Potential Dependence of Electrochemical Barriers from ab Initio

Letter pubs.acs.org/JPCL

Potential Dependence of Electrochemical Barriers from ab Initio Calculations Karen Chan†,‡ and Jens K. Nørskov*,†,‡ †

SUNCAT Center for Interface Science and Catalysis, Department of Chemical Engineering, Stanford University, Stanford, California 94305, United States ‡ SUNCAT Center for Interface Science and Catalysis, SLAC National Accelerator Laboratory, Menlo Park, California 94025, United States ABSTRACT: We present a simple and computationally efficient method to determine the potential dependence of the activation energies for proton−electron transfer from a single ab initio barrier calculation. We show that the potential dependence of the activation energy is given by the partial charge transferred at the transition state. The method is evaluated against the potential dependence determined explicitly through multiple calculations at varying potential. We show that the transfer coefficient is given by the charge transferred from the initial to transition state, which has significant implications for electrochemical kinetics.

The “charge-extrapolation” method for determining constant potential barriers from a constant charge calculation is based upon two assumptions:15 • The “chemical” and electrostatic contributions to reaction energetics are separable. • The electrostatic contributions to proton−electron transfer are described by a basic capacitor model. For a constant charge calculation from state 1 to 2 (e.g., from the initial to transition state), corresponding to work functions Φ1 and Φ2, energies E1(Φ1) and E2(Φ2), and interfacial charges q1 and q2, respectively, the corresponding energy change between the two states at constant work functions, Φ1 and Φ2, are given by15

T

he theory of electrochemical electron transfer has had a long history,1 from the phenomenological Butler−Volmer kinetics2,3 to the classical theories of Marcus and Hush4−6 and their quantum extensions.7−9 Because of the complex nature of the electrochemical interface, attempts to determine electrochemical barriers explicitly using density functional theory have emerged only in the past few years.10,11 Such calculations assume adiabatic electron transfer, which applies for processes close to the electrode surface.9,12 A major challenge in the ab initio calculation of electrochemical barriers is that simulations are done at constant charge, while real electrochemical systems operate at constant potential.10,13 Simulations at constant charge in finite-sized unit cells give rise to a change in interfacial charge density and potential as charge transfer occurs. The smaller the cell, the larger the resultant change in potential. To circumvent this issue, one can extrapolate to the constant potential case by calculating reaction energetics in larger and larger sized unit cells, i.e., the “cell-extrapolation” method.10 Such a scheme has been applied to study the hydrogen evolution and oxidation reaction14 but would be prohibitively costly to apply to more complex electrochemical reactions. Recently, we introduced a “charge-extrapolation” method to determine simple proton− electron transfer barriers at constant potential, using only a single barrier calculation done in a single size cell at constant charge.15 This method is based on a capacitor model for the electrostatic contribution to reaction energetics, and the interfacial charge links the constant charge barriers to the constant potential ones. In the present work, we develop an extension to the model that gives the dependence of the barriers as a function of potential from a single barrier calculation. © XXXX American Chemical Society

E2(Φ1) − E1(Φ1) = E2(Φ2) − E1(Φ1) +

(q2 − q1)(Φ2 − Φ1) 2 (1)

E2(Φ2) − E1(Φ2) = E2(Φ2) − E1(Φ1) −

(q2 − q1)(Φ2 − Φ1) 2 (2)

q is defined to be the number of excess electrons in the slab, equal to the net charge of ions in solution; see the computational methods section of ref 15. Taking eq 2 − eq 1 and setting ΔE (Φ) = E2(Φ) − E1(Φ) at a given Φ and Δq = q2 − q1, we obtain ΔE(Φ2) − ΔE(Φ1) = −Δq(Φ2 − Φ1)

(3)

Received: February 18, 2016 Accepted: April 18, 2016

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DOI: 10.1021/acs.jpclett.6b00382 J. Phys. Chem. Lett. 2016, 7, 1686−1690

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and Φ2 to be the independent variable, we obtain the reaction energetics as a function of potential ΔE(Φ) = ΔE(Φref ) − Δq(Φ − Φref )

(4)

or, in differential form

dΔE = −Δq dΦ

(5)

