J. Phys. Chem. 1987, 91, 3417-3425
3417
Theory of Electron-Transfer Processes via Chemisorbed Intermediates A. K. Mishra and S. K. Rangarajan* Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560 01 2, India (Received: July 17, 1986)
A model Hamiltonian to describe adsorbate-mediated electron transfer is proposed and analyzed for the (electrochemical) currents. This Hamiltonian differs fundamentally from the ones used earlier (cf. Schmickler). Three situations are treated, viz., (I) a lone adsorbate picture; (11) presence of an inactive neighboring adsorbate; and (111) electrostatic effects due to an adsorbate lattice. In all three cases the mediation is via a single adsorbate only. Several limiting cases are considered and related to the earlier results of Marcus and Schmickler.
1. Introduction A description of an electron-transfer process, mediated by chemisorbed intermediates, is essential for understanding the catalytic (or inhibiting) effects of adsorbates,l-I0 and is also a prerequisite for characterizing more involved processes like electron transfer on a chemically modified electrode”-” or through a m ~ n o l a y e r or ’ ~film-covered ~~~ electrode surfaces.2G24 Adsorbates can influence the electron-transfer rate between a redox couple located at O H P (or beyond) and the electrode directly, by participating in the transition process, or indirectly, by changing the micropotential at the site of the redox couple,2s-28 besides altering the electrode characteristics such as density of states, Fermi level, work function etc. As the indirect influence of the adsorbates on the reaction rate can be accounted for, at (1) Adzic, R. R.; Tripkovic, A. V.; Grady, W. 0. Proceedings of the Symposium on Electrocatalysis; The Electrochemical Society: Pennington, NJ, 1982, p 254. (2) Watanabe, M.; Shibata, M.; Motoo, S. J. Electroanal. Chem. 1985, 187, 161 and references therein. (3) Peter, L. M.; Durr, W.; Bindra, P.; Gerischer, H. J. Electroanal. Chem. 1976, 71, 3 1. (4) Gerischer, H.; Pettinger, B.; Lubke, M. In Electrocatalysis; Breiter, M., Ed.; Electrochemical Society: Princeton, NJ, 1974. ( 5 ) AM El-Halim, A. M.; Juttner, K.; Lorenz, W. J. J. Electroanal. Chem. 1980, 106, 193. (6) Horanyi, G.;Rizmayer, E. M. J. Electroanal. Chem. 1982,140,347. (7) Adzic, R. R.; Tribkovic, A. V.; Markovic, N. M.. J. Electroanal. Chem. 1983, I 50, 79. (8) Schultze, J. W.; Brenske, K. R. J. Electroanal. Chem. 1982, 137, 331. (9) Binder, H.; Kohling, A. In From Electrocatalysis to Fuel Cells; Sandstede. G.. Ed.: Batelle Research Centre: Seattle. WA. 1977. (10) Adzic; R. R.;Triplaovic, A. V.; Markovic, N. M.’J. Electroanal. Chem. 1980, 114, 37. (1 1) Albery, W. J.; Hillman, A. R. Annu. Rep. Prog. Chem. Sect. C 1981, 7R 111,. I.
(12) Murray, R. W. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1984; Vol. 13, p 191. (13) Andrieux, C. P.; Saveant, J. M. J. Electroanal. Chem. 1980, I l l , 377. (14) Stickney, J. L.; Soriaga, M. P.; Hubbard, A. T.; Anderson, S. E. J . Electroanal. Chem. 1981, 125, 73. (15) Zak, J.; Kuwana, T. J. Electroanal. Chem. 1983, 150, 645. (16) Gorton, L.; Torstensson, A.; Jaegfeldt, H.; Johansson, G.J. Electroanal. Chem. 1984, 161, 103. (17) Schultze, J. W.; Mohr, S.; Lohrengel, M. M.. J. Electroanal. Chem. 1983, 154, 57. (18) Adzic, R. R.; Simic, D. N..; Despic, A. R.; Drazic, D. M. J. Electroanal. Chem. 1977, 80, 81. (19) Diaz, A. F.; Orozco Rosales, F. A,; Rosales, J. P.; Kanazawa, K. K. J. Electroanal. Chem. 1979, 103, 233. (20) Dogonadze, R. R.; Kuznetsov, A. M.; Ulstrup, J. Electrochim. Acta 1977, 22, 967. (21) Bockris, J. O M . ; Khan, S. U. M. Quantum Electrochemistry; Plenum: New York, 1979. (22) Kuznetsov, A. M..; Ulstrup, J. Sou. Electrochem. 1983, 19, 127. (23) Schmickler, W. J . Electroanal. Chem. 1977, 82, 65. (24) Schmickler, W.; Ulstrup, J. Chem. Phys. 1977, 19, 217. (25) Krylov, U. S.; Kiryanov, V. A.; Fishtik, I. F. J. Electroanal. Chem. 1980, 109, 115. (26) Kuznetsov, A. M.; Kiryanov, V. A. Sou. Electrochem. 1981,17, 1163. (27) Fawcett, W. R.; Levine, S. J. Electroanal. Chem. 1973, 43, 175. (28) Guidelli, R. J. Electroanal. Chem. 1974, 53, 205.
