2377
J. Phys. Chem. 1994,98, 2377-2381
Mechanism for Proton-Coupled Electron-Transfer Reactions R. I. Cukier Department of Chemistry, Michigan State University, East Lansing, Michigan 48823- 1322 Received: November 16, 1993" We introduce a model whereby the rate of an electron-transfer reaction can be modulated by an intervening proton-transfer reaction. The mechanism for this modulation is the assumed dependence of the electronic matrix element, which enables the electron to transfer from donor to acceptor, on the configuration of the proton(s) that are potentially undergoing a proton-transfer reaction. As the proton configuration changes, so does the electronic matrix element and, consequently, the electron-transfer rate depends on the proton configuration and dynamics. For this proton-coupled electron-transfer reaction, we find that the observed rate constant may have a significant isotope effect upon replacing the protons by deuterons and may also exhibit a temperature dependence above that expected for a conventional electron-transfer reaction.
I. Introduction Charge separation is a basic mechanism of biological and chemical energy conversion. Often, the electron-transfer (ET) reactions responsible for the charge separation are coupled to the motion of protons. The importance of this coupling between proton motion, which may be a proton-transfer (PT) reaction, and ET is a well-elaborated theme in the study of biological assemblies including PS I1,l4 cytochrome c oxidase,s-* and c y t o c h r o m e ~ . ~ For J ~ examples, the propagation of the initial electron/hole pair formed upon light absorptionin PS I1 is coupled to proton motion within the quinone pool,' and Okamura and Feherll have shown that such a circumstance may occur in photosynthetic bacterial reaction centers on the basis of an in increase in charge recombination rates upon DzO exchange. We shall refer to this coupling of PT and ET as proton-coupled electron transfer (PCET).I2 In response to the complexity of biological systems, attempts to systematize the study of PCET with the use of model compounds are warranted. One such approach has been carried outl2 by utilizing the ability of carboxylic acids to form cyclic dimers in low dielectric solvents.l3-1s This provides a mechanism to create a fixed distance, intramolecular, donor/acceptor electron-transfer couple, separated by a proton-transferinterface, as shown in Figure 1. The photoinduced ET from donor to acceptor exhibits a distinct isotope effect, upon deuteration of the proton interface. The ratio of proton-todeuteron ET rate constants, k H / k D , was found to be about 1.1 for both the charge separation (forward ET) and charge recombination (return to ground-state) reactions.'2 In this article we shall (1) suggest a mechanism for the coupling between proton transfer and electron transfer, (2) provide a theoretical framework with which to calculate PCET rates, and (3) explore the predictions of this theory with regard to what isotope effect to expect and whether any unusual temperature dependence to the rate constant should be anticipated. (By unusual temperature dependence we mean one above that to be anticipated for a conventional electron-transfer process.) The mechanism we propose is quite straightforward-we view the ET reaction as modulated by the PT reaction via the changing electronic coupling of the ET reaction engendered by the configuration of the protons (cf. Figure 1). As the donor and acceptor are separated by a large distance, through the protontransfer interface, we assume that the electron transfer is in the nonadiabatic regime where the electronic matrix element V is relatively small and appears in the ET rate constant expression as a prefactor, as provided by a Golden Rule calculation.16 We anticipate that this electronic coupling depends upon the proton configuration in the interface, V = V(x),where x measures the e
Abstract published in Aduance ACS Absrracrs, February 1, 1994.
-x-
Figure 1. Representative molecule for PCET with a proton-transfer interface formed by a dicarboxylic acid dimer. The substituents of the dicarboxylic acids are appropriatelychosen electron-transfer donors (D) and acceptors (A). The ET is initiated by photochemical methods, hence the excited-statedesignation for D.
