874
J . Phys. Chem. 1987, 91, 874-883
Adiabatkity of Electron Transfer at an Electrode John D. Morgan* and Peter G. Wolynes School of Chemical Sciences, University of Illinois, Urbana, Illinois 61801 (Received: July 2, 1986)
A simple description of the kinetics of electron transfer at an electrode is developed which includes the effects of both outer-sphere solvent dielectric relaxation and electron tunneling. From this description,an adiabaticity criterion is obtained which determines when the rate of the electron transfer is adiabatic, that is, determined by solvent relaxation dynamics, or diabatic, determined by electron tunneling dynamics. Rate constants are also obtained for the diabatic and adiabatic limiting cases and the intermediate regime in which the dynamics of both processes are important. The degree of localization of the electron transfer and the dependence of the rate on the electrode potential, the Miller indices of the electrode surface, the surface site of the reacting ion, and the distance of the ion from the electrode are examined. Possible extensions of the formalism to include other important factors are also discussed.
I. Introduction The rate of electron transfer from an ion in solution to an electrode depends on many different processes. Among the most important are the outer-sphere dielectric relaxation of the solvent and the tunneling of the electron from the ion through the solution to the electrode. A number of workers have studied the effect of outer-sphere relaxation’ and electron tunneling2 under the assumption that only that one process is responsible for the rate. The present work examines the question of the adiabaticity of the electron-transfer process, that is, the relative time scale of the outer-sphere relaxation and electron tunneling and thus their relative importance in determining the overall rate constant. A similar treatment for homogeneous electron transfer in the high activation energy limit was done by Zusman3 and the question of adiabaticity of electron transfer in heme proteins was addressed by Frauenfelder and W o l y n e ~ . These ~ ideas have been extended to the quantum region by Wolymxs A new feature of our analysis is the continuum of energy levels available in the electrode. In this paper, we will derive formulas for the rate constant in the high normal activation energy, activationless, and high abnormal activation energy limits for an effective Smoluchowski equation description. These appear in eq 23a-c along with their diabatic and adiabatic limits. The degree of localization of the electron transfer to the electrode and the dependence of the rate and adiabaticity on the electrode potential, the Miller indices of the electrode surface, the surface site of the reacting ion, and the distance of the ion from the electrode will also be examined. This analysis establishes in a pleasant way a rough relationship between tunneling matrix elements for the homogeneous and heterogeneous problem. Finally, we will discuss the possible extensions of this formalism to include other important factors which affect the rate and adiabaticity of the electrode reaction. The adiabaticity can be analyzed experimentally by combining information on the dependence of the electron-transfer rate constant on solvent relaxation timez3and the distance of an ion covalently atttached to an e1ectrode.ls In the adiabatic limit, the preexponential factor in the rate constant will be inversely proportional to the solvent relaxation time but will be independent of tunneling dynamics and thus independent of the distance of the ion from the electrode. In the diabatic limit, the preexponential factor will decrease exponentially with the distance of the ion from the electrode but will be independent of solvent relaxation time. In the intermediate regime, the rate constant will depend on both (1) For a review of the effect of outer-sphere dielectric relaxation on electrode kinetics, see Levich, V. G. Physical Chemistry; an Aduanced Treatise, Vol. 9B,Eyring, H., Henderson, D., Jost, W., Ed.; Academic: New York, 1970; Chapter 12. (2) Ulstrup, J. Charge Transfer Processes in Condensed Media; Springer-Verlag: Berlin, 1979. Khan, S. U. M.; Wright, P.; Bockris, J. O’M. Sou Electrochem. (Engl. Transl.) 1977, 13, 774. (3) Zusman, L. D. Chem. Phys. 1980, 49, 295. (4) Frauenfelder, H.; Wolynes, P. G. Science 1985, 228, 337. (5) Wolynes, P. G. J . Chem. Phys., in press.
