A Unified Model for the Electrochemical Rate Constant That

Sep 17, 2009 - A generalized Anderson-Newns Hamiltonian is employed to model electrochemical electron transfer kinetics via a potential energy surface...
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J. Phys. Chem. C 2009, 113, 17904–17914

A Unified Model for the Electrochemical Rate Constant That Incorporates Solvent Dynamics A. K. Mishra Institute of Mathematical Sciences, CIT Campus, Chennai 600113, India

David H. Waldeck* Department of Chemistry, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15260 ReceiVed: June 4, 2009; ReVised Manuscript ReceiVed: August 17, 2009

A generalized Anderson-Newns Hamiltonian is employed to model electrochemical electron transfer kinetics via a potential energy surface approach. Important novel features of this treatment are inclusion of the electrode induced broadening of the reactant level and the effect of solvent dynamics on the pre-exponential factor, which is determined by a numerical integration of the Kramers’ diffusion controlled barrier crossing rate expression. An interpolation scheme is provided to treat nonadiabatic, transition state theory adiabatic, and solvent dynamical effects in the weak coupling (nonadiabatic) and strong coupling (adiabatic) limits in a unified manner. The derived rate expressions are valid for arbitrary values of the electrode-reactant coupling term V, solvent polarization mode frequency ωr, and longitudinal solvent dielectric relaxation time τL. A comparison of the results obtained here with the earlier treatments of electron transfer reactions is presented, and the relevance of the present formalism for experimental systems is provided. The density of states based approach for the heterogeneous electron transfer reaction is briefly discussed. I. Introduction Electron transfer plays a central role in many well-known and emerging electrochemical technologies, for example, biosensors and bioelectrochemistry.1 In comparison to homogeneous electron transfer processes in solution, the involvement of an additional solid or liquid phase (i.e., electrode) in heterogeneous electron transfer reactions makes their microscopic description more challenging. On the other hand, the presence of an electrode provides additional control variables, enabling one to push the electron transfer in a desired direction. Overpotential is one such variable. In addition, one can manipulate the electronic interaction strength |V| between the electrode and a redox species by forming a self-assembled monolayer (SAM) on the electrode, making it possible to realize different mechanistic limits for the reaction: nonadiabatic, adiabatic transition state theory (TST), or solvent dynamic limits. Conventional theoretical treatments of the electron transfer rate use either the potential energy curve (pec) method or the density of states (dos) method. The pec method generates an activation energy for the electron transfer,2,3 which is used in a TST-like description of the rate constant, whereas the electron transfer rate in the dos method is determined by the overlap of the reactant and electrode densities of states.4,5 The dos based results are obtained when the electron transfer rate is determined using Green’s function or correlation function methods.6-10 In both approaches, the Condon approximation is applied and the rate constant contains a multiplicative pre-exponential factor that is the product of a nuclear frequency factor νn and an electronic transmission coefficient κel.5 In the nonadiabatic limit, the pec and dos methods lead to identical results, but for strong electrode-reactant coupling, a comparison between the pec and dos formalism is lacking. This work models the electron transfer through a generalized Anderson-Newns Hamiltonian which has been earlier employed for the adiabatic electron transfer reaction by Schmickler.11 We

extend this formalism to the nonadiabatic regime within the framework of the golden rule, Marcus theory treatment. In general, the Anderson-Newns Hamiltonian can be employed to provide both pec and dos based results for the electron transfer. However, the present work primarily focuses on the application of the pec approach to describing nonadiabatic, adiabatic, and the barrier (strong coupling) and cusp (weak coupling) cases of solvent dynamic controlled electrode kinetics. This description requires the determination of the nuclear frequency factor for all four cases, which has been achieved by numerical integration of the Kramers’ diffusion controlled barrier crossing rate expression. An interpolation scheme for the pre-exponential factor among these four limits is also provided. The remainder of the paper is separated into four sections. In section II, the background information for the electron transfer reaction is provided. Starting with the nonadiabatic electron transfer reactions, the various symmetries associated with the activation energy are described and the potential energy curve obtained from the generalized Anderson-Newns Hamiltonian is found. This section also includes a brief account of the dos method. Section III pertains to the theoretical developments. It discusses how to find the activation energy from single and multiple pec methods and determines the pre-exponential terms for nonadiabatic, adiabatic, and solvent dynamic controlled situations. The electrochemical rate constants are also obtained in this section. The numerical results and discussions are provided in section IV, and section V provides a summary and conclusions. II. Background IIa. Potential Energy Surface. Dogonadze and Kuznetsov12 have shown that the electrochemical electron transfer rate is a sum of the partial electron transition probabilities between the redox orbital r at energy r and the electron states {k} with energy {k} in the electrode. The noncrossing adiabatic upper

10.1021/jp9052659 CCC: $40.75  2009 American Chemical Society Published on Web 09/17/2009

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and lower pec’s for each one of the r T k electron transfers is obtained by solving a secular equation for a two-level system.13 The splitting between these two pec’s equals V ) 2∑k Vrk, where Vrk is the strength of electronic coupling between the reactant state r and the kth electronic state in the electrode.2 The energy difference between the equilibrium configuration and the maximum of the potential barrier formed by the lower adiabatic pec determines the activation energy. The location of the pec maximum lies at the point of intersection of two diabatic potential energy curves corresponding to the electron in the reactant and electron in the electrode configurations. The most significant feature of the analysis is that a manifold of potential energy curves, one for each k, is required to obtain the rate constant. For example, the rate constant for electron transfer from an electrode to a reactant in the nonadiabatic limit may be written as

kred(η) )