location of the protons in the interface. In particular, we will assume that this coupling is maximal for the symmetric (transition state) configuration of the protons. At this configuration, we may anticipate a larger degree of charge delocalization over the dicarboxylic acid ring and a concomitant enhanced electronic coupling. Preliminary quantum chemical calculations lend credence to this notion.17 While the specific conclusions we reach will depend upon the specific form of V(x), the mechanism suggested here for PCET does not. Of course, if V(x) does not depend on the proton configuration, then this mechanism will not be applicable. The plan of the rest of this paper is as follows: In section I1 we formulate a hamiltonian that will describe the coupling between proton andelectron transfer of the formjust outlined. TheGolden Rule evaluation of this rate is carried out, and shown to provide a useful expression for the PCET rate constant. In section I11 we evaluate the rate constant expression based on several models of the potential surface characterizing the proton wave functions. The isotope and temperature effects to be anticipated are explored here. Section IV summarizes our conclusions. 11. Proton-Coupled Electron-Transfer Hamiltonian A Hamiltonian that incorporates the transfer of an electron from donor (state 1) to acceptor (state 2) with an electronic matrix element V ( x ) modulated by the proton's position, x , is18
H = V(&
+ h(x) +
where for definiteness we write V(x) = Vo exp(-ylxl). Here x is measured relative to the minimum of the proton potential curve (where the electronic coupling is assumed to maximize) and y is the decay constant for thematrixelement. The proton potential u(x) and associated hamiltonian h(x) = t ( x ) + u ( x ) are as yet unspecified. The proton potential can be thought of as a doublewell potential, implying a hydrogen-bondedproton that is localized
0022-3654/94/2098-2317~04.50~0 0 1994 American Chemical Society
2378 The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 on one or the other side of the proton interface. The @ k , q k j are the momenta and coordinates of the solvent, the { i l k } their frequencies, and the (gkj the couplings of these solvent modes with the tunneling electron, and we have set the oscillator masses to unity for convenience. The u, (a= x , y , z ) are the Pauli spin matrices. The A term is minus the reaction free energy (A is positive for an exothermic reaction). The hamiltonian of eq 2.1 includes the standard spin-boson representation of an electrontransfer reaction. It also incorporates the effect of the proton’s influenceon the electronic coupling of donor and acceptor ( V ( x ) ) and how that coupling is modulated by the proton’s dynamics
Cukier to be independent, i.e.
The analysis of the solvent correlation function has been carried out many times.I6 We consider only the limit of a classical solvent, as this suffices at conventional chemical temperatures when the coupling is to the solvent polarization. The result is
(W).
To evaluate the rate constant describing the electron’s transfer, it is convenient to work in a basis of localized states, 1 and 2. All the terms of eq 2.1 are diagonal in this basis except for the term proportional to V(x) that is responsible for the transfer. If Vis sufficiently small for all x values, then the transition rate between state 1 and 2 is given by the Golden rule expression16
where E, is the solvent reorganization energy, given in terms of the solvent frequencies and solvent donor/acceptor coupling constants as (2.10)
w w‘
with u and u’denoting the quantum states associated with the x coordinate and w and w‘ with the (qk) coordinates. The energy eigenvalue E ~ d d= Ed + E1d is associated with the Hamiltonian
Using eq 2.9 in the rate constant expression of eq 2.5 then yields
.
v2
HI:
H,= (1141) = with the matrix element I(u,u? defined as and Ebw= E,
+ Elw with the hamiltonian H2:
H2 = (2142) = h(x)
+
&[5+
(d :!
qk
+
) ]-’
2 (2.3b)
k‘
Expressing the 6 function in eq 2.2 as
~ ( E I , -, Ez,,) = d(E1, - E,, + Ed - E,) = J dt 6(Ed- E, + t)S(t - ( E l , - E2w)) (2.4) and using the Fourier time representation of the second 6 function permits expression of eq 2.2 as
We havedefined these matrix elements as dimensionless quantities for convenience. If these matrix elements and the energies of the states in the potential u(x)can be evaluated, then the rateconstant can be obtained by carrying out the sums in eq 2.1 1. We discuss the structure of the rate constant expression and the evaluation of the Z(u,u? in the following section. 111. Analysis of the Rate Expression
A. Expression for the PCET Rate Constant. The analysis of the rate constant expression is facilitated with the definition G(uf,u)= ~
~ ~ T ~ - ( E ~ A + E ~ - E ~ )= ’ / ~ E & E T
h vh2 o21/,2/qk,Te-EAd/kBT
J - y d t e+i(eA)t/h( eMfzc/he-itHp/h
) (2.5)
The bracket denotes an average over the equilibrium ensemble for the solvent with weight PI
= exp(-HIq/kBT)/Tr eXp(-H,‘/kBT)
(2.6a)
exp(-E,/kBT)/Cexp(-E,/kBT)
(2*6b)
Pu
v
is the corresponding quantity for the x coordinate. In obtaining these results we have introduced the definitions
(3.1)
The effective activation energy EA,^ defined in eq 3.1 accounts for the requirement of energy conservation in the combined electron-proton system. The states in the proton well with energies &(E,) serve to provide a set of donor (acceptor) surfaces for the electron transfer that can increaseor decrease the activation energy EA^^ dependent on the particular u and u‘ involved. Noting the relation p ~ G ( v ’ , v= ) exp[-O(A/E,)(E, - Ed)]p,G( u , ~ ’ ) ,we may rewrite k as v‘
VI0
0
CpO v=o
[l
+ e-s(A/E~)(E,E,)]12(u’,~) G(u,u?
