Kinetics of Proton Discharge on Metal Electrodes: Effects of Vibrational

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Kinetics of Proton Discharge on Metal Electrodes: Effects of Vibrational Nonadiabaticity and Solvent Dynamics Yan-Choi Lam, Alexander V. Soudackov, and Sharon Hammes-Schiffer* Department of Chemistry, Yale University, 225 Prospect Street, New Haven, Connecticut 06520, United States

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S Supporting Information *

ABSTRACT: Proton discharge on metal electrodes, also denoted the Volmer reaction, is a critical step in a wide range of electrochemical processes. This electrochemical proton-coupled electron transfer (PCET) reaction is predominantly electronically adiabatic in aqueous solution and is typically treated as fully adiabatic. Recently, a theoretical model for this PCET reaction was developed to generate the vibronic free energy surfaces as functions of a collective solvent coordinate and the distance of the proton-donating acid from the electrode. Herein a unified formulation is devised to describe such PCET reactions in terms of a curve crossing between two diabatic vibronic states corresponding to the lowest two proton vibrational states, employing an interpolation scheme that spans the adiabatic transition state theory, nonadiabatic Fermi golden rule, and solvent-controlled regimes. In contrast to previous treatments, application of this formulation to the aqueous Volmer reaction highlights the importance of vibrational nonadiabaticity and solvent dynamics. The calculated transfer coefficients and kinetic isotope effects are in reasonable agreement with experimental measurements. These fundamental insights have broad implications for understanding electrochemical processes.

S

vibronic states decrease with increasing proton donor− acceptor distance.6 Thus, although the aqueous Volmer reaction is expected to be electronically adiabatic, it may range from vibrationally nonadiabatic to adiabatic, depending on the proton donor−acceptor distance, and the overall degree of vibrational nonadiabaticity is determined by the distances with dominant contributions to the current density. In addition, solvent relaxation dynamics may also influence reaction kinetics. 7−13 Herein, the Volmer reaction is formulated in a curve crossing framework, i.e., as a radiationless transition between two diabatic vibronic FESs corresponding to the lowest two proton vibrational states. The reactant and product are defined in terms of diabatic vibronic states that are obtained from diabatization of the adiabatic ground and first excited vibronic state FESs. Landau−Zener theory,14,15 which describes the effects of nonadiabaticity, is combined with the stable-states picture of chemical reactivity,16,17 which accounts for the effects of solvent dynamics. This unified formulation is used to obtain rate constant expressions valid for a range of vibronic couplings V and longitudinal dielectric relaxation times τL, analogous to those derived by Makarov and Topaler for electron transfer.13 The reaction is assumed to occur along a single reaction coordinate, namely, the collective solvent coordinate X.7 Assuming rapid equilibration along the proton donor− acceptor coordinate R on the time scale of the collective

everal important electrochemical processes, including water electrolysis, involve the deposition of hydrogen atoms on a metallic electrode by proton discharge from an acid: M(electrode) + HAz + +e− → M−Hads + A(z − 1) +

(1)

A variety of theoretical models have been proposed to describe this reaction, commonly known as the Volmer reaction.1−6 In our recently developed model,6 the electronic states are described in the framework of empirical valence bond theory. The electronic Hamiltonian is expressed in the basis of a single reactant state in which the proton is bonded to the acid molecule, and a continuum of product states in which the proton forms a covalent bond with the electrode. The solvent degrees of freedom are represented by a single interaction energy gap coordinate, denoted X. Diagonalization of the Hamiltonian results in an isolated electronic ground state separated from a continuum of excited electronic states by a finite energy gap. The electronic ground state energy is a function of the proton coordinate, the solvent coordinate X, and the proton donor−acceptor distance R, which is the distance of the acid donor atom from the electrode. The nuclear quantum effects of the transferring proton are described by quantization along the proton coordinate, resulting in ground and excited vibronic states (i.e., proton vibrational states associated with the ground electronic state), and the free energies of these vibronic states are functions of X and R.6 The free energy surfaces (FESs) calculated using our model indicate that the splittings between the ground and first excited © XXXX American Chemical Society

Received: July 8, 2019 Accepted: August 22, 2019 Published: August 22, 2019 5312

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The Journal of Physical Chemistry Letters solvent fluctuations, the current density is computed by integrating the product of the rate constant k(R) and the local acid concentration c HAz+(R ) over R:

∫

j = F k(R )c HAz+(R )dR

(2)

ÄÅ É ÅÅ ΔG≠ ÑÑÑ eff Ñ ÑÑ k(R )c HA+(R ) = c HA+(R → ∞)ν(R ) expÅÅÅÅ− ÅÅ kBT ÑÑÑ ÅÇ ÑÖ