processes. Since the adiabaticity is a strong function of electrode potential (see eq 18-22 and Figure 2), it may be possible to find an electrode system which changes from nearly adiabatic transfer in one potential range to nearly diabatic transfer in another range. 11. Effective Smoluchowski Equation The state of a system consisting of one ion in solution at an electrode is assumed to be described by the probability densities p,(x,t) where the subscript refers to the electronic state and x is a collective solvent variable described below. The free energy of polarization of a dielectric continuum with static dielectric constant cs and optical dielectric constant cop in the presence of an electric displacement D is assumed to be given by the Marcus free energy functionaL6
F[P,,Dl =
P, = the low-frequency polarization
(1b)
P(R) = total solvent polarization at the point R
(IC)
E(R) = total electric field at R
(14
Our notation differs from that of Marcus6 in that the two additional fields P and E in his treatment are expressed in terms of D (E, in his notation) and P,,. As in his treatment, the optical polarization P, = P - P, is assumed to be always in equilibrium with the total electric field with an optical dielectric constant cop. In the electrode problem, the collective solvent variable x , defined by
AD(R) = D,(R) - D,(R)
(2b)
is the “arc length” along the steepest descent path (with respect to F) in the function space of P, connecting the P, in equilibrium with DI (the electric displacement with the electron on the ion) to that in equilibrium with De (with the electron in the electrode.) This reduction of the infinite-dimensional description in terms of the field P,(R) to the one-dimensional description along the steepest descent path (with a different scale factor for x) was done for the case of cop = 1 (and thus P, = P) by Calef and Wolynes.’ ( 6 ) Marcus, R. A. J . Chem. Phys. 1956, 24, 966. J . Chem. Phys. 1956, 24, 979.
0022-3654/87/2091-0874$01.50/00 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 4, 1987 815
Adiabaticity of Electron Transfer at an Electrode
~ X Nw I / / /
Pa,,= transition probability from electronic state a‘ to electronic state a (3g) T~
= longitudinal dielectric relaxation time of solvent
(3h)
It should be noted that the choice of scale factor for x in eq 2a makes u2 dependent only on solvent properties, independent of the ion or electrode properties. The total probability density p I ( x , t ) for the transferred electron to be on the ion with the solvent coordinate x is simply the sum over the densities p,,(x,t) for all aI. Similarly, the total probability density pe is the sum of the pa,. The rate of change in pI is thus given by P ~ X J = )
Ci),,(x,t) =
C
c OORDI NAT E Figure 1. Intersection of continua of U,,and Uoe diabatic potential curves. R E A C T ION
This is also proportional to the reaction coordinate E in the treatment of homogeneous electron transfer by Z ~ s m a n . ~ The time evolution of p,(x,t) is assumed to obey an equation containing master equation terms for transfer between electronic states and a Smoluchowski equation term for the classical diffusional dynamics on the effective harmonic diabatic potential V&). A more precise treatment of the electronic dynamics would use the quantum Liouville equation, describing the time solution of both the diagonal density matrix elements pa and the off-diagonal elements pma,. Under the assumption that the time scale of change in the latter is fast compared with the change in the diagonal elements, this descriptior, reduces to the electronic master eq~ation.~ The electronic states a can be grouped into the states aI with the transferred electron on the ion and a, on the electrode. The different aIand a, states are distinguished by excitations of the other electrons in the electrode conduction band. The lowest energy state in each group corresponds to all the states below the Fermi energy being filled and all those above the Fermi energy being empty. The system can thus be represented as the intersection of a continuum of ion states with a continuum of electrode states as in Figure 1. The master-Smoluchowski equations for the pa are thus
= C
i)al(~,t)
( P a , a < ~ a
0, independent of x.: The diabatic rate approaches a constant equal to twice the activationless rate for transfer to the electrode in the high abnormal activation energy limit, while the rate of homogeneous electron transfer decreases again to zero in the same way as for normal activation energy. Again, this results from the presence of electrode states higher in energy than the Fermi level which will accept electrons with a lower activation energy than the transfer to the Fermi energy, so the “turnaround” behavior seen in homogeneous electron transfer in the abnormal activation energy regime does not occur in the electrode case.
