Theory of Proton Discharge on Metal Electrodes: Electronically

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Theory of Proton Discharge on Metal Electrodes: Electronically Adiabatic Model Yan Choi Lam, Alexander V. Soudackov, Zachary K. Goldsmith, and Sharon Hammes-Schiffer J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b02148 • Publication Date (Web): 22 Apr 2019 Downloaded from http://pubs.acs.org on April 22, 2019

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The Journal of Physical Chemistry

Theory of Proton Discharge on Metal Electrodes: Electronically Adiabatic Model Yan-Choi Lam, Alexander V. Soudackov, Zachary K. Goldsmith and Sharon Hammes-Schiffer* Department of Chemistry, Yale University, 225 Prospect Street, New Haven, CT 06520 *

Corresponding author email: [email protected]

Abstract The first step of the hydrogen evolution reaction, an important reaction for the storage of renewable energy, is the formation of a surface-adsorbed hydrogen atom through proton discharge to the electrode surface, commonly known as the Volmer reaction. Herein a theoretical description of the Volmer reaction is presented. In this formulation, the electronic states are represented in the framework of empirical valence bond theory, and the solvent interactions are treated using a dielectric continuum model in the linear response regime. The nuclear quantum effects of the transferring proton are incorporated by quantization along the proton coordinate. The ground and excited state electron-proton vibronic free energy surfaces are computed as functions of the proton donor-acceptor distance and a collective solvent coordinate. In the fully adiabatic regime, the current densities and Tafel slopes are computed from the ground state vibronic free energy surface. This theory is applied to the proton-coupled electron transfer reaction involving proton discharge from H3O+ in aqueous solution to a gold electrode. This theoretical model opens the door for future studies, including examination of the effects of vibronic nonadiabaticity, electronic friction, and solvent dynamics.

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I. Introduction The hydrogen evolution reaction (HER), 2H + + 2e  H 2 , is among the most widely studied electrochemical reactions, owing in part to its technological applications. Despite extensive research, its underlying mechanism is still not fully resolved.1–5 The first step of HER on a metal electrode is generally agreed to be the formation of a surface-adsorbed hydrogen atom, a reaction known as the Volmer reaction or proton discharge:

M  electrode   HA z + + e  M  H ads  A z 1

(1)

The Volmer reaction can also be viewed as an electrochemical proton-coupled electron transfer (PCET) reaction and hence can be described by a general theory for PCET,6–12 analogous to Marcus theory for electrochemical electron transfer (ET).13–17 This PCET theory has been applied to homogeneous PCET reactions in solution6–9,11,12 as well as to homogeneous electrochemical reactions, in which electron transfer (ET) occurs between the electrode and a solvated molecule and proton transfer (PT) occurs in solution.18–20 In contrast, the Volmer reaction involves interfacial PT from a proton donor in solution to an electrode surface site and ET from the electrode to form a surface-bonded hydrogen atom. Recently this general PCET theory was extended to describe this type of heterogeneous reaction and applied to proton discharge from an acid to a gold electrode surface in acetonitrile,21 thereby opening up new directions for theoretical studies. Because protons cannot tunnel over distances longer than ~1 Å, the proton donor prior to the Volmer reaction, as well as the conjugate base after the Volmer reaction, must be in close proximity to the electrode. The proximity of the proton donor to the electrode has several important consequences. One effect is that the local chemical and electrostatic potentials of the proton donor and its conjugate base may differ from their values in bulk solution. Their chemical potentials (i.e., the partial molar free energies) may differ from bulk solution due to partial desolvation and non2


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electrostatic interactions with the electrode. Moreover, the electrostatic potentials acting on charged particles may differ from bulk solution due to incomplete screening of the electrode charges. Therefore, work terms for bringing the proton donor to the electrode surface and removing its conjugate basis from the surface must be considered. In contrast to the nonadiabatic PCET reactions in references 18–20, as well as the Volmer reaction in acetonitrile,21 the Volmer reaction in aqueous solution is thought to be fully adiabatic due to strong electronic coupling between the donor and acceptor and short proton donor-acceptor distances.22–25 In this double adiabatic limit, the transferring proton responds instantaneously to the classical motion of the solvent nuclei, and the electrons respond instantaneously to the motion of the transferring proton. As a result, the reaction proceeds on the ground state free energy surface (FES). Significant attention has been directed toward the theoretical description of the Volmer reaction and related interfacial electrochemical PCET reactions,22,23,26–35 building on and complementing previous pioneering work.36–39 Grimminger and Schmickler proposed a model Hamiltonian to compute potential energy surfaces for electrochemical ET, PT, and concerted PCET,28,29 further developed by Koper to model the transition from sequential to concerted PCET.5 Beginning instead with a model Hamiltonian for electrochemical ET coupled to bond breaking,40– 42