The physical picture here is that the change in electronic charge gives the potential dependence of an electrochemical activation or reaction energy. Consider a generic proton−electron transfer: A* + H+ + e− → AH*

(6)

where * refers to an adsorbed species. For the reaction energy, Δq = qFS − qIS = 0 − 1 = −1, where FS refers to the final state and IS to the initial state. The transfer coefficient, β, of phenomenological Butler−Volmer kinetics2,3 is, in this model, given by the change in charge of the initial state to the transition state. The transition state TS corresponds to a complex [A*··H]+(1−β) + (1−β) e−, where a partial electron transfer of magnitude β has occurred. The corresponding Δq = qTS − qIS = (1 − β) − 1 = −β. We apply this scheme to three simple proton−electron transfers, Heyrovsky, Volmer, and reduction of *OH to H2O on Pt(111) H* + H+ + e− → H 2 + *

Figure 1. Three basic proton−electron transfer reactions considered: (a) Heyrovksy, (b) Volmer, and (c) OH to H2O, as calculated within 3 × 3 unit cells. Reprinted from ref 15. Copyright 2015 American Chemical Society.

(7)

* + H+ + e− → H* +

(8)

−

OH* + H + e → H 2O + *

Setting Φ1 as the reference work function, Φref (e.g., that corresponding to the standard hydrogen electrode potential16),

(9)

The constant charge calculations of such barriers in 3 × 3 unit cells are illustrated in Figure 1. The Heyrovsky barriers were

Table 1. Work Functions Φ, Changes in q, Charge-Extrapolated E for Various Reactions and Cell Sizes and Proton Densitiesa Heyrovsky size

IS n(H+)

ΦIS

ΦTS

× × × ×

1/9 1/12 1/18 1/24

3.44 4.28 5.14 5.41

4.81 6.36 5.59 6.35 6.17 6.42 − 6.41 fitted −Δq: Volmer

3 3 3 6

3 4 6 4

size 3 3 3 3 3 3 6

× × × × × × ×

2 3 4 6 4′ 6′ 4′

ΦFS

−(qFS−qIS)

−(qTS−qIS)

−(qTS−qFS)

ΔE (ΦIS)

ΔE(ΦFS)

Ea(ΦIS)

Ea(ΦTS)

Eb(ΦFS)

Eb(ΦTS)

0.75 0.78 0.81 0.84 0.74

0.44 0.5 0.55 − 0.35

−0.31 −0.28 −0.26 − −0.34

−1.17 −0.61 −0.06 0.25

1.01 1.00 0.98 1.09

0.34 0.49 0.74 −

0.94 1.15 1.31 −

0.48 0.52 0.53 −

0.95 0.73 0.60 −

Volmer 1ML H*

ΦIS

ΦFS

−(qFS−qIS)

ΔE (ΦIS)

ΔE(ΦFS)

1/6 4.03 1/9 4.31 1/12 5.07 1/18 5.73 1/6 4.03 1/6 4.03 1/6 4.03 fitted −Δq:

6.4 6.32 6.61 6.71 4.94 4.65 4.40

0.57 0.62 0.68 0.69 0.49 0.46 0.45 0.66

−0.7 −0.28 0.02 0.38 −0.75 −0.67 −0.71

0.64 0.96 1.06 1.05 −0.30 −0.38 −0.55

+

IS n(H )

+

size

IS n(H )

× × × ×

1/9 1/12 1/18 1/24

3 3 3 6

3 4 6 4

ΦIS

ΦFS

−(qFS−qIS)

ΔE (ΦIS)

ΔE(ΦFS)

0.68 0.67 0.81 0.84 0.72

−0.57 −0.41 −0.15 0.02

0.69 0.57 0.68 0.98

3.18 5.04 3.21 4.66 3.81 4.83 3.97 5.11 fitted −Δq:

OH to H2O size

IS n(H+)

ΦIS

ΦTS

ΦFS

3×3 3×4 3×6

1/9 1/12 1/18

4.31 4.92 5.56

4.45 5.95 5.12 6.14 5.69 6.36 fitted −Δq:

−(qFS−qIS)

−(qTS−qIS)

−(qTS−qFS)

ΔE (ΦIS)

ΔE(ΦFS)

Ea(ΦIS)

Ea(ΦTS)