0022-3654/87/2091-3417$01.50/0
least qualitatively, within the framework of conventional electron-transfer by renormalizing the relevant parameters, we consider here the more important case wherein the adsorbate orbitals directly mediate the electronic transitions. Since we are mainly interested in elucidating the effect of adsorbates on the reaction mechanism, we ignore the direct coupling between the electrode and the depolarizer in the solution phases. We assume the coupling between the adsorbate and the depolarizer to be weak, and therefore, handle it in a perturbative sense. On the other hand, a perturbative analysis obviously fails to describe the adsorbate-electrode subsystem because of the strong coupling between them. This is also the reason why the earlier formalisms for the bridge-assisted electron-transfer reactions, based on the secondorder perturbation t h e ~ r y ~ ~are - ’ inapplicable ~ to the present context. Schmickler’s earlier c o n t r i b ~ t i o nto~ this ~ problem, viz., electron transfer via intermediates, is notable and our work here is similar to his approach except for the following differences: (a) At the level of Hamiltonian, the descriptions of transitions among the electron states involved, viz., of the reactant, metal, and the intermediate are introduced differently. Whereas Schmickler’s Hamiltonian (refer to eq 8, ref 35) involves triple products of fermion operators like c , , A ~ + c ~ A c ~ A , ours (refer to eq 1) would have like cllr+cI.The change is necessitated because it is not proper to introduce triple fermion operator products in the second quantized Hamiltonian. We must note here that such triple products as in ref 35 violate particle number conservation. (b) At the level of analysis, we go beyond the approximations of Schmickler, viz., the broadening or chemisorption parameter A >> kT, and our results are valid for arbitrary strength of the adsorbate-metal coupling. In another interesting report,36Schmickler has treated a different problem, viz., electron transfer through surface states by a T-matrix approach. On the basis of isomorphism at the Hamiltonian level, we can transpose our results here to this situation also within the limit of linear response formalism. This is made possible only because the expression for the current deduced here is also valid under the limit most likely to be true in this case of surface states, viz., weak “(adsorbate) surface state-bulk state coupling”. This report36 too has the triple operator product anomaly. In the present paper, we develop first a formalism for the case (29) Dogonadze, R. R.; Kuznetsov, A. M. Prog. Surf. Sci. 1975, 6, 1. (30) Schmidt, P. P. In Spec. Period. Rep. (Electrochemistry) (a) 1975, 5, 21; (b) 1978, 6, 128. (31) Ulstrup, J. Charge Transfer Processes in Condensed Media; Springer-Verlag: New York, 1979. (32) Kharkats, Yu. I.; Madumazov, A. K.; Vorotyustev, M. A. J. Chem. Soc., Faraday Trans. 2 1974, 70, 1578. (33) Dogonadze, R. R.; Ulstrup, J.; Kharkats, Yu. J. Electroanal. Chem. 1973, 39, 47. (34) Doyonadze, R. R.; Ulstrup, J.; Karkats, Yu. J. Electroanal. Chem. 1973, 43, 161. (35) Schmickler, W. J . Electroanal. Chem. 1980, 113, 159. (36) Schmickler, W. J. Electroanal. Chem. 1982, 137, 189.
0 1987 American Chemical Society
3418 The Journal of Physical Chemistry, Vol. 91. No. 12, 1987
wherein the redox species is coupled to a single adsorbate, represented by a single orbital and the associated spin degeneracy. The effects of other adsorbates on the mediating adspecies have been ignored. Obviously, this model is fairly realistic in the low-coverage regime. In section 2, we describe the relevant model Hamiltonian. The expression for current is derived in section 3 and a discussion is given in section 4. As an extension of this formalism, we consider in section 5 the effect of lone inactive neighboring adsorbate characteristics on the current. By “inactive” we mean that neighboring adsorbate, identical with the central adsorbate as regards molecular or electronic properties, has no electronic overlap with the redox couple. The neighboring adsorbate affects the rate mainly via (i) the electrostatic interaction with the mediating adspecies-an effect which can easily be handled by renormalizing the adsorbate energy levels, (ii) substrate-mediated interaction with the reacting adspecies, the solvent-mediated interactions with the reacting adspecies and the redox couple and, (iii) changing the equilibrium solvent configuration. We investigate next, in section 6, how the inactive adsorbates, distributed over the whole electrode surface, change the mediating adsorbate characteristics, and hence, the reaction rate. We have, following Muscat and N e w n ~ , ~assumed ’ that (i) adsorbates interact with each other through electrostatic forces, (ii) they are distributed uniformly over the electrode surface, and (iii) they form a square lattice, wherein the size of the unit lattice obviously depends on the coverage factor. The adsorbates are considered to be identical and are represented by a two-orbital picture. Both the orbitals of the mediating adsorbate are coupled to the depolarizer. The electric field produced by the inactive adsorbates is assumed not to penetrate into the solution phase beyond the OHP.25338 This field affects the “mediating adsorbate” in two ways. First, the field shifts its energy levels. Next, the field provides a coupling mechanism between the orbitals of the mediating adsorbate, thus leading to intraadsorbate charge redistribution, i.e., to the polarization of the reacting adsorbate. Clearly, such a model helps us in investigating (i) the coverage dependence (introduced in the formalism via the abovementioned electric field) of the current and, (ii) the effects of the mediating adsorbate polarization on the transition rate. Finally, the summary and the conclusions are provided in section I.