Santos and Schmickler proposed a model for electrochemical hydrogen adsorption that accounts

explicitly for the electrode electronic structure, particularly the interactions of its d-band with the valence orbitals of hydrogen.32,33,43 The absence of electron correlation in their initial model was later rectified using density functional theory (DFT) calculations.22,24,31 Herein we present a different theoretical description of the Volmer reaction, representing the electronic states and their interactions with the solvent in the framework of empirical valence bond (EVB) theory.44–46 The nuclear quantum effects of the transferring proton are incorporated

3



by quantization along the proton coordinate. Thus, the electron-proton vibronic free energy surfaces are computed as functions of the proton donor-acceptor distance and a collective solvent coordinate. This general description is outlined in Section II. In Section III, this approach is illustrated by computing the FESs for the aqueous Volmer reaction on Au(111) electrodes at different applied potentials using a series of well-defined approximations and physically reasonable parameters. The conclusions are presented in Section IV.

II. Theory A. Model Electronic Hamiltonian The electronic Hamiltonian H el for a system composed of a proton donor molecule at a distance R from an electrode surface can be expressed in a basis of general diabatic states corresponding to the reactant and product. Construction of these states in the framework of EVB theory is illustrated in Figure 1. The reactant is described by a single diabatic state  a , corresponding to the proton bonded to the donor molecule and the electrode with delocalized electrons occupying the continuum of one-electron levels k up to the Fermi level k F . The product is described by a continuum of states with a covalent bond between the hydrogen and the metal surface. Analogous to the standard two-state EVB model for proton transfer,44–46 this covalent bond in each of the product states corresponds to a chemical bond with two electrons shared between an orbital representing the electrode valence band state k with energy  k and an orbital localized on the hydrogen. In each product state, all of the one-electron levels in the metal are occupied by two electrons with opposite spins except a single level k that is involved in a covalent bond with hydrogen and therefore is formally occupied by one electron. Thus, by construction, 4


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each many-electron product state can be uniquely identified by the occupied electrode level k and will be denoted  k . Note that these states are simply used as a basis to describe the product of the Volmer reaction characterized by the hydrogen covalently bonded to the electrode surface.

Figure 1: Schematic representation of the reactant and product electronic diabatic states for the Volmer reaction. Note that the Fermi level  F is at the top of the valence band. Two dotted arcs connecting the donor and hydrogen orbitals in the reactant state and hydrogen and metal orbitals in the product states designate a conventional two-electron covalent bond. The intramolecular coordinates r and R correspond to the proton coordinate and the distance between the donor group A and the electrode surface, respectively. In this basis of many-electron diabatic states, the gas-phase electronic Hamiltonian H el(g) is represented by an infinite-dimensional matrix with diagonal matrix elements corresponding to the potential energy surfaces U a (r , R ) and U k (r , R ) for the reactant and product diabatic states, respectively, and off-diagonal elements Vk corresponding to the electronic couplings between the reactant and product states: H el(g)  U a ( r , R )  a  a   U k ( r , R )  k  k k

  Vk  R    a  k   k  a  k

5


(2)


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where the coordinates r and R correspond to the proton coordinate and the distance from the donor group to the electrode surface (i.e., the proton donor-acceptor distance), respectively, as depicted in Figure 1. The summations in the second and third terms of the above expression extend over the product states corresponding to the occupied electronic levels of the metal electrode, with the highest occupied level being the Fermi level k F with energy  F . In our definition of the reactant and product diabatic states, the electronic couplings Vk correspond to the resonance integrals between the metal levels and the proton donor orbital. Therefore, the electronic couplings Vk are assumed to be independent of the proton coordinate r and to decay exponentially with the proton donor-acceptor distance R. This electronic coupling has been shown to be nearly independent of the proton coordinate for a simple model system corresponding to PCET within a phenoxyl-phenol complex.47 The matrix elements U k (r , R ) describe the unperturbed energies of the product states in the reduction reaction where one electron is formally transferred from the electrode orbital k with energy  k to the hydrogen orbital to form a bond between the electrode surface and hydrogen. The Coulomb and exchange interactions between the electron in the hydrogen orbital and electrons in any of the delocalized electrode states with significant contributions from the appropriately oriented surface atomic orbitals (i.e., typically metal d-orbitals oriented perpendicular to the electrode surface) are assumed to be the same. In this case, the energies of the product states differ from each other by the differences in energies of the electrode states from which the electron was transferred. Formally, these energies can be expressed as U k (r , R)  U F (r , R)  k

6


(3)