Eb(ΦFS)

Eb(ΦTS)

0.74 0.75 0.75 0.64

0.00 −0.01 0.00 −0.03

−0.74 −0.76 −0.76 −0.47

−0.98 −0.69 −0.43

0.24 0.23 0.17

0.09 0.08 0.05

0.09 0.07 0.05

−0.10 −0.08 −0.07

1.01 0.69 0.43

All E and Φ are listed in electronvolts; q values are in atomic units e. The initial state proton density, IS n(H+), is given as number of protons in the water layer/surface Pt atom. The fitted −Δq was determined through linear least squares fitting. a

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The Journal of Physical Chemistry Letters calculated on a surface covered with a monolayer of H*. In the case of the Volmer reaction, we considered both bare and H-covered surfaces. We refer to ref 15 for the details of the DFT barrier calculations, and relevant convergence tests for the systems considered have been done in ref 17. Table 1 lists the results of the charge-extrapolation scheme (eqs 1 and 2), applied to the barriers and reaction energies calculated at constant charge and in various unit cell sizes. Cell sizes where more than one proton concentration was considered are labeled with an apostrophe. The work functions, Φ, can be related to an absolute potential via U = (Φ − Φref) /e. Here, ΔE = EFS − EIS refers to the reaction energy, Ea = ETS − EIS to the activation energy of the forward reduction reaction, and Eb = ETS − EFS to the activation energy of the backward oxidation reaction. For example, for the Heyrovsky reaction, the tabulated Ea(ΦTS) for the 3 × 4 cell refers to the forward activation energy at a fixed Φ = 5.59 eV. For the Volmer reactions, only reaction energies are included because no observable barriers were found in most cell sizes considered. Figure 2 shows the extrapolated energies as a function of work function Φ. ΔE, Ea, and Eb are color-coded based on the state they are extrapolated to, i.e., the IS or FS. The relationships are linear, as expected from eq 5. Slight scatter can arise, in part, from uncertainty in the Bader partitioning used to determine the interfacial charges, q. For a given constant charge calculation, the errors in the two extrapolated energies arising from Δq = q2 − q1 should have a different sign (compare eqs 1 and 2). Thus, a line fitted through all points should provide some error cancellation and serve as an accuracy check for the potential dependence determined using the method presented in this work. We note that the extrapolation scheme is not restricted to a determination of charge through Bader analysis. In principle, other charge partitioning schemes can also be applied, but they should be benchmarked to the full cellextrapolation scheme as performed in ref 15 for a Bader analysis. Figure 3 now overlays the potential dependence as determined through eq 4 on top of the data of Figure 2. Thin solid lines connect the two extrapolated energies calculated for each barrier/reaction energy calculation performed at constant charge (where eqs 1 and 2 were applied); thin dotted lines extrapolate out beyond these energies. In general, within the range of the work functions of the constant charge calculation, the agreement to the fitted potential dependence is reasonable, with differences of 1 ML H* could also contribute to the overall scatter. Now we turn to a discussion of the calculated Δq. For reaction energies ΔE, Table 1 shows a −Δq = −(qFS − qIS) ≈ 0.7 (except for the densest n(H+) = 1/6 as discussed above). This 0.7 charge does not arise from errors in charge partitioning; the cell extrapolation method of ref 15, which requires only reaction energies and work functions, also yielded the same charge. Fractional interface charge could arise from the selfinteraction error in conventional density functional theory.18−22 The implication of this error on the gap between the highest occupied and lowest unoccupied molecular orbitals (HOMO− LUMO gap) has been discussed in ref 23. Where the Fermi level does not straddle the HOMO and LUMO of the water layer, artificial charge transfer across the interface occurs. This phenomenon has also been observed in ref 24. However, in 1688

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Figure 4. Parity plot between the fitted β from a series of extrapolated energies (Figure 2) and that from β = −Δq using the constant charge calculation.