(3)

γ LZ(R , v) ≡

(7)

Note that the vibronic coupling V denotes the coupling between the reactant and product vibronic states and is typically significantly smaller than the electronic matrix element. Due to different recrossing possibilities, the reaction yield ϒ depends differently on the transition probability PLZ in the normal and abnormal regions,20−23 where the first derivatives F1 and F2 have different and the same signs, respectively:

The integrand can be expressed as

l 2P LZ(R , v) o o o o , o LZ o o o 1 + P (R , v) ϒ(R , v) = m o o o o o o 2P LZ(R , v)[1 − P LZ(R , v)], o o o n

where the first term on the right-hand side is the bulk acid concentration, ν(R)is a pre-exponential term, and ΔG≠eff is the effective activation free energy that includes the work term needed to bring the acid from bulk solution to a distance R from the electrode surface. In this formulation, the reaction occurs at all distances R with weightings given by the local acid concentration. Because describing the reaction in the curve-crossing formulation leads to cusp barriers, the reaction rate constant can be significantly affected by solvent dynamics around the reactant and product minima, but not around the crossing.9 The inverse of the pre-exponential factor ν(R) can therefore be written using the stable-states picture formalism:16,17 1 1 1 = LZ + SC ν(R ) ν (R ) ν (R )

2π |V (R )|2 ℏ|v||F1(R ) − F2(R )|

F1(R ) < 0 (normal region) F2(R ) F1(R ) > 0 (abnormal region) F2(R )

(8)

The Landau−Zener pre-exponent is calculated by averaging over the thermal Maxwell−Boltzmann distribution of velocities:20−22 ÄÅ ÉÑ 2 ÑÑ ∞ ÅÅÅ 1 v LZ ÑÑdv ν (R ) = v ϒ(R , v)expÅÅÅ− 2 Ñ ÅÅ 4ω λkBT ÑÑÑ 4πωλkBT 0 Ç Ö

∫

(9)

where ω denotes the effective frequency along the reaction coordinate near the reactant minimum. As shown in the Supporting Information, the adiabatic and nonadiabatic limits21,22 of eq 9 in the normal region are equal to the adiabatic transition state theory (TST) pre-exponent, νTST = ω/2π, and the Fermi golden rule (FGR) pre-exponent, νFGR ∝ V2, respectively. Figure 1 depicts the dependence of the preexponent on the vibronic coupling V and solvent relaxation time τL. To calculate the diabatic FES and vibronic coupling at each proton donor−acceptor distance R, an adaptation24 of the diabatization scheme first proposed by Baer25 was used. The assumption of a single reaction coordinate circumvents potential complications in the diabatization procedure.25 As a

(4)

where νLZ denotes the Landau−Zener pre-exponent valid in the limit of fast solvent dynamics and νSC denotes the solventcontrolled pre-exponent.7 Equation 4 indicates that the time scale of the reaction can be approximated as the sum of the time scale associated with nonadiabatic transitions at the crossing, given by 1/νLZ, and the time scale of solvent dynamics, given by 1/νSC. This approximation is valid only for thermally activated processes, i.e., when the activation free energy is substantially larger than the thermal energy kBT. The rate constant expressions for nearly activationless and activationless processes11 are not considered here, as the Volmer reaction is thermally activated under experimental conditions.18 The time scale of solvent dynamics is proportional to the solvent (effective) longitudinal relaxation time τ L as follows:7,12,19 yz i 1 1 1 zz = τL 4πλkBT jjjj + |ΔG(R ) − λ| z{ ν (R ) k |ΔG(R ) + λ| SC

(5)

where ΔG(R) is the reaction free energy at distance R and λ is the solvent reorganization energy. Because eq 4 is only valid for activated processes, where |ΔG(R)| ≠ λ, eq 5 leads to a finite time scale for solvent dynamics. In the limit of fast solvent dynamics, the Landau−Zener formalism is used to account for the effects of vibronic nonadiabaticity. The probability of a nonadiabatic transition at a single crossing is a function of the vibronic coupling V, the velocity at the crossing point v, and the first derivatives of the diabatic FESs at the crossing point, denoted F1 and F2: P LZ(R , v) = 1 − exp[−γ LZ(R , v)]

Figure 1. Schematic representation of the unified formulation used in this work to account for both nonadiabaticity and solvent dynamics. In the limit of fast solvent dynamics (bottom of figure), the preexponent is given by νLZ, which in the normal region interpolates between the nonadiabatic FGR limit νFGR (bottom left) and the adiabatic TST limit νTST (bottom right). In the limit of slow solvent dynamics (top), the pre-exponent is given by νSC. Equation 4 interpolates between νLZ and νSC. This model is valid for thermally activated processes in the curve-crossing formulation.