2. Description of the System through Model Hamiltonian The Hamiltonian describing the adsorbate-mediated electrontransfer process must obviously contain terms representing the metal states, chemisorbed species, the orbitals of the electroactive species participating in the transfer process, relevant solvent modes, and interaction terms involving these components besides, of course, the external field effects. Even stipulating a Hamiltonian incorporating all the details presents a formidable many-body problem involving fermions and bosons, not to speak of solving this! This constraint, hence, forces us to setup a model Hamiltonian which, though simple in nature, is sufficiently realistic to describe the system faithfully. Having this criterion in mind, we represent the system by the following model Hamiltonian:
H = Ctknka + E[‘%+ U(naa)]n,u+ k,o
0
[ Va!&an+Cka +
Vk&kg+cmI
+ €rnr +
[ Vo&oufCr
+
n
k.0
Vrac,+c,,l + hCw,b,+b,
+ hCw,g,,(b, + b,+)n,, + v,a
Y
hEw,g,,(b, + b,+)n, + hCw,(g,, + g d ( b , + b,+) + € 0 ( 1 )
”
”
In the above Hamiltonian, k, a, and r refer to the electrode, adsorbate, and reactant orbitals, respectively. c:(ci) is the fermion creation (annihilation) operator and n, is the occupation number operator for the ith orbital. u is the spin variable. b,+(b,) is the boson creation (annihilation) operator for uth solvent polarization (37) Muscat, J. P.; Newns, D. M. J . Phys. C. 1974, 7, 2630. (38) Fishtik, I. F.; Krylov, V. S. Sou. Electrochem. 1980, 16, 546.
Mishra and Rangarajan mode. Energy of the kth substrate orbital is tk. t, is the energy of the adsorbate orbital and it includes the external field and image effect. We write it as e , = @ - I + e(+, - $,) + tim where @ is the substrate work function, I is the ionization potential of the adsorbate, +,, is the outer potential of the metal, and qd is the potential at the adsorbate site. timis the image energy and it includes the contributions from the interaction of an electron in the adsorbate with its own image as well as with the image of adsorbate core charge. Here, the zero of energy is taken at the Fermi level of the electrode. 6 is the intraadsorbate repulsive term, as modified by the image effects (U = U - 2tim). Presently, the intraadsorbate repulsion term Unaano5is treated within the Hartree-Fock approximation. The term containing Vak’scharacterizes the adsorbate-electrode coupling. t, is the orbital energy of the reactant or depolarizer. V,, ( V,,) is the coupling coefficient between adsorbate and reactant. w, is the frequency of uth solvent polarization mode. In the present work, the harmonic oscillator model is used to describe the solvent polarization modes. The linear terms in the boson displacement operator 6, b,+, appearing on the right-hand side of (1) describe the interaction of solvent polarization mode with the adsorbate and the reactant. The coupling coefficient of the adsorbate orbital with the uth polarization mode is g,, and the same for the reactant orbital is g,. g, and g,, are the respective coupling strengths of the adsorbate and reactant cores with the uth mode. to is the energy of the reactant in the oxidized form.
+
3. Evaluation of Current via Linear Response Formalism Our next task is to evaluate the microiscopic current associated with the electrochemical reaction of the reactant R. The quantity which one needs to evaluate for this purpose is the average value of the rate of change of electronic occupancy of the reacting orbital r, Le., (dn,/dt). The microscopic current is related to this quantity through the e x p r e s s i ~ n ~ ~ . ~ ~
I = -e(
2)
(3)
In general, it is difficult to evaluate (&,/at) completely. But invoking the condition that the coupling term H I = C[V,&,,+c,
+ Vrn~,+~,,I = VI + Vi’
(4)
U
which provides the mechanism for the transitions involving the reactant’s electron is weak, we can bring into the formalism the notion of perturbation. Thus, the use of linear response formalism leads to the following expression for for the evaluation of (ri,)35,39 the current (5)
where the time evolution of the operator VI and the expectation value corresponds to the unperturbed system, characterized by the Hamiltonian Ho = H - HI. We note that the expression 5, which is quadratic in V,, does not envisage any constraint on the magnitude of the remaining parameters appearing in Ho. Since the ri, contains the information both for the forward (anodic) and backward (cathodic) exchange processes, the two terms appearing inside the commutator in ( 5 ) correspond to the anodic and cathodic currents. As these two are related to each other via the detailed balancing condition, it is sufficient to evaluate only one type of, say e.g. anodic, current expression. We write the anodic current contribution as
The above expression for the current can be easily evaluated (39) Duke, C. B. Tunnelling in Solids; Solid State Physics, Supplement 10; Academic: New York, 1969; p 207.
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987 3419
Theory of Electron-Transfer Processes if (i) the time correlation function involving the two different spin states of the adsorbate is assumed to be zero and, (ii) we treat the solvent modes in the classical oscillator limit. Next, as done by Schmickler,j5 we use the Franck-Condon approximation to separate the averages over the electronic and solvent subsystems, and first evaluate the fermionic averages considering the coordinate variables characterizing the solvent as the fixed parameters. The anodic current expression now becomes
I, =
where
where the subscript F denotes the ferminoic average (with fixed qY)and the subscript B denotes the average over the solvent system. Since the state r now evolves as a free particle, we can directly write cr(t))F
(C?(o)
= eXP[
1
it
+ ZhwYg,,q,) (n,)
-;(ef
(8)
where
g,, = 2112g,,, i = (a,r,c,O}
(9)
which follows from giu(bv + bu+) = 211zgiuqu
(10)
1 --J eicflh [ I - f ( t ) l a
(11)
Next, we write (c,,(o)
c,,+(f))F
=
Im G:;U(~) dt
-m
The C,Rd“(c)can be easily evaluated under HF and we write it as w(z) = Y
where 7,
=
t,
+ U(n,,)+ A
(13)
and
An equivalent integral representation for the Im C;:(t) as -1m G:;“(e)
= j/zJmexp[i[t - (7, -m
is given
CR
€1 -
(27)
‘/zChw,gr,Z - Chw,gcYgl,
(28)
Y
where Z = 0 corresponds to the oxidized state, I = R denotes the reduced state, -I/2CVhwgI: is the corresponding solvation energy and -~Yhw,gc,g,uis the solvent-mediated interaction between the adsorbate core and Ith state of the reactant, can rewrite the parameter D as
D = exp [-@(e
Z” = E,” + go” + E,” (18) Changing the order of integration in ( 1 6 ) from ( t , qy, e, T ) to (e, T , t , q y ) and evaluating the various integrals, we obtain the following expression for the anodic current
(26)
€0
where gRy is the coupling constant for the reactant vth polarization mode interaction when the former is in the reduced form. With the help of these transformations, E, can be identified with the solvent reorganization term l/zCuh~,(gRv - go,)’ corresponding to a direct heterogeneous electron-transfer reaction. Also, introducing the free energies of the reactant in the oxidized and the reduced states, viz. Y
and
-
= gRu - go,
gru
FI =
where
(25)
Similarly, g,,,,the coupling coefficient between the reactant orbital and uth solvent mode, can be approximated as
Y
(15)
erfc(-iz)
Within the physical limitations of the model Hamiltonian (1) and the linear response formalism for V,,, expression 19 for the anodic current is exact when the boson fields are treated in a classical manner. The physical meaning of various terms appearing in eq 19 can be better understood in terms of various parameters appearing in the conventional electron-transfer theories. For this purpose, we approximate e, as the difference in the energies of isolated redox couple (reactant) in the reduced and oxidized form; i.e.