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where U F (r , R )  U k F (r , R ) is the surface corresponding to the highest occupied level of the electrode (i.e., the Fermi level), which corresponds to the lowest-energy product state. Moreover,

k   k   F are the energies of the occupied electrode states relative to the Fermi energy, and the energies  k are assumed to be independent of the coordinates r and R. Note that all k are negative because the product states are defined only for the occupied states below the Fermi level. The interactions with a polar solvent may be added to the gas-phase electronic Hamiltonian in the framework of the dielectric continuum treatment. Within this treatment, the solvent degrees of freedom are described by a single collective coordinate X, denoted the interaction energy gap coordinate, which is related to the difference in electrostatic interaction energies of the solute charge distributions in the reactant and product states with the inertial polarization field of the solvent. In our solute model, consisting of a proton donor molecule and a metal electrode, all product states have very similar charge distributions, and thus the solvent electrostatic interaction terms can be introduced using the standard prescription for the two-state electron transfer model.16,48,49 The resulting surrogate electronic Hamiltonian for the solvated system with the proton donor at a distance R from the electrode surface is 2  1  H el    X     Ga  R, E   Iˆ  U a  r , R   a  a  4    U F (r , R)  k  X  GF ( R, E )  k  k

(4)

k

  Vk  R    a  k   k  a  k

where Iˆ designates the identity operator,  is the solvent reorganization energy, and E is the applied electrode potential. Ga ( R, E ) is the equilibrium free energy of the proton donor in the reactant state at distance R from the electrode surface relative to the potential-independent free energy in bulk solution. This quantity is equivalent to the work term for bringing the proton donor

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from bulk solution to a distance R from the electrode, and it depends on R and E explicitly due to different electrostatic and non-electrostatic interactions within the electrical double layer at the interface compared to such interactions in bulk solution. Thus, the zero of energy for the electronic Hamiltonian is defined as the free energy of the reactant state with the proton donor in bulk solution. The quantity GF  R, E  is defined as the difference in equilibrium free energies of the reactant and product species at distance R from the surface of the electrode held at the applied potential E, without including the gas-phase intrinsic electronic energy bias. The details of the parameterization of these terms will be discussed in the following section. The electronic Hamiltonian can also be recast in the following equivalent form, which is convenient for further manipulations:

1 2  H el  U a (r , R)  F  r , R, E   Ga  R, E    X     Iˆ  H el 4  

(5)

where

H el   F (r , R, E )  X   a  a

  k  k k  Vk   a k  k  a k



(6)

k

is the reduced electronic Hamiltonian with shifted diagonal matrix elements, and

F  r , R, E   U F  r , R   GF  R, E 

(7)

with U F (r , R)  U F (r , R)  U a (r , R) defined as the gas-phase intrinsic electronic energy bias.

B. Ground Electronic State Adiabatic Free Energy Surface Given the form of the electronic Hamiltonian in Eq.(5), the ground electronic state adiabatic free energy surface or potential of mean force (PMF) at a fixed applied potential E is given by

8


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0  r , R, X ; E   U a (r , R)  Ga  R, E   F  r , R, E  

1 2  X     0  r , R, X ; E  4

(8)

where 0 is the lowest eigenvalue of the Hamiltonian H el given in Eq.(6). Using Löwdin partitioning,50 also known as Feshbach-Fano partitioning51 in scattering theory, 0 can be obtained as the lowest root of the following nonlinear equation (see SI for details):   F ( r , R , E )  X   k

Vk

2

  k

0

(9)

where the summation is over the occupied states below the Fermi level ( k  0 ). This equation can be rewritten in a more conventional form by introducing two functions related to the density of occupied electronic states in the electrode, as used widely in chemisorption theory.52 Specifically, we introduce the width function related to the weighted density of occupied electrode states 2           Vk     k 

(10)

k

and its Hilbert transform, known as the level shift or chemisorption function,      

          d   

(11)

where  ( ) is the Heaviside step function and  denotes a principal value integral. With the use of these two functions, Eq. (9) acquires the form:

     i      0   F (r , R, E )  X  

(12)

As illustrated in Fig. 2, this equation always has a single negative real root 0 because      0

    is a smooth real function with no poles in the integrand of Eq. (11). This for   0 and  isolated localized state below the continuum is analogous to the well-known Fano-Feshbach 9



resonance state appearing in the general model describing the interaction of a discrete state with a continuum of states.51 In our model it represents the discrete ground electronic state separated from the continuum of excited states by a finite energy gap. In practice, the lowest negative root 0 can be found by the numerical solution of the nonlinear equation given in Eq. (12) using an appropriate

   . model for the density of occupied states 

Figure 2: Graphical solution of Eq. (12). For any value of H aa  F ( r , R, E )  X , in the negative region where   0 (red shaded region in the plot), there exists only a single real root 0 representing the ground electronic state in our model. In the positive region where   0 (blue shaded region in the plot), there is a continuum of roots representing a continuum of virtual states. The solid and dashed black lines are representative examples of two different cases, H aa  0 and

H aa  0 , respectively.