∫-∞∞ f() ∑ |Vrk|2δ(

2π p

k

[

]

≡

∫-∞∞ kred(, η) d

4πλβ ×

2π 2 |V| p

β ∫-∞∞ f()F() 4πλ

(1)

[

exp -β

×

]

(λ + F -  + η)2 d (2) 4λ

where F() ) ∑k δ( - k) is the electrode dos. Correspondingly, we can write the rate constant for the oxidation reaction as

kox(η) )

2π 2 |V| p

β ∫-∞∞ (1 - f())F() 4πλ

[

exp -β

×

]

(λ +  - F - η)2 d (3) 4λ

The activation energies for the oxidation and reduction reactions are a function of , namely,

∆Gqox(, η) )

(λ +  - F - η)2 ; 4λ ∆Gqred(, η) )

(λ + F -  + η)2 4λ

(4)

These formulas for the activation energies have a number of implications. First, the oxidation reaction becomes activationless at η ) λ +  - F and the reduction reaction becomes activationless at η ) -λ +  - F. In addition, a comparison of the two formulas implies that

∆Gqred(, η - 2λ) ) ∆Gqox(, η)

Clearly, as η′ > λ + ′, η ) 2(λ + ′) - η′ is less than λ + ′. Equation 6 highlights that ∆Gqox(, η) is symmetric with respect to η variation, with ηq ) λ +  - F being the point of symmetry. This symmetry property means that one can obtain q (, η) for η < (λ +  - F), from the value of the value of ∆Gox q ∆Gox(, η) for η > (λ +  - F), and vice versa. A similar q relation exists for ∆Gred

(7)

where β ) (kBT)-1, η is the overpotential, and λ is the solvent reorganization energy. The last line in eq 1 highlights the fact that the total rate constant contains contributions from all energy states of the electrode. If we replace {|Vrk|2} by an effective average |V|2, then kred becomes

kred(η) )

∆Gqox(', η′ > λ + ') ) ∆Gqox(', η ) 2(λ + ') - η′) (6)

q q ∆Gred (', η′ < -(λ - ')) ) ∆Gred (', η ) -2(λ - ') - η′)

- k)

(F -  + η + λ)2 d exp -β 4λ

Next, the activation energy for oxidation when η > λ +  - F can be obtained from the activation energy when η < λ +  F. By defining  - F ) ′, the activation energy becomes

(5)

Although the Dogonadze formalism is derived for the nonadiabatic (|V| f zero) limit, the resulting rate expressions (eqs 2 and 3) can be modified (albeit ad hoc) so that they interpolate between nonadiabatic and adiabatic (large |V|) limits. The first change that must be made is to replace the preexponential factor (2π/p)|V|2(β/4πλ)1/2 by the product νnκel,5,14,15 and the second step is to account for the lowering of the activation energy by the reactant-electrode coupling V. The form of the nuclear frequency factor νn and the electronic transmission coefficient κel may change with the model used. IIb. Generalized Anderson-Newns Model and Potential Energy Surface. An alternative approach to generating potential energy curves for heterogeneous adiabatic electron transfer reactions has been formulated by Schmickler.11,16,17 This approach is nonperturbative and is based on a generalization of the Anderson-Newns model.18 In contrast to most models, this treatment accounts for the broadening of the reactant energy level by virtue of its coupling with the continuum of electronic states in the electrode. Before proceeding further, two main characteristics of the Anderson-Newns Hamiltonian should be emphasized. First, this Hamiltonian is valid for any strength of the reactant-electrode coupling. Hence, it can be used to model both the adiabatic and nonadiabatic coupling limits of the heterogeneous electron transfer reaction. Second, the electronic coupling exists between the reactant orbital and the continuum of electronic states in the electrode. As a consequence, the electron transfer involving any electrode state and the reactant orbital is permissible in this model. An important result of the Schmickler treatment is that the adiabatic electron transfer reaction is modeled through a single pec corresponding to the ground state. This result should be contrasted with the need for a manifold of potential energy surfaces in the Dogonadze analysis.19 As a single potential energy curve suffices for obtaining the electron transfer reaction rate, the  integration appearing in the rate expression, for the Dogonadze treatment, is now absent. In addition, when the overpotential η lies in the normal region, the connection between the electrochemical electron transfer reaction and homogeneous electron transfer reaction is more transparent, with the main difference being that the Gibbs free energy of the reaction can be controlled through η. The most dramatic difference between the homogeneous electron transfer and heterogeneous electron transfer occurs for the activation energy at the onset of the activationless region and beyond. In the Schmickler approach, the Marcus inverted region, manifest for homogeneous electron

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Mishra and Waldeck

transfer reactions, does not occur, since the activation energy is taken to be zero in this regime; i.e., the rate constant is solely determined by the pre-exponential factor in this regime. In the Schmickler treatment,11 the potential energy u(q) of the system as a function of solvent polarization coordinate q may be expressed as

1 Re(w(z()) 2√πP

Fr(, λ, ∆, (η) )

where “Re(w(z())” denotes the real part of (w(z()).

w(z() ) e-z( erfc(-iz() 2

u(q, F) )

(14)

(15)

2

q + (r + q - F)〈nr(q)〉 - zq + 4λ ∆ ln[(F - r - q)2 + ∆2] + C (8) 2π

where r is the reactant orbital energy and equals

and

Q+ ) F -  + λ - η;