+ C h w , g , , q , ) l ~ l- AI71 d r
Substituting ( 8 ) , ( 1 l ) , and ( 1 5 ) in ( 7 ) , writing down the solvent average appearing in (7) explicitly, and summing over the spin variable u, we get44
e-z2
+F g R
+ EAZ
]
(29)
Next, P , which varies linearly with the temperature, depends only on the coupling coefficients corresponding to the adsorbate and reactant orbitals-solvent modes interactions. More specifically, P is a function of solvation energetics associated with the adsorbate and reactant electrons and the solvent-mediated, indirect interaction between these two electrons.
3420 The Journal of Physical Chemistry, Vol. 91, No. 12, 1987
The I,(€) appearing in (1 9a) can be considered as the partial current associated with the transition of the electron from the reactant to an electronic level with energy t. In the next section, we will see that Q can be expressed in terms of the saddle point coordinates q,S of the reaction hyperspace for the above transition process.
4. Limiting Cases and Discussion An insight into the nature of the current expression (19, 19a) corresponding to the adsorbate-mediated electron transfer can be gained by comparing I,(€) with the current associated with the direct electronic transition in the absence of adsorption from the reactant to a state with energy t in the electrode. The current for a such a transition
where Vis the reactant-electrode coupling constant, p(t) is the electronic density of states in electrode
Mishra and Rangarajan
In ( 3 3 , the coordinate t’ of the saddle point is obtained by the relation (37) The problem of locating the saddle point through the above relation has been considered in detail in the existing literature31@and we may refer to them for further details. After considering the general expression for the anodic current, we now concentrate on the various limiting forms of the expression 19. 4.1. Direct Electron Transfer to Electrode. As such, a direct coupling between the reactant and the electrode has not been provided in the Hamiltonian (1). But this case can be realized by reinterpreting the adsorbate orbital u as one of the metal states k’, and subsequently recognizing V, as Vvr. Since the metal states are not coupled among themselves, we can put V,k, Le., Vpk equal to zero in the new scheme, which in turn implies that A = 0. Further, as the coupling between metal states and the solvent = E,,, = 0. These transmodes is negligible, we have E,, = formations lead to &tu
P=O;
Q=t-tk,
(38)
from (19), (29), (32), (38), and the relation4’ F /’ = F /(gc”-o) - -
(32)
From (19a) and (32), we observe that the mediation due to the adsorbate effects the activation parameter for the direct transition viz., -l/p In D’by changing the free energies of the oxidized and reduced states of the reactant (cf. eq 28 and 32). Apart from this, the partial current I,(€) differs from the Za’(t) by a multiplicative factor (33) which depends in a rather complicated way on temperature (via P, of eq 22) and overpotential (via t, and Q,cf. eq 2 and 24). The temperature dependence of the multiplicative factor will again lead to a change in the activation parameter corresponding to t transition. Since this change is not linear in p. the direct t, activation energy will now apparently depend on temperature. Next, since Re w ( z ) , which incorporates the modifications due to the overpotential, is a slowly varying function of its argument, the additional effect of the overpotential will be rather weak. Further, using the saddle point method to evaluate the integration appearing in (19), we can show that in general the above comparison between the partial current Z,(t) and I*’(€) is also valid for the adsorbate-mediated total anodic current and the anodic current corresponding to the direct transition. We rewrite ( 1 9) as
-
We obtain (after deleting the spin degeneracy effect)
-
which is the microscopic anodic current for the direct e, tk8 tran~ition.~~,~~ 4.2. Homogeneous Electron-Transfer Rate. To arrive at the usual homogeneous electron-transfer rate in the nonadiabatic limit,30we now consider a “one-electron-two-level ( a , $ ‘ system with initially the electron being localized at the level r. In such a case, the probability (1 - f ( c ) ) that the adsorbate orbital a is vacant (cf. eq 11, 19) becomes unity. Next, since the present two-level system is not coupled to any other electronic state, we put A = 0. Also for simplicity, we take E,, = E,, = 0. Now using t, transition can be these limits the rate constants for the t, written from (19) as
-
Wra
where
-
[Note: 1 / 2 in (33) is introduced to remove the spin degeneracy effect. We have also taken IV12 IVar12for simplicity.] In eq 34 g(c) = In [ ( l - f ( t ) ) D ] and Re w ( z ) , as already mentioned, is a slowly varying function of t . Now, the saddle-point analysis leads to I* =
Evaluation of the integration in (41) leads to the homogeneous electron-transfer rateg0
-P(tlr/
-
E,‘
4 E,‘
+ E;)2
]
(43)
(35) with (40) Dogonadze, R.R. In Reactions of Molecules at Electrodes; Hush, N. S . , Ed.;Wiley-Interscience: New York, 1971; p 135.