Figure 5. (a) H3O+ oriented down with n(H+) = 1/6; (b) H3O+ oriented planar with n(H+) = 1/9.

in recent theories of charge transfer that consider solvent reorganization, ion transfer, or bond-breaking.28−30 In our present model, β is a measure of the amount of charge that is transferred between IS and TS; therefore, a smaller β is suggestive of a transition state that is initial-state-like, and a larger one a final-state-like transition state. To the best of our knowledge, the current work presents the first determination of β from ab initio calculations that has been evaluated with explicit calculations at varying potential. β has a significant effect on electrochemical kinetics, since turnover frequencies have an exponential dependence on the activation energy. In summary, we have presented an ab initio method to determine simple proton−electron transfer barriers as a function of potential. We show that the potential dependence of activation energy is determined by the partial charge transferred in the transition state. This allows us to predict the approximate potential dependence of the activation energy from a single constant charge calculation. Future work will focus on evaluating the effect of water structure on reaction energetics and using higher-order methods to investigate the charge delocalization error at the electrochemical interface. We will also extend the current theory to the case of more complex reactions that involve the orientation of adsorbate dipoles, where there is an additional contribution to the electrostatic component in the reaction energies.

Figure 3. Figure 2 overlaid with the potential dependence as predicted by eq 4. Thin solid lines connect the two extrapolated energies calculated for each constant charge barrier/reaction energy calculation (where eqs 1 and 2 were applied); thin dotted lines extrapolate out beyond these energies.

the present work, we have chosen systems where the Fermi level does straddle the HOMO and LUMO,23 and −Δq for the reaction energies still deviates from 1. Further work using higher-order methods, e.g. hybrid25 or self-interaction correction functionals,18,26,27 will be performed to evaluate this discrepancy. We emphasize, however, that the present method is not dependent on the functional used to determine the energies and charges. In the case of the activation energies, there is a significant difference in β = −Δq = −(qTS − qIS) between the Heyrovsky (∼0.5) and OH reduction to H2O (∼0). In classical theories of outer-sphere electron transfers, β has been suggested to be close to 1/2.1,6 Deviations of β from 1/2 have been predicted

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based in part on work supported by the Air Force Office of Scientific Research through the MURI program 1689

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(22) Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Insights into Current Limitations of Density Functional Theory. Science 2008, 321, 792− 794. (23) Björketun, M. E.; Zeng, Z.; Ahmed, R.; Tripkovic, V.; Thygesen, K. S.; Rossmeisl, J. Avoiding Pitfalls in the Modeling of Electrochemical Interfaces. Chem. Phys. Lett. 2013, 555, 145−148. (24) Quaino, P.; Luque, N. B.; Soldano, G.; Nazmutdinov, R.; Santos, E.; Roman, T.; Lundin, A.; Groß, A.; Schmickler, W. Solvated Protons in Density Functional Theory−A Few Examples. Electrochim. Acta 2013, 105, 248−253. (25) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid Functionals based on a Screened Coulomb Potential. J. Chem. Phys. 2003, 118, 8207−8215. (26) Klüpfel, S.; Klüpfel, P.; Jónsson, H. The Effect of the PerdewZunger Self-interaction Correction to Density Functionals on the Energetics of Small Molecules. J. Chem. Phys. 2012, 137, 124102. (27) Lehtola, S.; Jónsson, H. Variational, Self-consistent Implementation of the Perdew-Zunger Self-Interaction Correction with Complex Optimal Orbitals. J. Chem. Theory Comput. 2014, 10, 5324−5337. (28) Laborda, E.; Henstridge, M. C.; Batchelor-McAuley, C.; Compton, R. G. Asymmetric Marcus-Hush Theory for Voltammetry. Chem. Soc. Rev. 2013, 42, 4894−4905. (29) Schmickler, W. A. Theory for Nonadiabatic Electrochemical Electron-Transfer Reactions Involving the Breaking of a Bond. Chem. Phys. Lett. 2000, 317, 458−463. (30) Koper, M. T.; Schmickler, W. A Kramers Reaction Rate Theory for Electrochemical Ion Transfer Reactions. Chem. Phys. 1996, 211, 123−133.

under AFOSR Award FA9550-10-1-0572. This material is based in part on work performed by the Joint Center for Artificial Photosynthesis, a DOE Energy Innovation Hub, supported through the Office of Science of the U.S. Department of Energy under Award DE-SC0004993.