V. Sink Density for Electron Transfer to an Electrode The sink density So defined in (1 8c) is related to the corresponding quantity SoHin homogeneous electron transfer. The matrix element Ak is the transition matrix element from the ion to the band state Ik). If Ik) is decomposed in terms of Wannier states Ir) at the lattice sites r,I2 So becomes
where N is the number of lattice sites in the electrode, and G,+(E) defined by G,(E) = lim (rl(E - i6E - k)-llr’) 6E-0
(21)
is the lattice Green function between lattice sites r and r’, where He is the Hamiltonian of the electrode. The matrix element A, gives the transition amplitude from the ion to the Wannier function of site r and may be expected to resemble the transition amplitude A between two ions in homogeneous electron transfer. The latter has been found to be reasonably well represented by13 A
A. exp(-nr) A. = 2.1 eV
(28a) (28b)
= 0.72 A-1 (28c) for ions at a distance r apart in solution. Although Wannier functions are not expected to be very similar to atomic orbitals of solvated ions, the transition matrix element for long-distance electron transfer depends primarily on the height of the barrier (in this case, roughly half the LUMO-HOMO gap of the intervening ~olvent’~) that the electron tunnels through and is rather K
(12) Economou, E. N. Green’s Functions in Quantum Physics; SpringerVerlag: Berlin, 1983. (13) Hopfield, J. J. Proc. Nutl. Acad. Sci. U . S . A . 1974, 71, 3640.
insensitive to the form of the initial and final orbitals. The lattice Gree? function Gd(E) is the matrix element of the r_esolvent operator G = ( E - &)-I of the electrode Hamiltonian He between Wannier states (rl and Ir’), analytically continued from the lower half of the complex E plane. In terms of Hamiltonian eigenstates lk)l2
where P denotes the Cauchy principal value. Thus, the imaginary part of Gd(E) is a matrix element of the projection operator onto the eigenspace corresponding to E , which is characteristic of the expansion coefficients of the energy eigenstates Ik) in terms of Wannier functions. Since the Wannier states are not eigenstates of He,the total transition probability to the electrode comes from transition amplitudes to the separate Wannier states, coupled by Grr4E). For a perfect crystal with Born-von Karman Boundary conditions, G,,, is translationally invariant and thus depends only on the quantity Ar = r‘ - r. The diagonal element G , is simply rg(E), where g ( E ) is the density of states of the energy E. In homogeneous electron transfer, the factor ( 2 k B 5 2 ) - ’ takes the place of Grr, in the sink density. There are many possible ways to approximate Gd with different levels of sophistication. We will look at two simple models which in some sense represent two limiting cases, the tight binding model and the free electron model.I2 Although neither of these methods are particularly accurate in quantitatively predicting the details of the band structure, both models reproduce the shape of most of the Fermi surface reasonably well in the case of noble metals. Another possible method which will not be used here is to use experimentally determined Fermi surfacesI4 to numerically calculate Im Grr, from the expression in eq 26. The tight binding Green function for s-bands of an fcc lattice can be expressed analytically in terms of complete elliptic integrals of the first and second kind for arguments which are algebraic functions of the energy. Although there is no simple general analytical form for arbitrary lattice sites, these formulas can be generated for a particular lattice site by recursion relations over the lattice sites15 and analytical continuation with respect to energyI6 from above the top of the band (where G,+ is purely real) to the Fermi energy (where Im Gr+ is nonzero). For the free electron model, the lattice Green function has a simple closed form for all lattice sitesi2
for an fcc lattice with lattice constant a, where g(EF) is the density of states at the Fermi energy. The above discussion describes the calculation of the bulk Green function G‘>,However, the coefficients in the expansion of the band functions for atoms near the surface of an electrode will differ from those predicted by the Bloch theory, due to the breakdown of translational invariance at the surface. This means that the lattice Green function between Wannier states near the surface will differ from that in the bulk. One way of treating this is to use the method of images.” (14) Cracknell, A. P.; Wong, K . C. The Fermi Surface; Oxford University Press: London, 1973. Ashcroft, N. N.; Merman, N. D. Solid State Physics; Holt, Rhinehart and Winston: Philadelphia, 1976. (15) Inoue, M. J . Math. Phys. 1974, 15, 704. Morita, T. J . Phys. A . 1975, 8, 478. ( 16) Morita, T.; Horiguchi, T. J . Math. Phys. 1971, 12, 986.
880
The Journal of Physical Chemistry, Vol. 91, No. 4, 1987
The surface Green function G$ is required to vanish for lattice sites on a nodal surface corresponding to the closest layer of atoms which would be added if the lattice was extended past the true surface of the electrode. This is accomplished by subtracting from the bulk Green function G$ an image term corresponding to reflecting the lattice site r’ across the nodal surface to the (nonexistent) lattice site rR’.