(6)

where 5313

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estimate of 11 kcal mol−1 as an upper limit.26 This value is reasonable because proton discharge occurs at the interface within the inner Helmholtz layer, which has a lower effective dielectric constant than the bulk solution. At short proton donor−acceptor distances, the reaction is sufficiently endergonic that the adiabatic ground state FES has a single stationary point, namely, a reactant minimum (Figure 2A). This endergonicity is due to three factors.6 The first is the strong steric repulsion between the electrode-adsorbed H and H2O in the product state. The second is that the electronic coupling, which stabilizes the reactant state but not the product state, increases exponentially with decreasing proton donor−acceptor distance.6 The third is the strong electrostatic attraction between H3O+ and the negatively charged gold electrode in the reactant state. Despite the lack of a product minimum on the adiabatic ground state FES, diabatization results in reactant and product diabatic FESs with well-defined minima. Consequently, the reaction rate constant in the curve crossing formulation can be defined even in the absence of a product minimum on the adiabatic ground state FES. With increasing proton donor−acceptor distance, the reaction becomes less endergonic, and the adiabatic ground state FES has a transition state between two well-defined minima (Figure 2B). As the proton donor−acceptor distance increases, the gap between the two adiabatic surfaces at the avoided crossing decreases, the derivative coupling increases, the vibronic coupling V decreases, and the diabatic FESs become more similar to the adiabatic FESs (Figure 2). Moreover, the vibronic coupling V is smaller for deuterium (D) than for hydrogen (H) and decreases more rapidly with increasing donor−acceptor distance for D than for H (Figure S1). The pre-exponent given in eq 4, as well as the various limiting cases, is plotted as a function of the proton donor− acceptor distance R in Figure 3. The solvent longitudinal relaxation time τL was taken to be that of bulk water, 0.462 ps.27 The estimation of the effective solvent frequency ω is explained in the SI. At very short proton donor−acceptor distances, the reaction is sufficiently endergonic to be in the abnormal region, as defined in eq 8. Note that this behavior can occur even when the magnitude of the reaction free energy is less than the reorganization energy, |ΔG(R)| < λ, because the diabatic FESs are not exactly parabolic. In this region, the vibronic coupling V is high, resulting in adiabatic suppression21,22,28 and low pre-exponents (Figure 3, small R values). Adiabatic suppression, in which the rate constant of an adiabatic process in the abnormal region decreases with increasing vibronic coupling, is more pronounced at shorter proton donor−acceptor distances and for H compared to D due to larger vibronic couplings (Figure 3A,B). At longer proton donor−acceptor distances, the reaction is in the normal region. This transition from the abnormal to the normal region eliminates adiabatic suppression. The preexponents are consequently larger than those in the abnormal region and are predominantly controlled by solvent relaxation dynamics because νTST > νSC for the parameters used here (Figure 3, R values where black and brown curves are the same). This inequality is likely to be valid for most solvents less polar than water around ambient temperature. These solvents have longer longitudinal relaxation times τL and therefore smaller solvent-controlled pre-exponents νSC. As the electrode potential becomes less cathodic (i.e., at less negative potentials), the reaction becomes more endergonic, and the

further approximation, only the adiabatic vibronic ground and first excited states were considered. The derivative coupling d12 between these two adiabatic vibronic states, denoted |Ψ1⟩ and |Ψ2⟩, at specified values of X and R is given by the following expression: ∂ |Ψ2⟩ (10) ∂X The details for calculating the derivative coupling within this model are given in the Supporting Information. The diabatic vibronic states |Φ1⟩ and |Φ2⟩ are defined as linear combinations of adiabatic states satisfying the condition ⟨Φ1|∂/∂X|Φ2⟩ = 0. For two adiabatic states with free energies >1 and >2 , the diabatization transformation matrix is a rotation matrix U, defined by a single rotation angle θ:24 d12(X , R ) = ⟨Ψ|1

ijG1(X , R ) V12(X , R )yz i yz 0 jj zz = Ujjj >1(X , R ) zzU −1 jj zz jj zz j V (X , R ) G ( X , R ) z j z > 0 ( X , R ) 2 2 k 12 { k {

(11)

ij cos θ(X , R ) −sin θ(X , R )yz zz U = jjjj z j sin θ(X , R ) cos θ(X , R ) zz k {

with

θ (X , R ) = θ ( X ≠ , R ) −

∫X

X ≠

d12(Y , R )dY

(12)