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REFERENCES

(1) Kuznetsov, A. Charge Transfer in Chemical Reaction Kinetics; Cahiers de chimie; Presses polytechniques et universitaires romandes: Lausanne, 1997. (2) Butler, J. Studies in Heterogeneous Equilibria. Part II; the Kinetic Interpretation of the Nernst Theory of Electromotive Force. Trans. Faraday Soc. 1924, 19, 729−733. (3) Erdey-Gruz, T.; Volmer, M. The Theory of Hydrogen High Tension. Z. Phys. Chem. 1930, 150, 203−213. (4) Marcus, R. A. On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer. I. J. Chem. Phys. 1956, 24, 966−978. (5) Marcus, R. A. Electron Transfer Reactions in Chemistry. Theory and Experiment. Rev. Mod. Phys. 1993, 65, 599−610. (6) Hush, N. Adiabatic Theory of Outer Sphere Electron-Transfer Reactions in Solution. Trans. Faraday Soc. 1961, 57, 557−580. (7) Levich, V. Present State of the Theory of Oxidation-reduction in Solution (Bulk and Electrode Reactions). Adv. Electrochem. Electrochem. Eng. 1966, 4, 249−371. (8) Dogonadze, R. R.; Kuznetsov, A. Theory of charge transfer kinetics at solid-polar liquid interfaces. Prog. Surf. Sci. 1975, 6, 1−41. (9) Schmickler, W. A Theory of Adiabatic Electron-transfer Reactions. J. Electroanal. Chem. Interfacial Electrochem. 1986, 204, 31−43. (10) Rossmeisl, J.; Skúlason, E.; Björketun, M. E.; Tripkovic, V.; Nørskov, J. K. Modeling the Electrified Solid-Liquid Interface. Chem. Phys. Lett. 2008, 466, 68−71. (11) Nie, X.; Esopi, M. R.; Janik, M. J.; Asthagiri, A. Selectivity of CO2 Reduction on Copper Electrodes: The Role of the Kinetics of Elementary Steps. Angew. Chem., Int. Ed. 2013, 52, 2459−2462. (12) Nørskov, J.; Studt, F.; Abild-Pedersen, F.; Bligaard, T. Fundamental Concepts in Heterogeneous Catalysis; Wiley: Hoboken, NJ, 2014. (13) Schnur, S.; Groß, A. Challenges in the First-Principles Description of Reactions in Electrocatalysis. Catal. Today 2011, 165, 129−137. (14) Skúlason, E.; Tripkovic, V.; Björketun, M. E.; Gudmundsdóttir, S.; Karlberg, G.; Rossmeisl, J.; Bligaard, T.; Jónsson, H.; Nørskov, J. K. Modeling the Electrochemical Hydrogen Oxidation and Evolution Reactions on the Basis of Density Functional Theory Calculations. J. Phys. Chem. C 2010, 114, 18182−18197. (15) Chan, K.; Nørskov, J. K. Electrochemical Barriers Made Simple. J. Phys. Chem. Lett. 2015, 6, 2663−2668. (16) Trasatti, S. The Absolute Electrode Potential: An Explanatory Note (Recommendations 1986). Pure Appl. Chem. 1986, 58, 955−966. (17) Skúlason, E.; Karlberg, G. S.; Rossmeisl, J.; Bligaard, T.; Greeley, J.; Jónsson, H.; Nørskov, J. K. Density Functional Theory Calculations for the Hydrogen Evolution Reaction in an Electrochemical Double layer on the Pt(111) Electrode. Phys. Chem. Chem. Phys. 2007, 9, 3241−3250. (18) Perdew, J. P.; Zunger, A. Self-interaction correction to densityfunctional approximations for many-electron systems. Phys. Rev. B: Condens. Matter Mater. Phys. 1981, 23, 5048−5079. (19) Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L., Jr. Densityfunctional theory for fractional particle number: derivative discontinuities of the energy. Phys. Rev. Lett. 1982, 49, 1691. (20) Mori-Sánchez, P.; Cohen, A. J.; Yang, W. Many-electron SelfInteraction Error in Approximate Density Functionals. J. Chem. Phys. 2006, 125, 201102. (21) Ruzsinszky, A.; Perdew, J. P.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E. Density Functionals that are One-and Two-are not Always Many-Electron Self-Interaction-Free, as Shown for H. J. Chem. Phys. 2007, 126, 104102. 1690

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