C. Quantization of the proton Quantization of the proton coordinate r yields the vibronic free energy surfaces

  R, X ; E  , which incorporate the nuclear quantum effects of the transferring proton. In the

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double adiabatic limit, which is appropriate when the ground electronic state corresponding to the adiabatic free energy surface 0  r , R, X ; E  is well separated from the continuum of virtual excited states, the vibronic surfaces are obtained by solving the Schrödinger equation for the proton moving in the adiabatic potential represented by the PMF 0  r , R, X ; E  : Tˆp   0  r , R , X ; E      r | R , X ; E     R , X ; E     r | R , X ; E   

(13)

where Tˆp is the kinetic energy operator for the proton, and   and  are the eigenfunctions and eigenvalues corresponding to the proton vibrational states and energy levels. Note that the proton can be treated as moving on the PMF because the entropic component of the PMF does not depend significantly on the proton coordinate.53 Herein the proton is assumed to move in one dimension, but the extension to three-dimensional proton motion is straightforward. According to the fully adiabatic picture of the Volmer reaction with inclusion of the quantum effects of the proton motion, the reaction occurs on the two-dimensional ground state vibronic free energy surface 0  R, X ; E  , which depends parametrically on the applied potential

E. In this limit, the excited state vibronic free energy surfaces   R, X ; E  are much higher in energy and may be disregarded, and the rate constant can be calculated using transition state theory (TST) or other approaches that account for dissipative solvent dynamics. The nonadiabatic effects due to the continuum of electronic states corresponding to electron-hole excitations in the electrode can be included by introducing the electronic friction54–61 influencing the dynamics along the classical coordinates R and X. Inclusion of vibrationally nonadiabatic effects typically requires consideration of only the first excited state vibronic surface 1  R, X ; E  , which could become close in energy to the ground state. In this case, the problem can be formulated in terms of an effective two-state model, and the nonadiabatic effects may be incorporated using one of the 11



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various interpolation schemes for rate constants spanning the nonadiabatic and adiabatic regimes.48,62–64 The present work will focus on the fully adiabatic regime, and more sophisticated treatments that include nonadiabatic effects will be the direction of future work.

III. Model Calculations for Volmer Reaction on Gold Electrode The theory outlined in the previous section is general and can be applied to the Volmer reaction for any proton donor in a polar solvent near the metal electrode. In this section, the model will be parameterized for the Volmer reaction in water near a gold (Au) electrode using available experimental and computational data.

A. Gas-Phase Diabatic Potential Energy Surfaces In the simplest implementation, the potential energy functions for the reactant and product diabatic states U a ( r , R ) and U F (r , R ) are modeled as the sum of bonding and non-bonding interactions between the metal surface, the transferring proton, and the proton donor. The bonding interactions are modeled with one-dimensional mirrored Morse potentials for a collinear

M  H  A system, with r and R defined as in Figure 1, while the non-bonding interactions are modeled with Buckingham and Lennard-Jones potentials (see the SI for details): 2

U a ( r , R )  DAH 1  e  AH  r  R  rAH    U anb ( r , R )  U anb ( R  rAH , R ) 2

U F ( r , R )  DMH 1  e   MH  r  rMH    U Fnb ( r , R )  U Fnb ( rMH , R )

(14)

Here DAH and DMH are the bond dissociation energies for the A—H and M—H bonds, respectively, and  AH and  MH are the parameters related to the frequencies of these bonds.

U anb ( r , R ) and U Fnb ( r , R ) denote the potential energies due to non-bonding interactions between

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the solute and electrode surface in the reactant and product diabatic states, respectively. The terms U anb ( R  rAH , R ) and U Fnb (rMH , R) with the proton coordinate fixed at the reactant and product equilibrium positions, respectively, are added to align the minima of the reactant and product potentials at zero energy, so that the intrinsic electronic energy bias (i.e., the difference in the dissociation energies of the A—H and M—H bonds for the Morse potentials) could be absorbed into the free energy bias GF ( R, E ) in Eq.(7).

B. Effects of the applied potential and electrostatic environment of the double layer As mentioned above, the chemical and electrostatic potentials of the species involved in the reaction are not the same near the electrode surface as in bulk solution. Ga  R , E  , defined as the work term for bringing HA+ from bulk solution to the distance R from the electrode, is the sum of an E- and R-dependent electrostatic contribution and an R-dependent non-electrostatic contribution. The electrostatic work term for a mono-cationic proton donor HA+ is given by es  R, E  , where s  R, E  is the electrostatic potential in solution at a distance R from the electrode surface relative to bulk solution and e is the elementary charge. The non-electrostatic work terms for HA+ and A, which reflect the differences in the non-electrostatic components of solvation in bulk solution and near the electrode surface, are denoted WHA  R  and WA  R  , respectively. These work terms can be extracted from molecular dynamics simulations or estimated using model nonbonding potentials from standard molecular mechanical force fields (see SI for details). Hence, the

Ga  R, E  work term for the HA+ species is given by:

13



Ga  R, E   es  R, E   WHA  R  .