In eq 11, G1 and G2 are the diabatic FESs, and V12 is the vibronic coupling between the reactant and product diabatic states. The integration constant θ(X≠,R) is specified at X≠, which is the point of maximum derivative coupling, and is set to −π/4, ensuring that the diabatic FESs G1(X) and G2(X) cross at this point (Figure 2).24 The adiabatic FESs for proton discharge from H3O+ to a gold electrode were computed using the model previously established6 with the parameters given in Table S1. The reorganization energy of 10 kcal mol−1 is consistent with the

Figure 2. (A) FES (left axis) for the adiabatic vibronic states (dashed black lines) and diabatic vibronic states (solid lines, reactant state in red, product state in blue). The adiabatic FESs were calculated as elaborated in ref 6 using the parameters in Table S1 at electrode potential E = −0.35 V vs RHE and proton donor−acceptor distance R = 3.0 Å. The diabatic FESs were computed using eq 11. The purple solid line represents the derivative coupling d12 (right axis) as a function of the solvent coordinate. (B) Same as (A), except at proton donor−acceptor distance R = 3.1 Å. 5314

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Figure 4. Concentration-weighted rate constants as functions of the proton donor−acceptor distance R calculated at electrode potential of −0.35 V vs RHE (solid lines) or −0.15 V vs RHE (dashed lines) for H (red) and D (blue). Note that the discontinuities occur when the reaction transitions from the abnormal to the normal region because eqs 4 and 9 are not valid for reactions in the activationless or nearly activationless regime.

rate constant is near its highest value. As the proton donor− acceptor distance increases further, the vibronic coupling V decreases sufficiently such that νLZ < νSC, and the rate constant decreases as a result. Consequently, the dominant contributions to the current density occur predominantly at proton donor−acceptor distances between the transition from the abnormal to the normal region and the transition from the solvent-controlled to the FGR regime (Figures 3 and 4). Because the latter transition occurs at shorter donor−acceptor distances for D than for H, the dominant contributions to the current density arise from shorter donor−acceptor distances for D than for H (Figure 4). At less cathodic potentials, the transition from the adiabatically suppressed abnormal region to the normal region occurs at longer proton donor−acceptor distances (Figure 3C). At the same time, the electrostatic attraction between H3O+ and the gold electrode and, therefore, the local H3O+ concentration decrease less rapidly with proton donor− acceptor distance. These two factors cause the dominant contributions to the current density to occur at longer proton donor−acceptor distances (Figure 4, dashed lines). The current densities were computed using eq 2 for a series of electrode potentials for both H and D (Figure 5). The

Figure 3. Pre-exponents as functions of the proton donor−acceptor distance R, calculated using parameters given in Table S1. The preexponents correspond to the nonadiabatic FGR limit (red dashed line), the adiabatic TST limit (blue dashed line), the Landau−Zener expression given by eq 9 (purple solid line), the solvent-controlled limit given by eq 5 (brown dashed line), and the overall pre-exponent given by eq 4 (black solid line). Pre-exponents calculated for (A) H3O+ at electrode potential of −0.35 V vs RHE, (B) D3O+ at electrode potential of −0.35 V vs RHE, and (C) H3O+ at electrode potential of −0.15 V vs RHE. Note that the discontinuities in the Landau−Zener and overall pre-exponents occur when the reaction transitions from the abnormal to the normal region because eqs 4 and 9 are not valid for reactions in the activationless or nearly activationless regime.

transition from the abnormal region to the normal region occurs at longer proton donor−acceptor distances (Figure 3C). At still longer proton donor−acceptor distances, the vibronic coupling V decreases sufficiently that the Landau−Zener preexponent becomes smaller than the solvent-controlled preexponent, νLZ < νSC, and the pre-exponent approaches the FGR limit (Figure 3, large R values where black, red, and purple curves are the same). This approach to the FGR limit occurs at shorter proton donor−acceptor distances for D than for H because the vibronic couplings are smaller and decrease more quickly with distance for D (Figure S1). Expressing the local H3O+ concentration at proton donor− acceptor distance R as a fraction of the bulk concentration, i.e., P(R ) ≡ c H3O+(R )/c H3O+(R → ∞), the scaled integrand in eq 2 can be plotted as k(R)P(R) for H and D (Figure 4). At very short proton donor−acceptor distances, the reaction is in the abnormal region and the pre-exponent is lowered by adiabatic suppression, resulting in a small rate constant. As the proton donor−acceptor distance increases, the reaction becomes less endergonic until the reaction is in the normal region. At the proton donor−acceptor distance where this transition occurs, the pre-exponent is in the solvent-controlled regime, and the