Morgan and Wolynes 7
’
,,U’
6 -
-5
-
10
-
u14
1 up3 I
The resulting surface Green function G$ will no longer be translationally invariant but will reduce to the bulk Green function G$ for lattice sites sufficiently far from the surface. Clearly, this can only be done for surfaces which are planes of symmetry of the crystal, such as the 100 and 110 surfaces of an fcc lattice, since otherwise the reflected site rtR will not correspond to a site in the extended lattice. Surface Green functions for other surfaces will not be as simply related to the bulk Green function. This general approach of expressing the probability of electron transfer to a large collection of atoms in terms of transition amplitudes to individual atoms, coupled by the Green function, can be applied to systems other than a macroscopic electrode, such as a metal cluster. It would be interesting to analyze the onset of bulk properties in the electron transfer to clusters of increasing sizes by calculating Green functions for the clusters and looking Some properties of $luster at the resulting sink densities S~’uster(r). which may be interesting to examine as a function of cluster size are the dependence on distance of the reactant from the cluster and the degree of localization of the electrode transfer, as discussed in the following sections for the electrode case.
VI. Dependence of Sink Density on Surface, Site, and Green Function The sink densities for electron transfer to an electrode were calculated from eq 26 by using the bulk and surface Green functions for the tight binding and free electron models for two sites on each of the 100 and 110 surfaces of silver as a function of the distance of the ion from the electrode and the number of lattice sites summed over in eq 26. The interpretation of the two surface sites on each surface occupied by the ion is facilitated by recognizing that an fcc lattice can be regarded as a simple cubic lattice in which every other site is omitted. The simplest types of surface sites that the ion can occupy correspond to either a lattice site of the extended fcc lattice (below referrred to as lattice sites) or one of the omitted simple cubic points not in the fcc lattice (below referred to as interstitial sites). The simplest sites on a 100 surface are thus the 110 lattice site and the 100 interstitial site. On the 110 surface, the simplest sites are the 101 lattice sites and the 100 interstitial site. A comparison of the sink density for lattice and interstitial sites on the two surfaces in the tight binding and free electron models show many interesting features. However, the differences are generally quite small, that is, within a factor of 2-3 and thus may not be easily observable experimentally. To more easily see these differences and to compare with homogeneous electron transfer, the sink densities So(r)are divided by the sink density SoH(r)for homogeneous electron transfer between ions at the same separation r. These appear in Figures 3 and 4. Here, the ion-ion transfer matrix elements are assumed to be the same as those from the ion to the Wannier states of the electrode and the scaled final state equilibrium coordinate y20 is also assumed to be the same. This value of y ; corresponds to assuming the two ions to be at contact and thus neglects the dependence of y,O on ion-ion distance. The resulting S O ( r ) / S O H (increase r) with distance from the elecrode, indicating that So(r)drops off more slowly with distance than SoH@). This is easy to understand since the ion “sees” a larger number of surface atoms at roughly the same distance when it is farther from the electrode than it does close to the electrode. ( 1 7) Jackson, J. D. Classical Electrodynamics, 2nd ed.;Wiley: New York, 1975.
2 -
‘ t
1
0 0
2
DISTANCE
6
4 OF ION
FROM
8
10
12
SURFACE
Figure 3. Ratio of electrode sink density to homogeneous sink density S,,(r)/SaH(r)vs. distance of ion from electrode (A) for bulk (0) and surface (0)Green functions using tight binding model for 100 surface, 100 site (-); 100 surface, 110 site 220 surface. 100 site (-*-); and 220 surface, 101 site (---) of Ag. (e-);
It
i
d
0 0
2
DISTANCE
Figure 4.
4 OF
ION
6
8
FROM
SURFACE
10
12
Same as Figure 3, but for free electron Green functions.