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(15)

Estimating the electrostatic potential s  R, E  requires a model of the electrical double layer (EDL). Herein an extended Gouy-Chapman-Stern EDL model is utilized to calculate s  R, E  (see SI for details). Note that any suitable EDL model may be used in conjunction with the general theory presented in Section II, and the model used herein was chosen solely for its simplicity. The total free energy difference GF  R, E  in Eq.(7) can be calculated using a simple thermodynamic cycle. Consider the Volmer reaction with a general acid HA+ in bulk solution, as given in Eq.(1), when the electrode is held at the potential ERHE , the potential of the reversible hydrogen electrode (RHE), for an acid HA+ with the half-reaction HA+ + e− → ½H2 + A. Because the reaction free energy of the RHE half-reaction is zero at E  ERHE , the reaction free energy of the Volmer reaction at this applied potential is the same as that for the reaction obtained by subtracting the RHE reaction from the Volmer reaction. The resulting equation, M + ½H2 → MHads, describes the reaction of the dissociative chemisorption of hydrogen on a metal electrode with the reaction free energy GHads , a quantity that can be measured or computed using standard quantum chemical methods.26,27on Thus, for bulk solution at R   , the total free energy bias is

GF  R  , E   GHads  e  E  ERHE  .

(16)

Including the work terms for both A and HA+, the full expression for GF  R, E  is given by

GF  R, E   GHads  e  E  ERHE   es  R, E   W  R  , where W  R   WA  R   WHA  R  .

14


(17)

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C. The weighted densities of states for the d- and sp-bands of Au electrode Following Newns,65 the d-band of the gold electrode is modeled using a semi-elliptical density of state (DOS) centered at c with a half-width of  d . Assuming that the coupling is constant within the band, Vk  Vd , the width function for the d-band is modeled by

   c  2 | Vd |2  d         1   d   

2

(18)

where     is a Heaviside function. Similarly, the sp-band is modeled using a semi-elliptical DOS centered at the Fermi level with a half-width of  sp . Assuming that the magnitude of the coupling is constant within the band, the width function for the sp-band is 2

2 | Vsp |  sp        

 sp

  1    sp

  

2

(19)

With these approximations, the corresponding level shift functions given in Eq.(11) can be evaluated analytically, yielding the following expressions for the d- and sp-components in the case of fully filled d-bands ( c   d  0 ): 2      2 | Vd |      c d  d2 

   c   d    c   d  

 sp | Vsp |2   2 2    sp         2 arcsin          sp    s2p  sp  

(20)

This case is directly relevant to the gold electrode. In the case of a partially filled d-band, the

    is more complicated and is given in the SI. Note that for   0 , the closed expression for  d right-hand sides of Eqs. (20) are smooth functions that decrease monotonically from zero to negative infinity with increasing  , ensuring that a single negative real root of Eq. (12) always exists in this range.

15



D. Parameters The Morse potential parameters were obtained from a combination of calculations and experimental data. The parameters of the Morse potential for H on a Au(111) top site were obtained by DFT (see SI for details). According to these calculations, the equilibrium bond length is rAuH = 1.60 Å, the dissociation energy is DAuH = 52 kcal mol-1, and the stretching frequency is ωAuH = 1919 cm-1, leading to βAuH = 1.70 Å-1. The parameters for H3O+ were computed from its experimentally measured OH stretching frequency (3536 cm-1)71 and homolytic bond dissociation energy, which was calculated with a thermodynamic cycle involving the gas phase proton affinity (165 kcal mol-1)66 as well as the first ionization potentials of H2O (291 kcal mol-1)67 and the hydrogen atom (313.6 kcal mol-1)68. These values resulted in DOH = 142.5 kcal mol-1 and βOH = 1.94 Å-1, and the equilibrium bond length was chosen to be the previously reported value of rOH = 0.976 Å.69 The projected DOS for the d- and sp- bands of bulk Au were also obtained from DFT calculations (see SI for details). The parameters  c ,  d , and  sp were approximated as −90, 60, and 200 kcal mol-1, respectively. Our DFT calculations indicate that hydrogen chemisorption on the top site of Au(111) is more endergonic than on a face centered cubic (fcc) site by 0.21 eV (see SI for details). Despite this finding, the top site was chosen to serve as the proton acceptor because the Au-H bond is further from the electrode surface, resulting in a shorter proton transfer distance. Because previous calculations estimate the adsorption free energy of H to be +0.41 eV for a half a molecule of H2 on an fcc site,22,31 GHads was estimated to be +0.62 eV. Because at pH=0, ERHE is −0.244 V vs. SCE for H3O+ in aqueous solution,70 and EPZC for Au(111) has been measured to be +0.24 V vs. SCE,71 ERHE is −0.484 V vs. EPZC, the reference used in the EDL model (see SI). The remaining quantities required for the model calculations are treated as fitting parameters, albeit

16


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within certain boundaries. In particular, 11 kcal mol-1 has been estimated to be an upper limit for the reorganization energy λ, owing to the experimental observation of a barrierless Volmer reaction on Au electrodes.72 This moderate value of reorganization energy is consistent with our assumption that the H3O+ ion is delivered to the electrode surface via the Grötthus mechanism and retains most of its solvation shell. Moreover, the hydrogen bond between one H2O molecule and H3O+ within the Eigen cation (H9O4+) must break prior to proton transfer to the electrode surface. For this reason, solv the work term WH O is not expected to be much larger than the enthalpy of one hydrogen bond in 3

the Eigen cation (~13 kcal mol-1).73 Finally, the electronic couplings Vd and Vsp are assumed to decay exponentially with distance with the decay constant   = 1.0 Å-1:18

V  R   V  R0  exp      R  R0   .