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987
Theory of Electron-Transfer Processes
4.3. Strong Adsorbatesubstrate Interaction. Obviously, the strong adsorbate-substrate interaction is characterized by a large A, and hence, by a large z. The current for this case is obtained by considering the asymptotic expansion of Re w(z) in (19). Thus, we have
This result has also been obtained by Schmickler employing the saddle-point method under the condition PA >> 1 .35 Clearly, the proper validity condition for expression 46 is that the scaled parameter A/2P1f2is much greater than unity. As pointed out earlier, the Hamiltonian used by Schmickler contains questionable triple products of fermion operator^^^.^' and thus violates the particle number conservation for the fermions. Even so, the above correct limit has been obtained by Schmickler only by deleting certain expectation values in the final expression. To establish the formal identity between (46) and Schmickler's result, we substitute D from eq 29, multiply the right-hand side of (46) by the number of adsorbate-reactant pair and rewrite Q as
Q = t - t, - ChwAySg,,
(47)
Y
where qyS, the saddle point of the reaction hyperspace is
comparing ( 5 2 ) with (46), we observe that in case the solvent is represented by a single polarization mode, the form of the current, irrespective of the magnitude of the adsorbate-substrate interaction strength, is identical with the current expression when the solvent is modelled in terms of multipolarization modes and the coupling between the adsorbate and the electrode is strong. 4.6. Electron Transfer via Surface States. Schmickler has treated the problem of electron-transfer reaction wherein surface states act as intermediates, by employing the T-matrix approach, and has obtained the expression for the cathodic current.36 Our method is applicable to this apparently different problem by considering small A limit (cf. eq 49). Presently, we make no comparison with the Schmickler result, not so much because a different method is employed, but because the Hamiltonian in ref 36 is different and, in our opinion, incorrect in view of the unphysical HT term.41 Apart from this, it appears that while deriving expression 12 of ref 36 from expression 8 of ref 36, an erroneous limit 1 x lim -- *KY) (54) *+I+ y2 x2
+
has been used in place of the exact limit lim x-o+
(In arriving at (48),we have used the eq 26-28.) If in the resulting expression we substitute g, = 0 (as is the case with the Schmickler formalism), we recover the expression for the current given by S c h m i ~ k l e r . ~[The ~ current expression (in the present limit) obtained by using the transition term involving the triple fermion operator product is given by eq 15 of ref 3 5 . In the subsequent analysis therein, the (1 - ( n r x ) )appearing in the above mentioned equation has been approximated to unity, which is virtually equivalent to describing the transition term by a bilinear operator (fermion) product.] From (30) and (46), it is also obvious that if we employ the saddle-pint analysis (cf. eq 19,34) in the present case, the resulting activation energy is of a form identical with the activation energy coresponding to a direct electron-transfer process between reactant and the electrode. 4.4. Current Corresponding to Weak Adsorbate-Substrate Interaction. The anodic current for this case is obtained by expanding Re w(z) in the powers of A in the expression 19. The current linear in A is thus written is
(E (
X - nm)
y2
+ x2
(55)
4.7. Effect of the Intraadsorbate Correlation on the Current. While evaluating the current expression 19, we have treated the intraadsorbate repulsion term within the Hartree-Fock limit. Using the formalism developed elsewhere by us:* we can evaluate the effect of this correlation on the current characteristics in a straightforward manner. The Hamiltonian including this correlation is obtained by replacing COU(n,,)n,, by Unauna,in expression 1. Next, we have to evaluate the correlation function (c,,(O) C,,+(t))F so that the current can be obtained from the relation 7 . As the knowledge of the adsorbate Green's function Gt;"(t) is required for the determination of (c,,(O) c , , + ( t ) ) ~we , need to know the C$'(t) beyond the Hartree-Fock limit for studying the correlation effects on current.42 It has been shown that when the two-body operator Unauna,is present in the Hamiltonian, the system evolves in a higher order operator product space, and truncation of this space or the associated basis set h is required for obtaining the closed form expressions for various GFs. Presently, we approximate the basis set h as42 Ih) =
Re-w-
3421
[IC,,),
I~"B)? IC,,,),
I ~ u ) l
(56)
The adsorbate G F is then given as 2;/2)
--(";)1/2)
de ( 4 9 )
4.5. Single Polarization Mode Model for the Solvent. Interestingly, when the solvent fluctuations are modelled in terms of a single boson mode, the parameter P vanishes identically. Further, Q and E, now become (here, we drop the polarization index Y and write w, as coo)
where (cf. eq 1 3 )
From ( 5 7 ) , we have - Im Gr;e(c) =
With these limits, the current in the present case is written as
which is a sum of two Lorentzians when A is energy-independent. Replacing the terms on the right-hand side of (59) by their integral ~~~~
where
(41) See eq 5, ref 36, for HT.As a consequence of this HT, it can be. shown that the expression reported in eq A3 of ref 36 for the expectation value of the second-order term in the T-matrix expansion is not correct in the strict mathematical sense. For example, in the said expression, even the term proportional to z-' vanishes if the expectation value is evaluated properly. (42) Mishra, A. K.; Rangarajan, S. K. J . Ektroanal. Chew., in press.
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987
3422
Mishra and Rangarajan
representation (cf. eq 15) and then carrying out the analysis exactly as prescribed in section 3, we obtain the modified expression for the anodic current as =
IAM
(nap)zAl
+ (l -
(60)
(nar))1,42
where IAl
=
-
IA(S,
-+
Z,-O( n a a ) )
where
I,, = I ~ ( z , z,-U(naa)+@ (61) and I, is given by expression 19. Since the further analysis of the modified current expression can be carried out like that of I A , we are not considering these details here. We finally mention that ( n a g ) (, n a p )appearing in (19) and (60) can be replaced, as a first approximation, by the values obtained in the absence of reactant.