(9)

F is the electrode Fermi energy, λ is the solvent reorganization energy, z is the reactant ion’s charge, and ∆ is the interaction energy between the electrode band states and the reactant state. See the Appendix for a discussion. The quantity 〈nr(q)〉 is the average occupation probability of the reactant orbital and is given by

)

F

(

F - r - q 1 1 + tan-1 2 π ∆

)

lim Fr(, λ, ∆, (η) )

lim Fr(, λ, ∆, (η) ) λf0

∆f0,λf0

(12)

∑ |Vrk|2δ(' - k)

(18) (19)

Fr(, λ, ∆, (η) ) δ( ( η)

(20)

Equation 18 gives a Gaussian distribution of redox species energy levels, and the maximum values for the oxidized and reduced forms are separated by 2λ (for a given ). This limiting description is the one most commonly used in the literature. It is interesting to note that when the reactant-electrode coupling is very weak, eqs 2 and 18 allow one to recast kred in terms of a product of reactant and electrode density of states, namely,

kred(η) )

where

∆(') ) π

∆ 1 π ( -  ( η)2 + ∆2 F

]

(11)

in which the imaginary part of the reactant Green’s function Grr is taken to be

∆(') (' - r - q)2 + ∆(')2

[

and

lim

Im Grr(') )



( - F - λ ( η)2 β exp -β 4πλ 4λ

(10)

The second line in the above expression follows by substituting the reactant dos

1 Fr(', q) ) Im(Grr(')) π

Q- )  - F - λ - η (17)

Use of the + sign gives the dos for the oxidized form, and use of the - sign gives the dos for the reduced form. Important limiting cases for the reactant density of states are

∆f0

∫-∞ Fr(', q) d'

(16)

with

r ) η + λ(1 - 2z) + F

〈nr(q)〉 )

z( ) (-Q( + i∆)/2√P

P ) λ/β;

(13)

k

The present work uses the wide-band approximation, so that ∆ may be considered energy independent,20 and consequently, its argument ′ is dropped. C in eq 8 is a logarithmically diverging constant. As the activation energy is a difference between two energy quantities, C cancels out in the final result. Finally, the potential energy expression 8 is valid for an arbitrary strength of the electrode-reactant interaction. This energy expression will be subsequently used to calculate the activation energy. IIc. Anderson-Newns Model and Density of States Approach for Electron Transfer. A thermal averaging of the q dependent reactant dos given in eq 11 leads to the following expressions for the redox species density of states7,8

2π 2 |V| p

∫-∞∞ f()F()Fr(, λ, -η) d

(21)

A similar rate expression can be written for the oxidation process. In contrast to eq 18, the limiting Fr in eqs 19 and 20 have no temperature dependence. The dos in eq 19 is valid when the coupling of the reactant with the solvent tends to zero. III. Theory IIIa. Activation Energy: Single PEC. In what follows, the activation energy is determined from a plot of U(q, 0, η), which is defined by

U(q, F, η) ) u(q, F, η) - u(q ) 0, F, η)

(22)

The use of u(q ) 0, F, η) produces only a lateral shift in the potential energy and does not change the value of the activation energy. In eq 22 and in all subsequent calculations, the zero of

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Figure 1. Potential energy curves, U(q) versus q, are shown for different values of ∆ (cf. eq 22) (λ ) 0.8 eV, η ) 0 V, F ) 0.0 eV, z ) +3, and T ) 298 K). The two curves for ∆ ) 10-5 eV (dashed red line) and ∆ ) 10-3 eV (dashed black line) overlap completely on this graph. The solid green curve is for ∆ ) 10-2 eV, the solid red curve is for ∆ ) 0.1 eV, the solid black curve is for ∆ ) 0.3 eV, and the dashed green curve is for ∆ ) 0.5 eV. The cartoon illustrates that the left-hand well corresponds to the reduced species (+2e) and the right-hand well corresponds to the oxidized species (+3e).

the energy scale is set at F, i.e, F ≡ 0. Figure 1 depicts the pec’s for different values of ∆ and with η ) 0, λ ) 0.8 eV, F ) 0, and z ) 3. The first minimum qmin1 on the left side in these plots corresponds to the “electron on the reactant”, and the second minimum qmin2 (on the right) corresponds to the “electron in the electrode” configurations. The maximum of the potential energy occurs at qmax. The activation energies for the oxidation and reduction processes are found by the relations

∆Gqox(F ) 0, η) ) U(qmax, 0, η) - U(qmin1, 0, η)

(23)

Figure 2. Panel A shows two pec’s (eq 22) for η ) 0.8 V, λ ) 0.8 eV, F ) 0, and T ) 298 K. The red solid curve is for ∆ ) 0.0001 eV, and the black dashed curve is for ∆ ) 0.025 eV. Panel B shows the dependence of ∆Gqox on η for the same parameter set (eqs 23 and 26).

the pec (cf. eq 8). This limitation applies whether one is considering an adiabatic or a nonadiabatic electron transfer reaction. Though no methodology has been provided to arrive at the activation energy for the barrierless case within this approach, it has been proposed by Schmickler that the oxidation ox 11 activation energy remains zero for η > ηcr , namely,

∆Gqox(F ) 0, η g ηox cr ) ) 0

(26)

red Similarly, if ηcr is the overpotential at which the reduction reaction becomes activationless, then

and

∆Gqred(F ) 0, η) ) U(qmax, 0, η) - U(qmin2, 0, η)