Figure 5. Tafel plot calculated using eq 2 with the parameters given in Table S1 for H3O+ (red) and D3O+ (blue). The calculated data are shown as filled circles, and the lines represent quadratic fits to the calculated data points.

transfer coefficient α calculated from a quadratic fit of log j versus electrode potential for H3O+ varies slightly from 0.73 at −0.15 V to 0.69 at −0.35 V vs RHE, in reasonable agreement with experimental measurements.29 The computed H/D kinetic isotope effect decreases from 7.1 to 3.8 over the same potential range, compared to experimental measurements of 5315

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The Journal of Physical Chemistry Letters ∼3.35.6.30 This decrease in the calculated kinetic isotope effect for more cathodic potentials reflects the increasing dominance of contributions from shorter proton donor− acceptor distances (Figure 4), where the pre-exponent is determined primarily by classical solvent relaxation dynamics and therefore is less sensitive to nuclear quantum effects. This work differs from previous theoretical studies of the Volmer reaction in that we consider both vibrational nonadiabaticity and solvent relaxation dynamics. Only reactions on the electronic ground state FES are considered, implicitly assuming that the excited electronic states are too high in energy to have a significant effect on the reaction. Our expressions for the pre-exponents corresponding to the Volmer reaction extend previous work on the effects of nonadiabaticity and solvent dynamics on electron transfer7−10,12,13,28 and proton transfer31 by substituting vibronic coupling for electronic coupling. In contrast to the statistical theory by Straub and Berne,10 but similar to other works,12,13,28 the model herein defines the reaction in the framework of a curve crossing between diabatic states. The validity condition for this formulation is V ≲ 4kBT,7 where V is the coupling between the reactant and product vibronic states. According to our model, this reaction is electronically adiabatic but vibrationally nonadiabatic at larger proton donor−acceptor distances. At shorter proton donor−acceptor distances, for which this condition is not valid, formulating the reaction as a transition from the reactant to the product minimum on the adiabatic ground state FES provides a better description, with the effect of lowering the activation free energy by V relative to the curve crossing formulation used here. However, the adiabatic ground state FES may have a single minimum at these short proton donor−acceptor distances due to large vibronic coupling and high endergonicity, especially at less cathodic electrode potentials. In this case, the reaction cannot be defined in the fully adiabatic formulation. Landau−Zener theory requires the velocity along the solvent coordinate to be constant in the vicinity of the crossing. If this condition were not fulfilled, leading to Landau−Zener breakdown, the rate constants of reactions in the nonadiabatic FGR and solvent-controlled limits would not be significantly affected.19 Adiabatic processes in the abnormal region, however, would be facilitated by velocity relaxation, mitigating the effect of adiabatic suppression.19,32 For the Volmer reaction considered here, this mitigation would lead to greater contributions to the current density from the proton donor− acceptor distances where the reaction is in the abnormal region. This effect would be more pronounced at less cathodic potentials, where the transition from the abnormal to the normal region occurs at larger proton donor−acceptor distances (Figure 3). Consequently, the calculated transfer coefficients and kinetic isotope effects would be smaller. Herein we present a formulation describing PCET reactions in the framework of a curve crossing between two diabatic vibronic states, accounting for vibronic (i.e., vibrational) nonadiabaticity using Landau−Zener theory and solvent relaxation dynamics using the stable states picture formalism. Because this formulation utilizes an interpolation scheme that spans the fully adiabatic, nonadiabatic, and solvent-controlled limits, it is valid for a wide range of vibronic couplings and solvent relaxation time scales. We applied this formulation to the aqueous Volmer reaction, diabatizing the lowest two adiabatic vibronic free energy surfaces along the collective solvent coordinate for each proton donor−acceptor distance.

In contrast to previous fully adiabatic treatments of this reaction, our calculations indicate that vibrational nonadiabaticity and solvent dynamics play significant roles. The calculated transfer coefficients and kinetic isotope effects are in reasonable agreement with experimental measurements. This theoretical formulation provides fundamental insights into the Volmer reaction and can be applied to a wide range of other electrochemical processes.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.9b01984. Adiabatic and nonadiabatic limits of Landau−Zener preexponent, details about derivative coupling and estimation of effective frequency, table with parameters, figure showing dependence of vibronic coupling on proton donor−acceptor distance (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: sharon.hammes-schiff[email protected]. ORCID

Yan-Choi Lam: 0000-0001-7809-4471 Sharon Hammes-Schiffer: 0000-0002-3782-6995 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-181-0420. We gratefully acknowledge Dr. Marko Melander and Dr. Zachary Goldsmith for stimulating and helpful discussions.

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REFERENCES

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