5. Electrochemical Current in the Presence of an Inactive Neighboring Adsorbate We consider in the present section the rate of an electrochemical reaction, mediated by an adsorbate when the adjacent site on the electrode is occupied by an identical, inactive adspecies. For simplicity, we model the solvent classically in terms of a single polarization mode.29 The Hamiltonian for the system is written as C [ E a + O(ni,7)]nj,+
H = Ctknh L,"
1.C
+ hwogrqnr + ~
Cgiqnig
U O 1.u
h@O
+ - (2p 2
+
+
(62)
U O ( ~ Cgci)q O
-
I
+ e($,,,
-
$d)
+ cim +
z, -
C(n,.,)
z, -
R [
a
C(na,) 0
( R 2 + 4d2)'I2
]
(63)
~~
t
-
( T - ir), ( E , + hw&q - iA)
where
E, = E ,
+ gr
(71)
+ U(n,,) + A
(72) Changing the order of integration in (69) from (t,q,C) to (C,q,t) and thereafter first evaluating the t integration, we obtain
i'#i
+ hwog, q ) l ( n r )
L
- ir)2
( E , + hw&bq - iA) (T
E -
]-Is( 7+
(73)
Next, the integration over the variable q in the above expression can be carried over in a straightfoward manner by using the property of the 6 function. The anodic current expression now becomes
(64)
and (c,,(O) c,,+(t))F is to be obtained from the relation 1 1 where the retarded G F for the adsorbate a is now given as43(Note: (nan) = (rzb,,) for identical adsorbates) ~
hwog,q - iA) -
= gac + g b c + g o
The term inside the square bracket in (63) denotes the electrostatic interaction energy between the electron localized on the ith adsorbate and the net charge (including the image charges) associated with the other adspecies. R is the lateral distance between the two adspecies and d is the distance between the electrode and the adsorbates. z, is the adspecies core charge, coo, p , and q are the frequency, momentum, and coordinate variables, respectively, for the solvent polarization mode. gi and gci are the coupling coefficients corresponding to the interaction of electronic and core charge of the ith adsorbate with the solvent polarization mode. The remaining symbols appearing in (62) have their usual meaning (cf. section 2 and eq 1). Using the linear response formalism with respect to the weak coupling term between the depolarizer asnd the adsorbate a, it can easily be seen that the anodic current expression in the present ~ in (7) case is again given by eq 7. The (c,+(O) ~ , ( f ) )appearing is now given as (cf. eq 8) (r,+(O) c , ( ~ ) ) F= exp[-it/h(t,
(68)
Substituting (64), (1 l), (65), and (68) in (7), writing down the classical average over the solvent polarization mode explicitly, and multiplying the resulting expression by a factor 2 for accounting the spin summation, we have
I
The index i (=(a&]) in (62) labels the two adsorbates a and b. The depolarizer is electronically coupled to the adsorbate a only. E,, the orbital energy for both the adsorbates, is given as (cf. eq 2) E, =
as the ener-
C [vikci~+Cko+
a
+~
&,a
k.i.0
Vkicka+cinj + ernr + C I V a r c u o ++~ , V,,c,+c,,] 4')
Following Grimley4, we approximate E,, and gy-independent quantities, Le., Z,, = A - iA; z a b = 7 - i r
where
~~
(43) Mishra, A. K.; Rangarajan, S. K., preceding paper in this issue. (44) (a) In expression 16, summation over the spin variable is accounted for by multiplying the current expression by 2. Clearly, this corresponds to the nonmagnetic limit wherein (no,,)= (naB).In the case of magnetic solution,
we may separately coinsider the terms corresponding to u and 8 in a similar fashion. (b) The total current is obtained by multiplying the microscopic current by the adsorbate coverage factor % and the reactant's concentration at the OHP.
L
1-1
Theory of Electron-Transfer Processes
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987 3423
is the density of state of the adsorbate a when the solvent coordinate is fixed by the constraint imposed by the 6 function in expression 73.