(24) In Figure 1, the curves with very low ∆ have reductant and oxidant curves that cross at q ) 4 and form a cusp-like barrier. As ∆ increases, the curves are more strongly coupled. Hence, the barrier is reduced and the maximum forms a smooth, rounded top. In the limit of very high couplings, ∆ of 0.5 eV and higher, only one minimum is apparent. These curves display a behavior that is qualitatively similar to those found for homogeneous electron transfer reactions as a function of the coupling strength. In contrast to homogeneous reactions, electrochemical reactions can have the reaction Gibbs energy varied by the applied potential, i.e., overpotential. By increasing η, one arrives at an activationless region for the oxidation reaction at a critical ox . For oxidation, the activationless regime overpotential ηcr corresponds to the case where

qmin1 ) qmax

(25)

q is zero (see eq 23). A further increase in η makes so that ∆Gox qmin1 and qmax vanish; the potential energy curve has a single minimum at qmin2 (Figure 2A). In fact, for η not lying inside the normal overpotential region, it is not possible to determine ∆Gqox through eq 23 wherein critical points are determined from

∆Gqred(F ) 0, η e ηred cr ) ) 0

(27)

The variation of ∆Gqox with respect to η, determined according to eqs 23 and 26, is plotted in Figure 2B for ∆ ) 10-4 (red solid curve) and ∆ ) 0.025 eV (black dashed curve). The η variations in these plots correspond to normal and activationless regions. The activation energy for the barrierless region cannot be generated because only a single minimum is found. A comparison of these curves (and others that are not shown) reveals that smaller values of ∆ give larger values of ηox cr , beyond q remains zero. which ∆Gox IIIb. Comparison with Marcus and Dogonadze Treatments. In the Marcus and Dogonadze approach, the reactant is initially in a reduced state (〈nr〉 ) 1), irrespective of the value of η. During the course of reaction, it reaches the transition state, transfers an electron, and acquires the oxidized form (〈nr〉 ) 0). In the Schmickler treatment, a similar picture holds true for the normal overpotential range (cf. eqs 23 and 24). However, in the abnormal overpotential range, the lower adiabatic pec loses the maximum and one of the minima (Figure 2A). Hence, it is not possible to obtain the activation energy through the adiabatic pec, in this case. This is true both for the nonadiabatic and adiabatic reaction as well as for the heterogeneous and homogeneous electron transfer. To overcome this problem, the ansatz used in ref 11 makes the activation energy zero for η g ηox cr , which differs from the Marcus result for the inverted region

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Mishra and Waldeck overpotential limit. In this latter case, one proceeds by substituting 〈nr(q)〉 ) 0 in eq 8; a potential energy curve that corresponds to the reactant in the oxidized form can be found and has the form

u(q) )

Figure 3. Panel A shows two pec’s for η ) 1.6 V, λ ) 0.8 eV, T ) 298 K, and ∆ ) 0.1 eV. The red curve is computed using eqs 22 and 8, whereas the black dashed curve is computed using eqs 22 and 28 (〈nr(q)〉 ) 1). Panel B shows the dependence of ∆Gqox on η for the adiabatic and nonadiabatic cases; the red curve is for ∆ ) 0.0001 eV, and the black dashed curve is for ∆ ) 0.025 eV.

wherein the activation energy increases with increasing overpotential and it is symmetrical around ηq ) λ - F. To recapture this activation energy versus overpotential behavior in the abnormal region, two additional potential energy curves can be employed for small to intermediate coupling strengths. These two pec’s are constructed using the AndersonNewns Hamiltonian. First, consider the regime in which η g ox , for which eq 8 generates a pec with a single minimum ηcr (Figure 2A). To generate a second pec that corresponds to the reduced form of the reactant, we set 〈nr(q)〉 ) 1 in eq 8, and find that

u(q) )

In the limit that ∆ ) 0, this potential energy curve corresponds to the diabatic potential energy of the oxidized reactant and the electron in the electrode at the Fermi level. qmin1 and qmin2 in the abnormal overpotential range are determined by the location of the energy minimum by expressions 28 and 29, respectively, and the point of intersection of these two pec’s determines qmax. In the normal region, these three critical points are obtained through the pec generated by eq 8. Following this procedure, the activation energy corresponding to the set of parameters specified in Figure 2B is reevaluated and plotted in Figure 3B. These plots correspond to ∆ ) 0.0001 and 0.025 eV. Comparison of Figures 2B and q does not remain zero 3B reveals two differences. First, ∆Gox throughout the abnormal overpotential region; however, it becomes zero at the onset of the activationless process, ηox cr , and ox . Second, the activation energy remains zero until η ) 2λ - ηcr is nearly symmetric around η ) λ. This analysis enables one to recover the Marcus predicted behavior as well as the overpotential range for which the Schmickler ansatz of zero activation energy applies. Also, note that this zero activation energy range vanishes as ∆ f 0. On the basis of the above results and requiring that the activation energy be symmetric around η ) λ (as found in the Marcus theory), a procedure to find the activation energy in all the regimes can be identified. Accordingly, the activation energy expression (eq 26) is modified to become ox ∆Gqox(F ) 0, ηox cr e η e 2λ - ηcr ) ) 0