This delocalization effect will be examined more explicitly in the next section. It can be seen that the free electron and tight bonding models give very similar results, indicating that the sink density is quite insensitive to the exact model of the band structure of the electrode. The surface Green function gives a somewhat higher rate than the bulk Green function, with the tight binding model predicting a somewhat larger difference between the surface and bulk than the free electron model. The dependence of surface and site is qualitatively the same for all four models of the Green function. Close to the electrode, the primary difference is between lattice and interstitial sites, with a larger rate from an ion in an interstitial site. This difference is larger on the 110 surface. Far from the electrode, the rate depends more on the surface, with the 100 surface giving a larger rate, but now the lattice sites give a slightly higher rate than interstitial sites on the same surface. When the ion is on the surface, the differences in rate can be understood by considering the distance of the ion from the nearest surface atoms. This distance is 4 2 times as large for an ion on a lattice site than for an ion on an interstitial site at the same distance from the electrode. This is partially offset by the fact that there are four nearest neighbors for a lattice site on either surface, only one for an interstitial site on the 100 surface, and two for an interstitial site on the 110 surface. As a result, there is a larger difference between interstitial and lattice sites on the 110 surface than on the 100 surface. (The nearest atom distance is the dominant effect due to the exponential dropoff of the matrix elements Ar.)
The Journal of Physical Chemistry, Vol. 91, No. 4, 1987 881
Adiabaticity of Electron Transfer at an Electrode
IO'"
!-
\ \ \
-I
10
t
2
0
DISTANCE
4 OF
ION
6 FROM
8
IO
12
SURFACE
Figure 5. Sink densities So(r) (sd) for tight binding surface Green function for 100 surface, 100 site of Ag including Wannier states within 1,2, 3, and 4 lattice spacings of central atom (-) vs. distance of ion from
electrode (A) and reciprocal relaxation times T ~ - *for benzonitrile, DMF, and acetonitrile (in order of increasing rL-'),
4
1 'O"-,* OB ,o 0
0.2
0.4
I
0.6
OVE R P O T E NTl AL
Figure 6. Anodic rate constant k(r) (s-l) vs. overpotential (V, lower axis) and Fermi activation energy (DEF*,upper axis) for ions 2.0 A (-), 6.8 A and 10.0 8, (---) from 100 surface, 100 site of Ag in DMF. (.e.),
(e-).
When the ion is far from the electrode, the larger number of atoms having a similar transition amplitude makes the rate depend less on the site and more on the number of atoms per unit area on the surface. Since the 100 surface has 4 2 times as many atoms per unit area as the 110 surface, the 100 surface has a larger rate. The larger number of nearest surface atoms for an ion over a lattice site appears to result in a slightly higher rate than for ions over an interstitial site far from the electrode. The results in the remainder of the paper will be for sink densities and rate constants on the 100 site of the 100 surface of Ag obtained by using the tight binding surface Green function.
VII. Localization of Electron Transfer Figure 5 shows So(r) for the 100 surfaces, 100 site obtained by using the tight binding surface Green function for Wannier states within 1, 2 , 3, and 4 lattice spacings of the surface atom above which the ion is centered (referred to below as the central atom). This corresponds to restricting the sum in eq 26 to atoms within a hemisphere centered about the central atom and provides information about the localization of the electron transfer. It can be seen that the electron transfer is essentially just to the central atom when the ion is close to the electrode and is confined to 5 2 lattice spacings for ion distances up to at least 11 8,. It is also apparent that the electron transfer becomes less localized as the ion moves away from the electrode. VIII. Comparison with Solvent Relaxation Times The horizontal dotted lines in Figure 5 correspond to reciprocal relaxation times T ~ - I for three common polar solvents.23 The crossing points of these lines with the (converged) So(r) curve correspond to the distances at which S o ~=L1, as in the adiabaticity plot Figure 2. At BEF* 0.41 in the normal activation region, where the , distance at adiabaticity reaches its maximum of ~ 0 . 5 & , 7 ~the which S o ~ L 2 corresponds to the crossover from adiabatic electron transfer at short distances to diabatic transfer at long distances. It can be seen that this ranges from -5.5 8, for acetonitrile (short relaxation time) to -8.0 8,for benzonitrile (long relaxation time) with DMF intermediate at -6.8 A. (It should be recognized, however, that these distances correspond to the simplified model used in this paper and are primarily indicative of the rough magnitude of the solvent and distance dependence of the adiabaticity, not the precise distances at which the crossover occurs.)