(3.2)
0’=,+1
and noted that the proton and solvent coordinates are assumed
For exothermic reactions A > 0, and the factor in brackets in the second term of eq 3.2 rapidly approaches 1. If the ET reaction is in the “normal” regime where A - E, > 0, then, from the definition of EA^^ it is clear that G(u,v’) is a decreasing factor
Proton-Coupled Electron-Transfer Reactions
The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 2379
relative to G(u,u). In this form, the dependence of the matrix elements on u and u’ becomes critical, as the weight of the offdiagonal elements will be reduced by the G(u,u’) factor. Let us assume for now (vide infra) that theoff-diagonal matrix elements are small compared with the diagonal elements. Then, from eqs 3.1 and 3.2 we obtain
The two states here correspond to the tunnel doublet in a deep double-well potential. The matrix element can be readily estimated for a deep well, as the states 0 and 1 can be written as linear combinations of left and right localized states, and these states are well-approximated by the corresponding harmonic oscillator ground-state wave functions. Noting that the WKB approximation for the quartic potential of eq 3.7 yields the following expression for 60, the tunnel splitting:20
where we define kei as the electron-transfer rate constant with V(x) = V(x-0) = Vo:
(3.9)
(3.4) The ET rate constant is referenced, arbitrarily, to the value VO that is the maximal rate. Since the value of the matrix element will not be well established when comparing with a given experimental system, the reference value is not crucial. The prefactor of k,, in eq 3.3 is readily interpreted. Since we are assuming that the I(u,u? are negligible, we may write it as
where V ( x ) = V(x)/Voand the subscript u in the last equality in eq 3.5 denotes the quantum statistical average with respect to the proton states. Thus, we arrive at the simple expression
we may safely neglect the splitting in eq 3.8 for D = 1, as it is much smaller than the thermal energy. Expressing the states 0 and 1 in terms of the left-right states then reduces eq 3.6 to
k/k,, = (RI eqIxI(R) with ( x l ~= ) (a~)-1/2e-(x42/2a2
u(y) = h w o b 2 / 4 - y4/64D]
(3.1 1)
where d locates the right-side minimum of the double-well potential. This result shows that the protonic contribution to the rate is independent of temperature. Thus, no additional temperature dependence to the rate would be predicted over that anticipated for a pure ET reaction. Now we turn to the isotope effect that this deep-tunnel limit would predict. The integral in eq 3.10 is readily evaluated with eq 3.1 1 to be
k/kei = e1(ra)Lu4[l This rate constant expression corresponds to the thermal average of the electron-transfer rate for a given electronic coupling V(x). As just noted, the value of VO will not be known for a given experimental system, so we will consider the relative rate klk,, and consider what the differences between an ET and a PCET rate can be, as embodied in the ( p ( x ) )factor. , That requires an examination of the matrix elements Z(u,u) to which we now turn. B. Matrix Elements I( v,v). The matrix elements’ values rely on obtaining the wave functions of the states in the double-well potential. Commonly used potentials for proton tunneling in a symmetric hydrogen bond include the sum of two suitably displaced Morse potentials or a potential with constant, quadratic and quartic powers of the coordinate,x . Comparison of the results of ab initio calculations with this potential and with the sum of the Morse potentials representation has shown that they both provide good fits to quantum chemical res~1ts.I~Thus, we shall use the quartic representation
(3.10)
+ @ ( d / a- ya/2)]/2 + e[(ra)2+r4[l - @(d/a+ ya/2)]/2
(3.12)
with 3 the probability integral. Assuming that d is larger than a, the integral can be approximated as
k/k,, = e[(Ya/2)zqdl
(3.13)
The scale of a is the width of the proton’s wave function in the (harmonic) well of frequency WO. This is around 0 . 1 4 . 2 A for typical proton-transfer wells. (WO = 2000 cm-’ corresponds to a = 0.13 A). The scale of d comes from the specific geometry of the proton-transfer interface with typical distances d = 0.3-0.45 A. Thus, d > a is a safe assumption. For the scale of the dependence of V(x) on x we assume that y is roughly between l / d and 4/d that corresponds to a falloff from x = 0 to x = d of a factor of e-I to e4. Mass only enters the a parameter in eq 3.13, and it is straightforward to obtain