2

q + (1 - z)q + (r - F) + 4λ ∆ ln[(F - r - q)2 + ∆2] + C (28) 2π

Note that the q dependence appearing in conjunction with ∆ is logarithmic and hence is weak. The remaining terms in the above expression generate the diabatic potential energy corresponding to zero electronic coupling between the reactant and electrode, as introduced by Dogonadze et al., provided that electron transfer occurs only with the Fermi level in the electrode and the energy is measured with respect to the Fermi level. In this modified picture, the activation energy for oxidation in the abnormal region is determined by the two potential energy curves generated from eqs 8 and 28. For example, the potential energy curves obtained from eqs 22 and 8, along with the pec as obtained from eqs 22 and 28, are shown in Figure 3A. These plots correspond to η ) 1.6 V and ∆ ) 0.1 eV. The minimum at point 1 corresponds to the reduced form; the minimum at point 3 corresponds to the oxidized state; and the crossing of the curves at point 2 corresponds to the transition state, where electron transfer occurs. The ∆Gqox value, instead of being zero, is now finite and equals the energy difference between points 1 and 2. An analogous procedure can be used to find the activation q , in the high negative energy for the reduction reaction, ∆Gred

q2 ∆ - zq + ln[(F - r - q)2 + ∆2] + C 4λ 2π (29)

(30)

In addition, the activation energy for η > λ is obtained from the symmetry condition, namely,

∆Gqox(F ) 0, η′ > λ) ) ∆Gqox(F ) 0, η ) 2λ - η′) (31) The above equation is similar to eq 6 with ′ fixed at zero. Note ox occurs at η ) λ - F. One can that, when ∆ tends to zero, ηcr proceed similarly for the reduction reaction and finds the corresponding equations red ∆Gqred(F ) 0, ηred cr g η g -2λ - ηcr ) ) 0

(32)

and

∆Gqred(F ) 0, η′ < -λ) ) ∆Gqred(F ) 0, η ) -2λ - η′) (33) Thus, it has been shown how the activation energy, which is similar in form to that of the Marcus and Dogonadze treatment, can be obtained using the Anderson-Newns model. The activation energy versus overpotential plot recovers the Marcus

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J. Phys. Chem. C, Vol. 113, No. 41, 2009 17909

∆Gqox(, ηox cr e η e (λ + )) ) 0

(35)

and

∆Gqox(, η′ > λ + ) ) ∆Gqox(, η ) 2(λ + ) - η′) (36) Figure 4. Average population 〈nr(q)〉 (red curve) (eq 10) and potential enegy U(q) (black dashed curve) (eq 22) plotted versus q. The potential energy curve has been shifted upward (but not horizontally) for better comparison with the 〈nr(q)〉 plot (λ ) 0.8 eV, ∆ ) 0.1 eV, η ) 0.0 V, and  ) F ) 0).

result for the inverted region, characterized by the increase in the activation energy with the increasing overpotential and its being symmetric around ηq ) λ - F. In the next subsection, it is shown that, according to this model, the reduced species gets oxidized at the Fermi energy; however, a steady state calculation of the electron transfer rate or current requires one to perform (de facto) a sum over electronic states. In addition, the treatment given above assumes that the Anderson-Newns Hamiltonian can be applied to nonadiabatic electron transfer reactions. In order to tackle both of these problems, a potential energy curve corresponding to an arbitrary energy state  in the electrode needs to be generated. This challenge is taken up next. IIIc. Activation Energy: Multiple PEC. The formalism developed in section IIIb can be extended to obtain the pec corresponding to any specific energy level  in the electrode by replacing F in eqs 8, 10, 28, and 29 by  (cf. the Appendix). The resulting pec for three values of , namely, -0.8, 0.0, and 0.8 eV, and η ) 0 are plotted in Figure 13 of the Appendix. The curves overlap at low values of q. A consequence of this substitution is that instead of the reduced species getting oxidized at F, it gets oxidized around the transition point qmax corresponding to the energy level . When η is zero, qmax corresponds to the value of q at which r -  + q is zero. Figure 4 shows the variation in 〈nr(q)〉, given by eq 10, with respect to q when  ) F ) 0, ∆ ) 0.1 eV, and η ) 0. The oxidation does not occur sharply at the transition point qmax ) 4.0, because the reactant density of states acquires a substrate induced broadening (cf. eqs 11 and 12). Hence, the oxidation, as reflected by the variation in average population 〈nr(q)〉, occurs gradually as the reactant dos crosses the Fermi energy level with variation in q. From eqs 10-12, it follows that, when the value of r + q is sufficiently smaller than F, the reactant density of states completely lies below the Fermi level, and 〈nr(q)〉 equals 1. As q increases, a portion of the dos lies above the Fermi level, and the average charge becomes less. When r + q equals F, the Fermi level and reactant dos peak coincide, and 〈nr(q)〉 ) 0.5. The system is now in the transition region. A further increase in q progressively makes 〈nr(q)〉 tend to zero, and the electron has been transferred to the electrode.