(21)

The parameters mentioned above, and some others, are summarized in Table 1.

Table 1: Parameters Used for Model Calculations.

DAuH / kcal mol-1 DOH / kcal mol

52

-1

142.5

βAuH / Å-1

1.70

βOH / Å-1

1.94

rOH / Å

0.976

rAuH / Å

1.60

 c / kcal mol-1

-90

 d / kcal mol-1

60

 sp / kcal mol-1

200

ΔGHads / kcal mol-1

+14.3

ERHE / V vs. EPZC

−0.484

λ / kcal mol-1

7.5

17



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WHsolv / kcal mol-1 2O

0

WHsolvO / kcal mol-1

+15.0

3

E. Calculation of Tafel Plots and Kinetic Isotope Effects The lack of detailed knowledge about solvent dynamics at the electrode-solvent interface renders the application of reaction rate theories requiring solvent friction coefficients, such as Kramers-Grote-Hynes theory, problematic. Under certain assumptions, however, the observed current density will depend on only the potential-dependent activation free energy and H3O+ concentration at the dominant distance. This dominant distance is assumed to be R = RTS, corresponding to the saddle point of the ground state vibronic free energy surface 0  R, X ; E  . The first assumption is that the dependence of the pre-exponential factor on the applied potential E is negligible. The second assumption is that the Volmer reaction occurs over a narrow range δR around RTS, where δR does not vary significantly with E. In this case,39

 Gact  R TS , E    j  E   constant  cH O+  R , E   R exp   3 kBT   TS











(22)



where Gact R TS , E  0 R TS , X TS ; E  0 R TS , X 0 ; E is the free energy barrier, and X0 is the reactant minimum along the solvent coordinate X for the one-dimensional slice of the free energy





surface 0 R TS , X ; E . The H3O+ concentration at RTS depends on the free energy difference between H3O+ at RTS and in bulk solution as follows:

 0  R TS , X 0 ; E   0  R  , X      cH O+  R ; E   cH O+  R    exp   3 3 kBT   TS

Eq. (22) may then be rewritten:

18


(23)

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ln j  E   constant 

where X

TS

0  R TS , X TS ; E   0  R  , X    k BT

(24)

and  are the values of X at the saddle point and at the reactant minimum in bulk

solution, respectively, and the bulk quantity 0  R   , X    does not depend on E. The observed (apparent) transfer coefficient is therefore given by:

 app

TS TS kBT  ln j  E  1 0  R , X , E    . e e E E

(25)

Because the differences in solvent properties between H2O and D2O are minor, the difference in the pre-exponents for the Volmer reaction in H2O and D2O should be negligible. Thus, the kinetic isotope effect (KIE), which is the ratio of the current densities in H2O and D2O at electrode potential E, will be exponentially dependent on the difference in the isotope-dependent













TS TS TS TS activation free energies, Gact R ; E  0 R , X ; E  0 R , X 0 ; E , for H and D.

IV. Results Two-dimensional slices of the ground state electronic free energy surfaces  0  r , R , X ; E  at a series of fixed values of the proton donor-acceptor distance R and applied potential E were computed using Eq. (8). The parameters used to generate these plots and all other plots in this section are summarized in Table 1. In the two-dimensional (2D) free energy surfaces (FESs) as functions of the proton coordinate r and the collective solvent coordinate X, the well-defined reactant and product minima are located near X   , r  R  rOH and X   , r  rAuH , respectively (Figure 3).

19



Figure 3: Two-dimensional FESs as functions of r and X computed at R = 3.0 Å, T = 298 K, and E = −0.42 V vs. RHE. These plots were generated using the parameters in Table 1, in conjunction

20


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with Vd = 50, Vsp = 30 (top), Vd = 60, Vsp = 30 (center), or Vd = 50, Vsp = 40 (bottom) kcal mol-1. The energies at the reactant and product minima, as well as at the saddle point, are given in kcal mol-1. Note that the reaction proceeds from the reactant minimum in the top-right corner to the product minimum in the bottom-left corner of each FES, and the contour lines are spaced 1 kcal mol-1 apart.