F,,’ =
€0
- )‘’hug? - hw&&,
+ gbC)
(76)
F,’ and FR1are the free energies of depolarizer in the oxidized and the reduced forms. E,’ is the reorganization energy when the solvent is modelled in terms of single polarization mode. In arriving at 74, we have used the relation 26 and the single polarization mode limit of the expression 27. The current expression in the present case with a multiple polarization branch description of the solvent can also be obtained in the limit of strong adsorbate-electrode interaction by using the saddle-point method. The general expression for the anodic current in the present situation is still given by the expression 69 provided we make therein the following transformations in order to account for the multiplicity of the polarization modes
where
In the resulting expression, by first changing the order of integration from ( t , (qy],e) to (e, t , (q,,])and thereafter applying the saddle-point method twice-once for evaluating {qY]integrations and then t integration-we obtain the following expression for the anodic current
6. Effect of Mediating Adsorbate Polarizability and Uniformly Distributed Inactive Adsorbates on the Current Characteristics In the present section, we consider the effects of identical, inactive adsorbates on the reactive, Le., mediating central adspecies, and hence, on the electrochemical current characteristics. As pointed out in the Introduction, the adsorbates are assumed to form a square lattice and the inactive adsorbates interact with the mediating adsorbate only via the electrostatic forces. For definiteness, we consider here the monovalent adspecies, though the formalism developed here is equally valid for other types of adsorbates, of course within the limitation of the model considered here. We also model here the effect of the induced electronic polarizability of the mediating adsorbate on the reaction rate. As the induced electronic polarizability is associated with the intraorbital charge redistribution in the adsorbate, more than one orbital of the central adsorbate needs to be considered. For this purpose, we consider here the Ins), Inp,) orbitals and the associated spin degeneracies. Considering the orientation states of Inp,) and Inp,) orbitals, it can be presumed that their coupling with the electrode is weak, and hence, these states are not considered in the present model. The hybridization between InpZ) and Ins) orbitals is caused both due to the field produced by the other adsorbates and their images, and the external electrical field. We consider here a large concentration of electrolyte, and therefore these fields vanish beyond the outer Helmholtz plane region. The depolarizer is weakly coupled to both Ins) and Inp,) orbitals. As in the earlier cases, we develop a linear response formalism around these interactions for arriving at the current expression. We choose the origin of the coordinate system at the location of the “mediating” adsorbate. The position vectors of the other and their images lie at B1/2(l,m,-2d01/z) adsorbates are 6’/*(l,m,o), where 0 is the surface coverage, 1 and m are integers, and d is the distance between the adsorbates and the electrode. We let n be the total electronic charge, z , the core charge, and ~c the induced (electronic) dipole moment of the adsorbates. Then the electric field due to the other adsorbates at the origin and in z direction perpendicular to the electrode surface is given as3’
2ed(n - zc) 03/2U3,2(4d20)(89) €1
where p,’(e) is the adsorbate density of state corresponding to the saddle-point coordinates of the solvent polarization modes and is given as &’(e)
= --
e
K
where e, is the dielectric constant in the inner layer region, and m
U,(x) =
C
m
m=-m
(I2 + mz
+ x)-,
(90)
The summations in the above expression simultaneously exclude I=O,m=O. The potential at the origin due to the other adsorbates is given as37
- (E, + hXwYg,qvs- iA) V
f#J=
qlR,the coordinates of the saddle point on the reaction hyperspace, is
(85) The free energy of the depolarizer in the reduced and oxidized forms are
FR”= FO”
=
CR
€0
-% -
C ~ W Y ~ R-> ~ h w , g ~ , ( ~ , gbev) ,, Y
(86)
Y
1/2ChwvgO> - x h w u g O v ( g ~ c y + gbcu) Y
(87)
Y
respectively. The reorganization energy E, is given by = 1/2ChWv(gRv- g o y ) * Y
(88)
The model Hamiltonian describing the electrode, mediating adsorbate, depolarizer, u classical polarization modes of the solvent, and the relevant interactions is written as (cf. ref 37 also) H=
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987
3424
e, and tp are the s and pz orbital energies of the adsorbate and they include (i) the intraadsorbate repulsion effect in the Hartree-Fock limit, (ii) image effects and, (iii) external field effect. We write them as t,
= 4 -I
+ e($,
- $,)
+ elm + Ui(ni,), i = (s,p)
(93)
4 describes the shift in the orbital energies of the mediating adsorbate due to the field generated by other inactive adspecies. s and pz orbitals coupling coefficients with the electrode are given by v k s , Vkp Wis the coupling coefficient for the s and pz orbitals hybridization and is given as [cf. ref 371
w = eE,(sIzlpz) + (SI~extlPz)
(94)
The first term on the right-hand side of (94) is the coupling coefficient due to the field of other adsorbates and the second term is the coupling parameter associated with the external field. V,,, Vpr in (92) are the coupling coefficients corresponding to the mediated adsorbate-depolarizer electronic overlap. g,,, gpyare the adsorbate orbitals: vth solvent polarization mode coupling coefficients. cI,+, cia,and nirr(i = (s,p)) are the creation, annihilation, and occupation number operators, respectively, for the ith orbital of the adsorbate. The remaining parameters and operators in (92) have their usual meaning. Finally, the total electronic charge n and the induced dipole moment p (of the adsorbates) in eq 8 and 91 are given as37
n =
E@,,+ npu)
Mishra and Rangarajan coordinates, are obtained by using the Green's function approach. The expression linking the operator average ( B ( 0 ) and the G F G:A(t) is
(B(O) A+(t))F = - ~ J ~ e i f r ("I
vk, = Vkp E Vka,
gsv
=
gpv 1
gam
The superoperator
=
Vpr
(97)
Var
fi, in (104) is defined as
fiJ = [ X , H,]-; for any operator X , H , = H
- H,'
(105)
The bilinear product in (109) is defined as
(4Y)=
([*,
yl+)
(1 06)