For the reduction reaction, the activation energy is

∆Gqred(, η) ) U(qmax, , η) - U(qmin2, , η);

∆Gqred(, ηred cr g η g -(λ - )) ) 0

η > ηred cr (37)

(38)

and

∆Gqred(, η′ < -(λ - )) ) ∆Gqred(, η ) -2(λ - ) - η′) (39) The activation energies for the reduction and oxidation reactions follow the relation

∆Gqred(, η - 2λ) ) ∆Gqox(, η)

(40)

q q ∆Gred (η) and ∆Gox (η) for  ) 0.5 eV are plotted in Figure 5. In comparison to F ) 0, ηcr for the reduction and oxidation processes now occurs at a much lower |η|. To evaluate the rate constant for the multiple-pec case, determination of the activation energy for each (, η) pair is required. The activation energies for finite {} are obtained by ox for the oxidation and η g using eqs 34 and 37 until η e ηcr red ηcr for the reduction, respectively. Thereafter, ∆Gq(, η) ) 0 for the range of η given in eqs 35 and 38; beyond these values of η, ∆Gq(, η) is obtained by using the symmetry relations 36 and 39. Note that the activation energy plots in Figure 5 satisfy relation 40. This procedure for finding the activation energy agrees with that of the Dogonadze treatment, in the appropriate limit (see the Supporting Information). This subsection is concluded with two relations that show the interchangeability of  and η in the activation energy calculations, i.e.,

∆Gqox( ) 0, η) ) ∆Gqox(-', η - ')

(41)

and

The approach developed earlier for the determination of activation energy remains valid, with F getting replaced by . For the oxidation, the activation energy is

∆Gqox(, η)

) U(qmax, , η) - U(qmin1, , η);

η
0.8 V; however, this behavior is not evident in Figure 8. The reason is that, in the multiple pec formalism, even if the system is in the inverted region for a set of {}, it belongs to the normal overpotential region for another set of {}. Thus, the integrated rate constant does not display characteristics of the Marcus inverted region. Finally, at higher η, kox exhibits a saturation effect. This can be explained using the dos considerations.31 The plots of log(kbox) versus η for ∆ ) 10-5, 10-4, 10-3, 10-2, and 10-1 eV are given in Figure 9. The respective values of the 4 5 standard rate constant kbo et are 188.0, 1.125 × 10 , 1.077 × 10 , 5 6 -1 1.923 × 10 , and 4.189 × 10 s . Here, all of the remaining

Mishra and Waldeck

Figure 9. log(kbox/s-1) plotted versus the overpotential η for different ∆ values (eq 54) (λ ) 0.8 eV, ωr ) 1011 s-1, F(F) ) 0.2 eV-1, T ) 298 K, τL ) 10-12.5 s). The black dot-dashed line is ∆ ) 10-5 eV, the black solid line is ∆ ) 10-4 eV, the red solid line is ∆ ) 10-3 eV, the black dashed line is ∆ ) 10-2 eV, and the red dashed line is ∆ ) 0.1 eV.

Figure 10. log(kaox/s-1), obtained from a single pec, plotted versus η for different ∆ values (eq 53) (λ ) 0.8 eV, ωr ) 1011 s-1, T ) 298 K, and τL ) 10-12.5 s). The red solid line is ∆ ) 10-3 eV, the black dashed line is ∆ ) 10-2 eV, and the red dashed line is ∆ ) 0.1 eV.

parameters are the same as in Figure 8 and τL ) 10-12.5 s. As expected, the rate increases with increasing ∆ and attains a saturation value at large overpotential. The spacing between the curves at high η does not follow the pattern expected from the pec profiles in Figure 1. The reason behind this deviation is the nonlinear variation of the pre-exponential term with respect to |V|2 or ∆, as shown in Figure 6. With the aim of comparing these results with Schmickler’s single pec formalism2,11 for the adiabatic electron transfer rate, a ) with respect to η for ∆ ) 10-3, 10-2, the variation of log(kox -1 and 10 eV is shown in Figure 10. The respective standard 6 7 rate constants kao et are 7.126 × 10 , 1.278 × 10 , and 2.823 × a b is larger than kox . Next, for η 108 s-1. For all values of ∆, kox ox a g ηcr , kox remains constant and tends to the limiting value ωr/ b displays a more gradual 2π. In contrast, the rate constant kox approach to the constant value. The variation of the standard rate constant kbo et with respect to ∆ and τL is considered next. Figure 11A plots kbo et versus ∆ over the range 10-5 eV e ∆ e 0.01 eV at different values of τL, namely, τL values of 10-12.5 s (curve a), 10-11 s (curve b), 10-10 s (curve c), and 10-9.5 s (curve d), respectively. For low values of ∆, all of the results correspond to the nonadiabatic limit. As the magnitude of τL increases, kbo et increases more weakly with increasing ∆. In fact, for τL ) 10-9.5 s, the kbo et plot 4 -1 is almost flat between ∆ ) 0.00015 eV (kbo et ) 1.04 × 10 s ) bo 4 -1 and ∆ ) 0.01 eV (ket ) 1.86 × 10 s ). In the same interval 4 -1 to 5.46 × 104 s-1, 2.07 × of ∆, kbo et varies from 1.68 × 10 s 4 -1 5 -1 10 s to 1.64 × 10 s , and 2.11 × 104 s-1 to 1.92 × 105 s-1 for τL ) 10-10, 10-11, and 10-12.5 s, respectively. These plots are reproduced on a logarithmic scale in Figure 11B. Such an initial increase in koet and subsequently a plateau behavior when electrode-reactant coupling is enhanced has been observed experimentally, provided |V| is not too large.32,33 bo The log(kox ) for 0.01 eV e ∆ e 0.3 eV and for different bo values of τL are plotted in Figure 12. Unlike in Figure 11A, kox varies steeply with respect to ∆, and as the value of τL decreases,

Unified Model for the Electrochemical Rate Constant

Figure 11. (A) Plots of the standard rate constant ketbo versus ∆ for different τL (eq 54 with η ) 0). (B) Similar plots of log(ketbo) versus ∆ (λ ) 0.8 eV, ωr ) 1011 s-1, F(F) ) 0.2 ev-1, and T ) 298 K). The solid red line (a) is for τL ) 10-12.5 s, the solid black curve (b) is for τL ) 10-11 s, the dashed red curve (c) is for τL ) 10-10 s, and the dashed black curve (d) is for τL ) 10-9.5 s.