-
IX. Rate Constant vs. Potential for Ions at Fixed Distances The rate constants in D M F for transfer from ions at distances of 2.0, 6.8, and 10.0 %, are shown in Figure 6. It can be seen
that, while the three curves are nearly parallel for normal activation energy, the curve for 2.0 A rises sharply in the low abnormal activation region and then levels off quite abruptly. Examination of the adiabaticity along these curves provides an explanation. At the two larger distances, the reaction is diabatic over the entire potential range. With the ion at 2.0 8,,however, the reaction is adiabatic over the normal activation range and into the abnormal activation range up to BEF* 2. The rapid rise in the rate in the abnormal activation region for adiabatic electron transfer is predicted by eq 23. This rapid growth ceases, however, when the reaction becomes diabatic at PEF* 2. The similarity in the curves for normal activation energies reflects the dominant effect of the exponential dependence, while there is only a slightly slower growth in this region on the adiabatic curve at 2.0 A, consistent with the EF*'j2rather than EF*-'j2dependence in the preexponential factor.
-
-
X. Rate Constant vs. Potential for Ions at a Distribution of Distances In the previous section, it was assumed that the ion is at a fixed, well-defined distance from the electrode. This is reasonable for ions which are adsorbed to the surface or covalently connected as in the experiments of Weaver et a1.,18 but in other cases, the ion distribution may be less strongly peaked. It is thus interesting to compare the limit of an ion distribution uniform outside a distance of closest approach ro. As can be seen from Figures 3 and 5, from - 2 to 11 A, the sink density So(r) for the 100 site on the 100 surface drops by more than IO5,while SO(r)/SOH(r), the ratio of So(r)to the simple exponential dependence of SoH(r),changes by only a factor of - 2 . Therefore, the distance dependence of So(r)is quite well represented by a simple exponential over this distance range. Denoting the rate in the diabatic limit by kD(r)and in the adiabatic limit by kA, the rate constant as a function of distance k(r) can be approximated by
-
k(r) =
kD(d kD(r) 1+kA
-
kD(ro)e-22(('-d 1+
-e-2z(r.rd kD(rO)
(32)
kA
where 2s is the average slope of In k ( r ) vs. r over the first few angstroms from the electrode surface. This is very nearly equal to 2K, twice the dropoff rate of the matrix element Ar. The net rate constant per unit area kNETis given by the integral of k ( r ) over all distances, weighted by the radial distribution function g(r) (18) Li, T.
T.-T; Weaver, M. J. J . A m . Chem. SOC.1984, 206, 6107.
882 The Journal of Physical Chemistry, Vol. 91, No. 4, I987 E!
io4
I
104
L
9 7 5 I I , , I
3 I
t
I I
/ k,T I
0 I
,
1
3 5 7 9 1
1
1
I
1
1
1
l
Morgan and Wolynes the reaction would proceed on a two-dimensional harmonic potential (one outer-sphere and one inner-sphere coordinate) in each electronic state. Because inner-sphere relaxation is thought to be at most weakly damped, the Smoluchoswki equation will no longer be appropriate, since it corresponds to overdamped motion. It would, however, be possible to describe the motion in the solvent coordinate by the phase-space Fokker-Planck equation. This would also allow the inclusion of inertial effects in the outer-sphere relaxation, which have been investigated by Calef and Wolynes.' In this description, four solvent processes would compete with each other and with the electron tunneling in determining the overall rate, namely, the position and velocity relaxation along the inner-sphere and outer-sphere coordinates. In principle, at least, one or more of these processes may be faster than electron tunneling and the others slower, leading to a larger number of different limiting cases. In this paper, we have assumed that the solvent properties are not affected by the presence of the electrode. In a real system, if the electrode is at some potential other than the potential of zero charge, an electric double layer forms near the electrode.20B21 This consists of an increased concentration of ions of charge opposite to the electrode and a bias in the orientation of solvent dipoles in a layer close to the electrode (the inner Helmholz layer) and another layer oppositely charged with oppositely oriented solvent further away (the outer Helmholtz layer). This causes a very strong electric field in the first few Bngstroms from the electrode surface. One effect of the double layer will be to make the energy of the electron on the reacting ion a function of the distance from the surface. This means that the net rate constant at fixed electrode potential kNET will result from transfer from ions with different energies and thus different Fermi activation energies EF*. Another effect of the presence of the electrode is the distribution of reacting ions from the electrode. This will generally be significantly different from a step function. The reacting ion itself will be charged, either before or after the electron transfer or both, and thus the radial distribution function of the ion will be a function of electrode potential. The fact that the solvent composition and structure is significantly modified by the presence of a charged electrode also modifies the potential through which the electron tunnels. The double layer electric field introduces an electrode-potential-dependent slope to this barrier.