where the operators X and Y have Fermi characteristics. From the expressions 102 and 103, it is obvious that, for determining the anodic current, we have to evaluate the GFs, C;,' GF;", G$', and GZ". For this purpose, we presently employ the equation of motion approach and obtain t' E iA (107) = - E2 - p) 2iA( W E')
(95)
vsr
(103)
The retarded G F G;*(e) is defined as
U
The various creation-annihilation operator product averages in (95) and (96) can be approximated (as a first approximation) by their respective values obtained in the absence of the depolarizer. The formalism for the current evaluation can be considerably simplified if we assume that the coupling coefficient v k , equals to V,,, gsyequals to gpY,and V,, equals to Vpr,i.e.,37
-A€)) Im @A(€) d t
q6"
+ + +
+
Gs"" =
W - iA p) 2iA( W
+
(E'*
- E2 -
+
6')
q;"= Gs"" e' - E + iA GE?U = (er2 - E2 - p)+ 2iA( W + e')
(108) (109) (1 10)
where t'=t-Ei-+
(111)
From eq 8, 102, 103, and 107-1 10, the anodic current expression can be rewritten as
Substituting (97) in (92), we obtain after some rearrangements
H = Ceknku + C ( E i
+ 4)(nsu+ npu) + CE(npu- nsu) +
k-0
0
C [ vka(Cks+Csu + Cku'Cpu) + HCI + CW(Csn+Cpu + HC) + k.o c ern, + C [ var(Cpu+Cr + Csa+Cr) + HCI + C h w u g r u n r + Y
0
C h 4 L + E,u)q, + XEhWY(PU2 + SU2) (98) Y
where L
(99)
Y
E = (ep - ts)/2
(1 00)
After describing the model Hamiltonian, we evaluate the anodic current in the present context by employing the method prescribed in section 3. Using the linear response formalism with respect to the weak depolarizer-adsorbate coupling term
H,' = C [ V a r ( c p u + c r + csu+cr) +
Vra(Cr+Cpu
r
+ cr+csu)I
(101)
we can write the anodic current expression as (cf. eq 5-7) IA
=
'jJ 2m l v a-_ rlZ(l(csu(o)
+
Cs,+(t))F +
( C p ~ o )Cp,+(t))F
+
C ~ U +)F)(cr+(o) (~) Cr(t))F)B(classlcal)dt (102) (c,+(O) c,(t))F in (102) is already given by expression 8. The remaining expectation values in (102), which involve the creation and annihilation operators pertaining to the adsorbate orbitals, and which are to be evaluated for the fixed value of the solvent ( C p v ( 0 ) csu+(t) )F
(Csu(0)
(112) Writing the solvent average explicitly in (1 12) and after some manipulations, we have
J. Phys. Chem. 1987,91, 3425-3430
+ 2a2] - [a2+ (6 - A)2][6A - 2aW] 4r(a2 + A2) [a2 + (6 + A)2][6A - 2 a w - W[6A - 2a2]
where w ( z ) = exp(-z2) erfc (-iz); as in eq 25
W[6A
d, = d2 =
(115) zj =
-Qj+ iyj
(1 16)
+ A2) - A2 f 2iAWj’/2 a f i6 = [ E 2 + r = (a2 + b2)1/2 tp + = -+ + + a, tb2 = -+ + - a 2 2 4r(a2
2P’/2
(1 17) (118)
ES
6bl
3425
(119)
Y,= 16 - AI, Y2 = 16 + AI
(120)
Employing the integral representation
Y:
E,, D, and P are given by expressions 88, 29, and 22, respectively. Thus, presently we have derived the current expression in terms of various system parameters such as coverage factor 6, induced dipole moment of the adsorbate p , reorganization energy E,, various orbital energy levels and coupling parameters, etc. From (123), it is clear that the functional dependence of the current on these parameter is rather involved, though the form of the current expression is similar to the one obtained for the loneadsorbate, single orbital model (cf. eq 19). Therefore, the procedures discussed in section 4 can be applied to eq 124 for arriving at the various limiting cases. 7. Summary and Conclusions
in (1 13), interchanging the order of integration from (t, n,q,, e, 7) to (e, T, t, r Y q Y )and , thereafter evaluating the ayqUr t, and T integrations, we obtain the following analytical expression for the anodic current.
d,
+ c,a Re w ( z l ) + d2 - c2a Re w ( z 2 )- c, Im w ( z I )YZ
1
(1) Ignoring direct coupling between reactant and substrate, we consider the electron transfer only through the reactant coupling to the chemisorbate. We solve for the current as a function of various energy levels involved in the model, as well as the several coupling coefficients. This brings to the fore the appropriate scaling of energies/temperature in this problem. (2) The results of Marcus, both the homogeneous and heterogeneous versions, besides Schmickler’s are deduced as limiting cases, thereby defining the range of validity of the earlier results. (3) Our model Hamiltonian is different from that used by Schmickler for the same problem and we provide a critical discussion of the latter’s analysis; (4) The two-adsorbate results in chemisorption are transposed in this context and the dependence of the current on the neighboring adsorbates is studied. ( 5 ) Even though most of the results given here employ the HF expressions for adsorbate GFs for the sake of simplicity, the formalism is general enough to accommodate the correlation effects.
Electron Transfer through Film-Covered Surfaces. Case of Monolayer/Submonolayer-A Coherent Potential Approximation Formalism A. K. Mishra and S. K. Rangarajan* Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560 012, India (Received: July 17, 1986) The problem of electron transfer mediated by the chemisorbed layer with random occupancy on a two-dimensional lattice is taken up. The analysis employs linear response formalism and coherent potential approximation for evaluating the current. Two cases referring to (i) the delocalized land (ii) the localized electronic states in the chemisorbed layer are treated.
1. Introduction In an earlier report,l the problem of electron transfer via the adsorbed intermediates has been considered. Therein, the model system contained only a few (one or two) adspecies. In the present paper, we extend the formalism to a case wherein an aggregate of adspecies forms a submonolayer or monolayer film on the electrode. In general, the problem here is twofold, viz., (i) characterization of adsorbate layer through appropriate parameters (1) Mishra, A.
K.; Rangarajan, S. K., preceding article in this issue.
and (ii) investigation of the functional dependence of the electrochemical rate on these parameters. The first step concerns the equilibrium state of the adlayer whereas the electron dynamics involving the depolarizer and the adlayer states need to be considered for obtaining the required functional dependence. Herein, we develop a formalism at the microscopic level for describing the effect of a chemisorbed submonolayer or monolayer on the electrochemical reaction rate. The case when the electrons in the adlaver are delocalized and the case of the localized electronic states in the adsorbate’s film are considered separately. As considered earlier,, the direct coupling between the depolarizer
0022-3654/~7/2091-3425$01.50/0 0 1987 American Chemical Society
