueous solution to CoTPP adsorbed on a graphitic surface accompanied by electron transfer from the graphitic surface to CoTPP. (b) Calculated proton potential energy profiles for CoTPP immobilized on a C₉₆H₂₄ graphitic flake for the reactant (top) and product (bottom) diabatic electronic states at a range of $R$ values. In this system $R$ is the distance between cobalt and the hydronium oxygen. Figure adapted with permission from Ref.³⁶. American Chemical Society 2024.

Reorganization Energy

The inner-sphere reorganization energy was computed using the four-point scheme using Eq. (5). These calculations used equilibrium geometries obtained from periodic DFT calculations, replacing the extended graphene sheet by a C₉₆H₂₄ flake that was cut from the graphene sheet and optimizing the transferring proton in the gas phase to account for minor differences between the models. Using these geometries, the equilibrium and nonequilibrium quantities in Eq. (5) were calculated in the gas phase. For this system, the proton location controls the location of the transferring electron. The inner-sphere reorganization energy was computed to be ${\lambda }_{\text{i}}=0.83\text{eV}$. The outer-sphere reorganization energy for this system was assumed to be negligible because the PCET reaction does not cause any net change in charge at the catalytic site.

Proton Potential Energy Profiles

For the proton potential energy profiles, we generated average structures and proton coordinate axes for a series of proton donor-acceptor distances $R$ using the diabatic grid-based approach described in Section IIIB. Here $R$ is the distance between the cobalt and the oxygen of the hydronium ion. To maintain the correct electronic state at the different proton coordinates, we used CDFT with spin constraints. We split the system into three fragments: the C₉₆H₂₄ graphitic flake, the CoTPP, and the water cluster. The unpaired electron was constrained to be on the CoTPP for the reactant and on the graphitic flake for the product. The transferring proton was included in the water cluster fragment for the reactant and in the CoTPP fragment for the product. The resulting CDFT energies were then fit to sixth-order polynomials, which were used as the proton potential energy profiles, as shown in Figure 8b.

Electronic Coupling

The electronic coupling was extracted directly from the diabatic state coupling in the CDFT-CI calculations for the proton potentials at the proton donor-acceptor distance of 3.207 Å. The electronic coupling varied somewhat along the proton coordinate axis. We chose the upper bound of the electronic coupling, which should be viewed as only a qualitative estimate.

Reaction Free Energy

The reaction free energy for this PCET reaction contained multiple contributions as described in Section IIID and given by Eq. (10). The first term, $\Delta G_{\text{ads}}$, is the free energy of dissociative adsorption of H₂ on the electrode surface. The specific value for this system was obtained based on estimates from experiments as 0.55 eV for protons and 0.541 eV for deuterons, where the distinction is due to differences in zero-point energy and entropic terms. The work terms, $W_{{\text{HA}}^{+}}$ and $W_{\text{A}}$, were obtained by accounting for both nonbonded interactions and electrostatic interactions, which depend on the applied potential $E$. The nonbonded interactions between the CoTPP and ${\text{H}}_3{\text{O}}^{+}$ as well as between CoHTPP and H₂O were obtained through a series of constrained geometry optimizations in implicit solvent, fixing the positions of Co, the transferring hydrogen, and the hydronium oxygen, and then fitting the data to a Buckingham potential.⁷⁶ The electrostatic contributions were determined using an extended GCS model of the EDL. The electrostatic potential contributions were computed for a series of applied potentials $E$ and distances $R$. Additional computational details are provided in Ref.³⁶.

Proton Donor-Acceptor Distribution Function

The local concentration of the proton donor, $c_{{\text{HA}}^{+}}(R,E)$, was computed with Eq. (15) using $W_{{\text{HA}}^{+}}$, which depends on the applied potential $E$ as well as the proton donor-acceptor distance $R$, and the concentration of protons in the bulk set to 1 M (i.e., pH = 0).

PCET current density

For this system, the constant density of states approximation was not applicable, and therefore the graphene density of states was computed with periodic DFT. This computed density states was used in the calculation of the PCET rate constants at the different proton donor-acceptor distances.³⁶ The current density was calculated as a function of applied potential $E$ by numerically integrating Eq. (14).

D. Code Availability and Example Calculations

We have implemented a Python-based package for computing nonadiabatic PCET rate constants, pyPCET. This code is highly modular, and scripts for each of the systems described above have been created. Reference output files are also provided, showing the relevant outputs of these calculations. The pyPCET code, example scripts, reference outputs, and all necessary input data are available at https://github.com/shsgroup/pyPCET. Detailed documentation is provided in a README file. All examples given in this manuscript are contained in the examples folder of the repository.

Data Availability Statement

There are no new data in this manuscript.