*' This will have the effect of making the electron-transfer matrix elements A, a more complicated function of distance and introduce into them a dependence on the electrode potential. If these effects are included, a more precise calculation of the A, is appropriate which takes into account the form of the ion and Wannier orbitals. Another effect of the double layer field is to modify the effective force constant along the solvent coordinate. Since the double layer field is normal to the electrode surface, parallel to the AD of reaction, the energy of a polarization will be increased by a double layer field parallel to AD and reduced by one antiparallel to it. The force constant will thus be a function of electrode potential and will takes on its bare value only at the potential of zero charge. In view of the small differences between the predictions of the free electron and tight binding approximations, it is unlikely that a more refined calculation of the lattice Green function will yield a significantly different result, but the structure of the surface of the electrode leads to another modification of the matrix elements A, which may be significant. It has been assumed that the matrix element depends only on the distance between the ion and the center of the Wannier state r. However, for transfer to Wannier states in the top layer of atoms, the electron tunnels only through solvent, while for atoms below the top layer, some of the tunneling is through the metal of the electrode. If the A, are modified to include the effect of the double layer as discussed above, they should also be made a function of the distance of the
0
10'po,2
0.2
0
0.6
0.4
0.0
I ,o
OVERPOTE NTlAL
Figure 7. Net anodic k N E T (-) and cathodic k& rate constants (A s-I) assuming g ( r ) = e ( r - r,,) for ions over 100 surface, 100 site of Ag in DMF. (-a)
for the ion from the electrode, here assumed to be a step function. This yields
"
1
+ -e-av-r~~ .-L\ I
kA
1
kA kD(r0) -- -2ii I n 1 + - k ,
I
For ions which are not adsorbed or covalently attached to the surface, this estimate is likely to underestimate the rate since it neglects the peak in the ion-surface g ( r ) near the surface where k ( r ) is larger, but it may be in the right order of magnitude. The net anodic rate constant kNET for transfer to the 100 surface over the 100 site of Ag in D M F is shown in Figure 7 along with the corresponding cathodic rate constant k& = kNETCBAG. It can be seen that the qualitative shape of the potential debeing a superposition of rates from all distances, pendence of kNGNET, is intermediate between the largely adiabatic curve of an ion close to an electrode and the essentially diabatic curve of an ion at a large distance. It is interesting to note that the small upturn near BEF*= 0 has the effect of making log kNETvs. overpotential nearly linear between @EF* 3 in the normal activation region to @EF* 4 in the abnormal activation region.
-
-
XI. Refinements of the Theory The theory of electrode kinetics presented here is highly approximate in many respects. This was done for the purpose of focusing on the interplay between two specific factors entering into the overall electrode kinetics, namely, the outer-sphere dielectric relaxation and electron tunneling. In any real electrode reaction, a number of other factors are also important. In this section, we will discuss some of them and show how these effects can be incorporated into the formalism used in this paper. In addition to the outer-sphere dielectric relaxation which results from reorientation and induced dipole moments in solvent molecules far from the ion, which is reasonably well described by the simple continuum model, there is inner-sphere dielectric relaxation, which results from perturbation of nearby solvent molecules. This part of the dielectric relaxation is probably not very well described by the simple continuum model, since the discrete molecular nature of this solvent will be more apparent in solvent close to the ion. Some attempts have been made to describe this inner-sphere relaxation by undamped harmonic potentials in the two electronic states with minima at different configuration^.'^*^' In this case, (19) Bockris, J. O'M.; Khan, S. U. M.; Mathews, D. B. J . Res. Card. Hokkaido Unic. 1974, 22, 1.