(3.7)
where the dimensionless coordinate y is defined as y = x / a with a= and D = E B / h w o measures the barrier height EB in units of the energy hwo at the minima of the double-well potential. The pattern of the eigenvalues in this potential can be determined from the value of D. For cases where D 2, there will be one pair of tunnel-split states near the bottom of the well and another pair of states near the well barrier. Larger values of D imply more pairs of tunnel states below the well barrier. We consider first the simplest case of a pair of tunnel states well separated from other states in the well in the sense that these other states are thermally inaccessible. If the barrier E6 and the well energy hwo are several thousand wavenumbers, then this will be the case. Then, the thermal factor in eq 3.3 reduces the sum to
4 -
-
-
Using the numbers for y, a, and d as just obtained, leads to kH/kD 1.0-1.8. Thus, an isotope effect of up to 80% is indicated. That the ratio is always greater than one reflects the more concentrated wave function of the deuteron relative to the proton, which reduces the size of the matrix element for the deuteron relative to the proton. Thus a substantial isotope effect may be present for PCET, as has been shown experimentally.’z These estimates are of course very crude, and are subject to uncertainty due to the lack of knowledge of the behavior of V ( x ) . The expression using only the tunnel doublet states appropriate to the deep well will not exhibit a temperature dependence different than that of the ET reaction. So that there be a significant temperature dependence arising from the proton motion, the eigenvalues in the double-well potential must be spaced on a scale comparable with thermal energies. As we now discuss, the dicarboxylic acids may in fact exhibit such a level pattern. Potential surfaces for dicarboxylic
2380 The Journal of Physicat Chemistry, Vol. 98, No. 9, 1994
Cukier size to a t most of order 1, the thermal factors will reduce their contributions, so that just keeping two terms in the sum should be sufficient. In this case, we should write eq 3.3 as
k/k,, = 2pJ*(O,o) EB
+ p2Z2(2,2) = p0[2Z2(0,0)+ e-B(ErEa)Z2(2,2)] (3.16)
Thus, a temperature dependence from the population of the proton states will be in evidence in the PCET rate. As this temperature dependence is related to the states in the well, it has a completely different origin than that intrinsic to the ET process (i-e., the E T 0 activation energy as given in eq 3.4). The energy difference E2 -Eowill be around 500-1000 cm-l, as the state EO hwo/2 and - 2 6 2 6 E2 is around the barrier top. Thus the thermal factor and the relative sizes of P(0,O)and 12(2,2) are balanced, and a significant temperature dependence to the rate will be observed. We have based our analysis on the premise that the off-diagonal matrix elements Z(u,u? can be neglected relative to the diagonal matrixelements. Aquantitativeestimateof, e.g., theZ(0,2) matrix element for a deep tunneling case is readily made, as the u = 2 level would then too be well approximated as a linear combination of right and left localized harmonic oscillator states. Under this Figure 2. Representative double-well potential for the proton transfer assumption, it is straightforward to evaluate Z(0,2) and this yields when D = E ~ / h w o 2. The wave functions are drawn to represent Z(0,2) = c(y~)~l(O,O), with c an order-one numerical constant. states within thewelland at the topofthe barrier. Note theconcentration As long as y a is small, which is the case for a not too rapid decay of probability around x = 0 for the u = 2 state relative to that for the of V(x), then essentially Z(0,2)’s value is controlled by the u = 0 (or 1) states. The eigenvalues and wave functions are based on the work of ref 30. orthogonality of the involved wave functions and is indeed negligible in comparison with Z(0,O). As more off-diagonal terms acids have been extensively investigated by ab initio methods. involve wave functions that are more disparate, we can extend The initial barrier estimates on, e.g., formic acid dimers were this argument to these terms. Also recall that for exothermic very high, and higher