J. Phys. Chem. C, Vol. 113, No. 41, 2009 17913 an adiabatic electron transfer rate, modeled through a single pec,11 follows as a limiting case of the present analysis. In addition, the nonadiabatic electron transfer reaction rate was also shown to follow in the appropriate limit. Our earlier results28 for the pre-exponential factor have been generalized to account for arbitrary values of the solvent mode frequency ωr, longitudinal solvent dielectric relaxation time τL, and electrode-reactant electronic coupling |V|. A brief description of the dos based approach, as distinct from the pec approach considered here, was also provided. The unified formalism developed here has enabled us to numerically evaluate the electron transfer rate for nonadiabatic, TST adiabatic, and solvent dynamic controlled weak and strong adiabatic reactions while taking into account the effect of electrode induced broadening of the reactant orbitals. Calculations have also been performed for the situations intermediate to the above-mentioned four limiting cases. The results show the absence of the Marcus inverted region for the electrochemical electron transfer process. The results also reproduce the initial increase followed by a plateau behavior, observed experimentally for the standard rate constant, when |V| is progressively increased. This behavior is attributed to the transition of the electron transfer reaction from the nonadiabatic limit to the solvent dynamic weak adiabatic limit. Acknowledgment. D.H.W. acknowledges support from the US National Science Foundation (CHE-0718755 and CHE 0628169). Appendix

Figure 12. log of the standard rate constant ketbo plotted versus ∆ for different τL (λ ) 0.8 eV, ωr ) 1011 s-1, F(F) ) 0.2 ev-1, and T ) 298 K). The solid red line is for τL ) 10-12.5 s, the solid black curve is for τL ) 10-11 s, the dashed red curve is for τL ) 10-10 s, and the dashed black curve is for τL ) 10-9.5 s. bo the value of kox increases; e.g., for ∆ ) 0.3 eV, kbo et is around 106 s-1 when τL ) 10-9.5 s, and it is around 108 s-1 when τL ) 10-12.5 s. Thus, we see that the relaxation time controls the rate when ∆ is large enough. With these results, the task of calculating the rate constant kbox and the standard rate constant for a wide range of parameters is completed.

V. Summary and Conclusions Employing a generalized spinless Anderson-Newns Hamiltonian, which includes the solvent polarization modes in a harmonic approximation, and a numerical integration of Kramers’ diffusion controlled barrier crossing rate, the heterogeneous electron transfer rate was calculated as a function of the overpotential, electronic coupling strength, and solvent relaxation. The multioscillator description of the solvent was replaced by an effective single-oscillator model which is subsequently treated in the classical limit.11,16,34 In order to account for the contributions of the multitude of electrode electronic states to the electron transfer probability, the system was described through multiple potential energy surfaces, and the corresponding activation energies are evaluated for an arbitrary strength of electrode-reactant electronic coupling. This task was simplified by invoking symmetry relations that are satisfied by the activation energies. It was shown that the Schmickler result for

A redox couple interacts with the substrate band states and with the solvent polarization modes. The interaction with the electrode’s electronic states is responsible for the electron transfer. The solvent modes, which are modeled classically within harmonic approximation, and their coupling with the reactant leads to a redox solvation energy and contributes to the activation energy barrier for the electron transfer. Taking into account various system components and interactions among them, an effective Hamiltonian which is a generalization of the Anderson-Newns Hamiltonian (spinless case) for the electron transfer system can be written as

H)

∑ knk + (r + ∑ giqi)nr - z ∑ giqi + k

i

∑

(Vkrc†k cr

+

i

Vrkcr†ck)

k

1 + 2

∑ ωi(pi2 + qi2)

(A1)

i

k and r label the electrode and the reactant electronic states, respectively.  is the energy value. n, c†, and c respectively denote the number, creation, and annihilation operators for electrons. The index i characterizes various solvent polarization modes, having frequencies {ωi}, momenta {pi}, and coordinates {qi}. V and g respectively describe the reactant-electrode hybridization and reactant-solvent linear coupling strength. z is the ion core charge of the reactant. The multioscillator description of the solvent can be replaced by a single oscillator model using the coordinate transformation16,34

q)

∑ giqi i

(A2)

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Mishra and Waldeck

Figure 13. Plots showing pec’s for three different electronic levels in the electrode. The red curve corresponds to  ) -0.8 eV, the black dashed curve corresponds to  ) 0.0 eV, and the solid black curve corresponds to  ) 0.8 eV (λ ) 0.8 eV, ∆ ) 0.001 eV, and η ) 0.0 V).