(20) Bockris, J. O'M.; Reddy, A. K. N. Modern Electrochemistry, Vol. 1 and 2; Plenum: New York, 1970. (21) Bockris, J. O'M.; Khan, S. U.M Quantum Electrochemistry, Plenum: New York. 1979.
Adiabaticity of Electron Transfer at an Electrode
The Journal of Physical Chemistry, Vol. 91, No. 4, 1987 883
Wannier state from the surface. Although the model used in this paper is greatly simplified, the basic structure of the theory can be generalized to include many other factors. It may be useful in the future to examine the influence of these factors on the question of the adiabaticity of electron transfer at an electrode and its effect on electrode kinetics.
Acknowledgment. This research has been supported in part by N S F Division of Materials Research Grant 83-16981 and the donors of the Petroleum Research Fund, administered by the American Chemical Society.
Appendix: Derivation on &,(O) for Diffusion of a Harmonic Potential The most convenient form for the evaluation of &O) is G(0) = J d x Jdxo
POW
PO(X0)
S ( x ) S(X0) Q ( X , O l X O ) (A-la)
I
=
aQ(X,Olxo)
ax
?=,,
gives an expression for Q(x,olx0)
Q(x,Olxo)=
In 2
T~
sgn ( x - x o ) )
x
+
+
(A4 Substituting this into &(O) in eq A-la and using the scaled ~ U the ) value of (S) from eq 18a gives variable y = X / ( ~ ~ / and
So27
= - [(In 2) erfc2 yF + 2r'/'( erfc' yF 4 S ( X ) = SO
e(X
erf yF
(A-ld)
- XF)
The function Q(x,OJxo)satisfies the Laplace transformed Smoluchowski equation at z = 0.
So27L
= -[(In 4
1
e - y z s0Y d y .d2(2 erfc y , - erfc yF erfc 9)
J:dy
&
YF
dy eYz erf y
+
iFm
dy ey2 erfc2 y ) ] (A-7)
2)(erf2 yF + 1) - 2 ~ ' / ~ &dy eyz(erfyF(2 + YF
e r p y ) - erf y(2
+ erf2 yF)}]
where the integralz2 Integrating between 0 and x , multiplying by po-l(x), and integrating again gives
r 1 / 2 xdy -
ey'
erfc2 y =
&- dy, Amdy3
- s4- d y I r'/2
0
= 2 S r J 2 d 0 JRi2d+ 0
po-'(x)
&'dB
po(R) -
1
&'dP
o
~-Y~z-Yzz-Y12-2Y~~~+v2)
cos 0
[I
+ 2 cos 0 sin 0 (cos + + sin +)3/2]
po-I(%)(e(x0)- 8(xo - x))
U
(A-3) Evaluating at xo = 0 and using the symmetry of Q(x,Olxo)with respect to interchange of x and xo gives
Q(0,Olx) = Q(x,OIO) = Q(0,OlO)
-& 2 7L
x
+
a8 ( X- , O I O ) ~ ~ = O P O (JxdZ O 0)
dX po-'(x) erf
PO-'(?)
+
sgn ( x ) &'dP p O - ' ( X )
u2
(A-4) Replacing x by xo, substituting into eq A-3, and using the boundary values
=
T~
In 2
(A-5a)
1
= In 2
is used. Since erf(-y) = -erf y , the value of &O) depends only on bFI. This is reasonable since SCV,-YF) = SO~CV+YF) = So - S(-Y,YF) (A-9a) bSb(t),-yF) = -GS(-y(t),yF) (A-9b) (where thenew second argument of S is the location of the Fermi crossing point). So the time correlation function of dSb(t),-yF) is the same as for sSb(t),y,) and thus so is C,(O). (22) Gradshteyn, I. S.; Ryzhik, I. M. Table of Integrals, Series, and Products; Academic: New York, 1980. ( 2 3 ) Gennett, T.;Milner, D. F.; Weaver, M. J. J . Phys. Chem. 1985, 89, 2787. (24) Northrup, S. H.; Hynes, J. T. J . Chem. Phys. 1980, 73, 2700. (25) Hynes, J. T. J . Phys. Chem. 1986, 90, 3701. (26) Levich, V. G. Adu. Electrochem. Eng. 1956, 4 , 249.