level fully optimized gas-phase ab initio normal regime E T reactions, the G(u,u? factors are decreasing calculations, while leading to barriers around 3000 cm-l, are still relative to the G(u,u) factors. Thus, we expect that the diagonal high compared to those that would give closely spaced l e ~ e l s . ~ ~ , 2 ~approximation of eq 3.3 is sufficient for most purposes. Attempts to include the effect of the crystal field on the double well in crystalline dicarboxylic acids have been made, and these IV. Concluding Remarks reduce the barrier c o n ~ i d e r a b l y . ~ ~Several . ~ ~ experimental We have presented a mechanism that permits a proton-transfer have concluded that the barrier is actually quite reaction to influence the rate of an electron-transfer reaction. By low (EB 1000 cm-I), though it is important to note that the assuming that the electronic matrix element V(x), responsible barrier must be inferred from a theoretical model of the protonfor the electron transfer, depends on the proton(s) configuration, transfer mechanism. In fact, there are now a number of theoretical the electron-transfer rate will depend on this matrix element in studies that have been able to fit experimental data only with the a way that is connected with the dynamical behavior of theprotonassumption of a barrier in the neighborhood of 1000 cm-1.25-28 (s). This later behavior will depend on the potential that the Accepting a barrier value of 1000-2000 cm-1 for the proton proton experiences. It is interesting that a computationally useful potential and using a well separation of 2d = 0.9 that is typical rate constant expression (cf. eq 2.1 1) can be obtained without for the dicarboxylic acid geometry, then implies wo 800-1200 specifying the nature of the potential that the proton evolves cm-1. (This estimate is obtained by expressing the distance under (the specific form of h ( x ) ) . Furthermore, the form of between the minima of eq 3.7 as d = 2(EB/m)1/2/~o.)For this V ( x ) is arbitrary in this regard. Thus, as quantum chemical type of potential, there are states as shown in Figure 229330 and, input becomes available as to the form to choose for V(x),this importantly (1) the level spacing is now on the scale of the thermal information can be used to provide a more quantitative theory. energy and (2) the wave functions of the around-the-barrier states With the above-assumed form for V(x),we have shown that have a large weight in the region of the barrier, where the V ( x ) a characteristic isotope effect with regard to a proton versus factor is large. Thus, for this picture, which seems appropriate deuteron interface should be observed. The magnitude of the to thedicarboxylicacids,the possibility of more states contributing isotope effect can be consistent with that observed experimentally. to the sum in eq 3.3 is strong. To estimate the size of Z(2,2), the If the double-well potential for the proton motion is sufficiently first state that contributes around the well maximum, appeal to weak that there are states separated by energies of the order of Figure 2 shows that we may approximate the wave function as the thermal energy, then a temperature dependence to the PCET a rectangle of width 2d and height chosen to normalize the wave rate should be observed. The characteristic temperature for this function. Using this rectangular wavefunction in eq 2.1 1b then effect will be the separation of the levels within the double-well yields potential. Since the above-mentioned effects will be strongly dependent Z(2,2) = j m -m d x@k,2(x)eqM = on the actual double-well potential and the form of the electronic matrix element, attention should be focused on particular protontransfer interfaces where reliable ab initio calculations can be carried out for both these quantities. This value is of order 1; thus, especially as y gets larger (cf. eq 3.12), Z(2,2) becomes considerably larger than Z(0,O). FurtherAcknowledgment. The financial support of the National more, since the higher eigenvalue matrix elements are limited in Institutes of Health (GM 47274) is gratefully acknowledged.
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Proton-Coupled Electron-Transfer Reactions
References and Notes
The Journal of Physical Chemistry, Vol. 98, No. 9, 1994 2381
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