Subsequently, by averaging over electronic degrees of freedom while keeping the reaction coordinate q as a fixed parameter, the potential energy u(q) of the system, subject to the charge conservation criterion, is obtained as11,35

u(q) )

∫-∞ (' - F)Fr(', q) d' - zq + q2/(4λ) F

(A3) where Fr(′, q) is the reactant dos, and depends on the solvent coordinate q (cf. eqs 11 and 12). Substituting eqs 11 and 12 into eq A3 and carrying out the  integration, the potential energy expression 8 is obtained. Figure 13 shows plots of these pec’s for three different values of . Supporting Information Available: The variation of 〈nr(q)〉, the q-dependent average occupation probability of the reactant, is determined by the reactant density of states. The correlation between these two quantities is provided. In addition, the equivalence of the  dependence of activation energy in the limit ∆ f 0 obtained from eq 4 as well as from the formalism developed in section III are demonstrated. How to determine the solvent dynamic controlled preexponential factor employing Kramers’ difusion controlled barrier crossing rate is briefly discussed. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) (a) Armstrong, F. A.; Wilson, G. S. Electrochim. Acta 2000, 45, 2623. (b) Wang, J. Anal. Chim. Acta 2002, 469, 63. (c) Katz, E.; Willner, I. Angew. Chem., Int. Ed. 2004, 43, 6042.

(2) Miller, R. J. D.; McLendon, G. L.; Nozik, A. J.; Schmickler, W.; Willig, F. Surface Electron Transfer Processes; VCH Publishers, Inc.: 1995. (3) Kuznetsov, A. M. Charge Transfer in Physics, Chemistry and Biology; Gordon and Beach: Reading, MA, 1995. (4) Gerischer, H. Z. Phys. Chem. NF 1960, 26, 223; 1961, 27, 40, 48. (5) Miller, C. J. In Heterogeneous Electron Transfer Kinetics at Metallic Electrodes in Physical Electrochemistry, Principles, Methods and Applications; Rubinstein, I., Ed.; Marcel Dekker: NewYork, 1995; p 27. (6) Galperin, M.; Ratner, M. A.; Nitzan, A. J. Phys.: Condens Matter 2007, 19, 103201. (7) Mishra, A. K.; Rangarajan, S. K. THEOCHEM 1996, 361, 101. (8) (a) Mishra, A. K.; Rangarajan, S. K. J. Phys. Chem. 1987, 91, 3417. (b) Mishra, A. K.; Rangarajan, S. K. J. Phys. Chem. 1987, 91, 3425. (9) Mohr, J.; Schmickler, W.; Badiali, J. P. Chem. Phys. 2006, 324, 140. (10) Mohr, J.; Schmickler, W. Phys. ReV. Lett. 2000, 84, 1051. (11) Schmickler, W. J. Electroanal. Chem. 1986, 204, 31. (12) Dogonadze, R. R.; Kuznetsov, A. M. Prog. Surf. Sci. 1975, 6, 1. (13) Ulstrup, J. Charge Transfer Processes in Condensed Media; Springer: Berlin, 1979. (14) Newton, M. D.; Sutin, N. Annu. ReV. Phys. Chem. 1984, 35, 437. (15) Sutin, N. Prog. Inorg. Chem. 1983, 30, 441. (16) Mishra, A. K.; Schmickler, W. J. Chem. Phys. 2004, 121, 1020. (17) Schmickler, W.; Mohr, J. J. Chem. Phys. 2002, 117, 2867. (18) Newns, D. M. Phys. ReV. 1969, 178, 1123. (19) Dogonadze, R. R.; Kuznetsov, A. M.; Vorotyntsev, M. A. J. Electroanal. Chem. 1970, 25, App. 17. (20) Schmickler, W. Annu. Rep. Prog. Chem., Sect. C 1999, 95, 117. (21) Zusman, L. D. Chem. Phys. 1987, 112, 53. (22) (a) Zusman, L. D. Z. Phys. Chem. 1994, 186, 1. (b) Zusman, L. D. Chem. Phys. 1980, 49, 295. (c) Zusman, L. D. Chem. Phys. 1983, 80, 29. (23) Hynes, J. T. J. Phys. Chem. 1986, 90, 3701. (24) Gennett, T.; Milner, D. F.; Weaver, M. J. J. Phys. Chem. 1985, 89, 2287. (25) Khoshtariya, D. E.; Dolidze, T. D.; Zusman, L. D.; Waldeck, D. H. J. Phys. Chem. A 2001, 105, 1818. (26) Sebastian, K. L.; Ananthapadmanabhan, P. J. Electroanal. Chem. 1987, 230, 43. (27) (a) Calef, D. L.; Woylnes, P. G. J. Phys. Chem. 1983, 87, 3387. (b) Calef, D. L.; Woylnes, P. G. J. Chem. Phys. 1983, 78, 470. (28) McManis, G. E.; Mishra, A. K.; Weaver, M. J. J. Chem. Phys. 1987, 86, 5550. (29) Rips, I.; Jortner, J. J. Chem. Phys. 1987, 87, 6513. (30) Napper, A. M.; Liu, H.; Waldeck, D. H. J. Phys. Chem. B 2001, 105, 7699. (31) Mishra, A. K.; Bhattacharjee, B.; Rangarajan, S. K. J. Electroanal. Chem. 1992, 331, 801. (32) (a) Yue, H.; Khoshtariya, D. E.; Waldeck, D. H.; Grochol, J.; Hilderbrandt, P.; Murgida, D. H. J. Phys. Chem. B 2006, 110, 19906. (b) Khoshtariya, D. E.; Wei, J.; Liu, H.; Yue, H.; Waldeck, D. H. J. Am. Chem. Soc. 2003, 125, 7704. (33) Smalley, J. F.; Feldberg, S. W.; Chidsey, C. E. D.; Lindford, M. R.; Newton, M. D.; Liu, Y. P. J. Phys. Chem. 1995, 99, 13141. (34) Sebastian, K. L. J. Chem. Phys. 1989, 90, 5056. (35) Mishra, A. K. J. Phys. Chem. 1999, 103, 1484.

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