The 2D FESs are influenced by the magnitude of coupling to the d- and sp-band states, Vd and Vsp, in a manner unique to systems with a continuum of product diabatic states. In particular, an increase in either Vd or Vsp stabilizes the reactant region and the saddle point, while having virtually no effect on the product region (Figure 3). This variation of the reaction free energy with coupling is characteristic of heterogeneous electron transfer processes where electrons are transferred to or from an electrode.74 In contrast, for a two-state model, increasing the coupling greatly stabilizes the saddle point region and slightly destabilizes both the reactant and product regions, such that the reaction free energy of a thermoneutral reaction remains zero. For each value of the solvent coordinate X and proton donor-acceptor distance R, the onedimensional Schrödinger equation for proton motion along the proton coordinate r can be solved to compute the electron-proton vibronic states. At a fixed value of the proton donor-acceptor distance R, the quantization along r results in one-dimensional (1D) FESs for the ground and excited vibronic states, as shown in Figure 4. As with the 2D FESs from which they are calculated, increasing the coupling stabilizes the reactant region while exerting little effect on the product region.

21



Figure 4: One-dimensional FESs for the first three electron-proton vibronic states, obtained from the 2D FESs in Figure 3 by quantization along the proton coordinate r. Note that the reaction proceeds from the reactant minimum at positive values of the solvent coordinate X to the product at negative values of X.

22


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Figure 5: Two-dimensional FESs as functions of R and X computed at T = 298 K and E = −0.42 V vs. RHE, using the parameters in Table 1, together with Vd = 60, Vsp = 20 (top), or Vd = 65, Vsp = 20 (center), or Vd = 60, Vsp = 30 (bottom) kcal mol-1. Note that the reaction proceeds from the reactant minimum in the bottom right corner to the product minimum in the top left corner of each FES, and the contour lines are spaced 0.5 kcal mol-1 apart.

23



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Calculating the quantized 1D FESs in Figure 4 at different proton donor-acceptor distances R leads to quantized 2D FESs that are functions of the collective solvent coordinate X and the proton donor-acceptor distance R (Figure 5). An important feature is that the product minima occur at larger proton donor-acceptor distances than the corresponding reactant minima. In other words, the equilibrium position of H3O+ is closer to the electrode surface than the equilibrium position of H2O relative to an electrode surface with bound hydrogen atoms. This observation is in agreement with ab initio molecular dynamics simulations showing that the water molecule leaves the Pt(111) surface immediately after transferring a proton to the surface,75 and it may be attributed to two attractive forces present in the reactant state but not the product state. First, the negatively charged electrode surface exerts an electrostatic attractive force on the cationic H3O+ but not on the neutral H2O. Second, the exponential increase of the coupling with decreasing proton donor-acceptor distance R leads to greater stabilization of the reactant region at shorter distances. In contrast, as shown in Figure 3, the product region is not noticeably affected by changes in the coupling. For the same reason, the reactant minimum occurs at shorter distances with an increase in coupling, while the product minimum is virtually unaffected by this increase in coupling. The quantized 2D FESs exemplified in Figure 5 were computed at a series of different electrode potentials E using the parameters in Table 1, with the exception that the non-electrostatic solv work term for H3O+ was adjusted to WH O = 10 kcal mol-1 to improve agreement with the 3

experimentally measured Tafel slopes, in conjunction with Vd = 60 kcal mol-1, and Vsp = 30 kcal mol-1.





The potential-dependent saddle point free energies 0 RTS , X TS ; E in Eq. (24), which are equivalent to the observed activation free energies, were found to decrease monotonically from ca. 9.3 kcal mol-1 at ‒0.17 V vs. RHE to ca. 6.0 kcal mol-1 at ‒0.42 V vs. RHE, in reasonable 24


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agreement with experimental measurements76 (8.8 kcal mol-1 at ‒0.2 V vs. RHE, decreasing to 7.2 kcal mol-1 at ‒0.35 V vs. RHE). The transfer coefficient app calculated from a quadratic fit of these activation energies (Figure 6) decreases from 0.67 at ‒0.17 V vs. RHE to 0.49 at −0.42 V vs. RHE, in reasonable agreement with Hamelin et al.’s experimental Tafel plot measurements.71 The computed KIE decreases slightly from 2.3 at −0.65 V to 1.9 at −0.85 V (Figure 6). Both of these values are smaller than the KIE of ~ 4.3 measured for polycrystalline Au electrodes in 0.1 N aqueous HCl.77

Figure

6:

Calculated

apparent

activation

free

energies,

defined

as

0  R , X ; E   0  R , X    in Eq. (24), as functions of the electrode potential E TS

TS

solv computed at T = 298 K using the parameters in Table 1, except that WH O = 10 kcal mol-1, in 3

conjunction with Vd = 60 kcal mol-1, and Vsp = 30 kcal mol-1. The calculated data are represented as filled circles, and the line represents a quadratic fit to these data points, where H3O+ is shown in blue and D3O+ is shown in red.

The underestimation of the KIE may reflect a degree of nonadiabaticity in the Volmer reaction. The two aforementioned factors stabilizing the reactant region but not the product region (i.e., the electrostatic attraction and the coupling) are especially strong closer to the electrode. Consequently, at short proton donor-acceptor distances the reaction becomes sufficiently

25



endergonic that the 1D FESs along the solvent coordinate X lack a product minimum. The shortest distance at which a product minimum exists at an electrode potential of −0.65 V vs EPZC is ~3.13 Å. At this distance, the splitting between the ground and first excited vibronic states at the maximum of the ground state along the solvent coordinate X is ~0.78 kcal mol-1. At more cathodic potentials, the shortest distance at which a product minimum exists decreases, and the splitting between the ground and first excited vibronic states at this shorter distance is larger. The lack of information about solvent dynamics at the electrode-solvent interface prevents us from drawing conclusions about the degree of nonadiabaticity and the role that solvent dynamics may play. This model may be compared to the model proposed previously by Santos, Schmickler, and coworkers.22,24,31,78 Both models treat interactions with the solvent in the linear response regime and introduce the electrode potential in a similar manner. The main difference is that the previously published model describes the adsorption of a free proton, which is not covalently bonded to a donor atom, onto the electrode surface, and therefore proton adsorption involves bond formation without bond breaking. In contrast, the present model describes the formation of the metal-hydrogen bond concurrent with the breaking of the donor-hydrogen bond, and thus the proton transfers from the donating acid to the electrode surface. Thus, the model presented here explicitly includes the proton donor-acceptor distance and distinguishes between different acids. The calculations performed herein can be refined in several ways. First, it may be possible to compute the width functions (i.e., the weighted densities of states)  d    and  sp    for the d- and sp-bands from electronic structure calculations in conjunction with a partitioning scheme to define donor-localized electronic states and electrode bands.79,80 In principle, the width functions defined in Eqs. (18) and (19) can be computed and fit to a sum of several semi-elliptical functions instead of one each for the d- and sp-bands. Second, recent advances in molecular

26


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dynamics simulations allow an atomistic treatment of the electrode-solvent interface and of solutesolvent interactions. Such simulations can provide a more accurate description of the EDL and the impact of the applied potential on the structure of the EDL. These simulations are also able to describe phenomena such as hydrogen bonding between the solute and the solvent, which are not described in continuum descriptions of the solvent.81–83 A combination of input from such simulations and the model presented herein could provide more quantitatively reliable insights.

V. Conclusions This paper presents a theoretical description of the Volmer reaction. In this model, the electronic states are described in the framework of empirical valence bond theory, and the solvent effects are incorporated with dielectric continuum theory in the linear response regime. The ground electronic state free energy surface is computed as a function of the proton coordinate r, a collective solvent coordinate X, and the proton donor-acceptor distance R, which in this case is the distance between the acid donor and the electrode. The nuclear quantum effects of the transferring proton are incorporated via quantization along the proton coordinate, resulting in a twodimensional vibronic free energy surface that is a function of the solvent coordinate X and the proton donor-acceptor distance R. This model was applied to the Volmer reaction corresponding to proton discharge from H3O+ in aqueous solution on a gold electrode. Although the analysis herein focuses on the ground state vibronic free energy surface, the excited vibronic state surfaces are also computed within the double adiabatic approximation and could be used to incorporate nonadiabatic effects. Two interesting features of the free energy surfaces were observed in our calculations. First, increasing the electronic coupling stabilizes the reactant and saddle point regions while not 27



affecting the product region. This observation is in contrast with two-state models, where increasing the electronic coupling strongly stabilizes the saddle point region while slightly stabilizing both the reactant and product regions. Second, the proton donor-acceptor distance is substantially larger for the product minimum than for the reactant minimum, implying that the proton donor (i.e., the conjugate base) moves away from the electrode surface after proton transfer. Two factors contribute to this phenomenon: the electrostatic attraction between a negatively charged electrode and the cationic H3O+, and the increase in electronic coupling, and therefore stabilization, for the reactant with decreasing proton donor-acceptor distance. Neither of these two stabilizing factors apply to the product region: there is no electrostatic attraction between the electrode and neutral H2O, and the product region is not noticeably stabilized by an increase in the coupling. In addition, the current densities were computed for the fully adiabatic reaction occurring on the ground state vibronic free energy surface as a function of the applied potential E. The calculated transfer coefficients are in good agreement with experimental measurements, but the H/D KIEs are lower than experimental data, suggesting that vibronic nonadiabaticity may play a role in this reaction. Inclusion of nonadiabatic effects is expected to slightly increase the KIEs, thereby improving agreement with experimental measurements. In addition, refinements to the description of the diabatic electronic states, the electronic coupling between these states, and the electrochemical double layer are also expected to enhance the quantitative agreement with experiments. Other directions for future work include the investigation of the effects of vibronic nonadiabaticity, electronic friction, and solvent dynamics.

28


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Associated Content Supporting Information. Derivation of Eq. (9), electrochemical double layer model,

parameterization of non-bonding interactions, details of DFT calculations.

Acknowledgments This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-18-1-0420. Y.C.L. was supported in part by a Croucher Foundation Postdoctoral Fellowship.

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Theory of Proton Discharge on Metal Electrodes: Electronically

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