J . Phys. Chem. 1994,98, 3832-3837
3832
Effect of Overpotential on the Electronic Tunnel Factor in Diabatic Electrochemical Processes A. A. Kornyshev,td A. M. Kuznetsov,* and J. Ulstrup'gf I.E.V. Forschungszentrum Jiilich (KFA), 52425 Jiilich, Germany, The A.N. Frumkin Institute of Electrochemistry of the Russian Academy of Sciences, Leninskij Prospect 31, Moscow I 1 7071, Russian Federation, and Chemistry Department A, Building 207, The Technical University of Denmark, 2800 Lyngby, Denmark Received: October 11, 1993; In Final Form: January 29, 1994"
W e have investigated the charge and overpotential dependence of the electronic overlap integral in diabatic electrochemical electron transfer a t simple metal electrodes on the basis of a jellium model for the metal surface. The effect is found to be small for common metals and small molecules in contact with the electrode. It can, however, be appreciable for low-density metals or long-range electron transfer for which the electronic tunnel factor can vary by up to several orders of magnitude over a voltage variation of a few tenths of a volt. In such systems the tunnel factor variation may contribute a notable temperature- and potential-dependent correction to the transfer coefficient in the current-voltage relation.
1. Introduction Electron transfer (ET) between donor and acceptor centers in chemical and biological systems is the focus of much current research.14 Key issues are the electronic tunnel factor, the material between the donor and acceptor, and environmental polarization fluctuations. Perspectives are mapping of molecular ET mechanisms,'+ ET networks in composite systems,s-' and extension to intermediate-size structures such as scanning tunnel microscopy (STM) probes?-"J layer s t r u c t ~ r e s , ~and l - ~electro~ chemical sens0rs.1~ Attention to the electronic factor a t electrochemical metalsolution interfaces has been differently biased. The electronic coupling appeared early in electrochemical ET theory,I5-'8 but most attention has been given to the nuclear activation factor. Recent theoretical approaches to metal-electrolyte interfaces have focused on quantum chemical views of molecular adsorption on metallic clusters19and on surface electronic density functions.20-26 This has led to a fair understanding of the work function,20-21 interfacial differential capacitance,23-26 chemi~orption,2~928 and optical properties such as electrorefle~tance,~~~~~ plasmon resoand second-harmonic g e n e r a t i ~ n . ~ ~ . ~ ~ Variation of the metallic electron density profile with the electrode charge or overpotential in electrochemical or STM processes has no direct analogue in homogeneous ET. This is important in the context of bridge-groupassisted ETI4 and redox group immobilization in solid films33-35 and perhaps metalloproteins.3~4 Different electron tunnel effects can be envisaged. The metallic electron density may fall off rapidly compared with the wave function of the molecule to be discharged, as expected for high electron density metal electrodes and low molecular tunnel barriers such as in bridge-group systems.'+ In such cases the tunnel barrier is constituted primarily by the molecular part. On the other hand, strong electron localization on the discharging molecule and low-density electrode materials might display more conspicuous charge cloud expansion. In this work we investigate charge-induced electron cloud expansion or contraction in diabatic electrochemical ET. Weconsider which includes electronic coupling between the molecular wave function and individual delocalized wave functions of the electrode. It is shown that the tunnel factor can be approximately related t
I.E.V. Forschungszentrum Jiilich.
t The A.N. Frumkin Institute of Electrochemistry of the Russian Academy
to the electronic charge density when the level density varies slowly with energy. Jellium-like representation of the electrode charge cloud gives a new charge-dependent effect on the electronic tunnel factor, reflected in both its exponential and nonexponential parts. The effect is weak for normal metals and small discharging molecules but notable for long-range ET at electrodes with low electron density. 2. Electrochemical Current
We provide first a few elements of diabatic electrochemical E T theory.17J"38 We consider specifically a cathodic process, Le. E T from the electrode to a discharging molecular entity, either a mobile molecule or an immobilized redox center inside a protein or insulating filmcovering the electrode. Close to the equilibrium potential (frequently not far from the potential of zero charge (pzc)) the solvated molecular state is energetically higher than the electrode Fermi energy by a considerable fraction of the solvation Gibbs energy. Configurational environmental fluctuations bring the molecular acceptor energy close to the Fermi level where ET is most feasible. Reverse fluctuations similarly raise the molecular energy level, inducing the reverse, anodic process. At the equilibrium potential, both processes are equally probable and the average net current zero. The equilibrium is shifted in favor of the cathodic process by negative overpotentials 7, raising all electronic levels in the electrode by the energy e7 where e is the (numerical) electronic charge. In the absence of diffuse double-layer effects, eliminated by high background electrolyte concentrations, the whole overvoltage drop is in the spatial region between reactants and electrode; in other cases double-layer corrections must be incorporated. The net current density per unit area of electrode surface is given by the difference between the partial cathodic Uc) and anodic (ja) currents
Microscopic currents in diabatic E T are small, and contributions from each level can be considered independently. The total current density is then given by the average number of discharging molecules per unit surface area in the reaction zone, N, and the averaged current density per reacting molecular unit from independent electrode levels:
of Sciences. 8 The Technical University of Denmark. * Abstract published in Aduance ACS Absfracfs,March 1, 1994.
0022-3654/94/2098-3832$04.50/0
0 1994 American Chemical Society
Overpotential Effects on Electronic Tunnel Factor
The Journal of Physical Chemistry, Vo1. 98, No. 14, 1994 3833
(3) p(c) is the electronic density of states (energy-’; e.g. for a free electron gas p ( c ) = (mV/27r2h3)l/where m is the effective electronic mass, €0 the energy at the bottom of the conduction band, V the electrode volume, and 2rh Planck’s constant). f ( e ) = [ 1 + exp(e - ep)/k~Tl-lis the Fermi function, t~ the Fermi energi, kB Boltzmann’s constant, and T the temperature. Wc(e;q)is the transition probability per unit time (rateconstant)for ETfromagivenlevelofenergy ein theelectrode to the discharging molecule. W,(e;r])denotes, similarly, the anodic transition probability for ET from a reduced molecule to a vacant electrode level. Wc(t;q) and Wa(e;r]) take the following general f0rm1~,~”38
where a is the distance of most efficient overlap between electrode and molecule. We disregard the weak dependence of a on e. Aa is the reaction zone extension perpendicular to the electrode surface, and A a normalization constant determined by the potential surfaces used, while F,’ is the activation Gibbs energy. For displaced harmonic surfaces
I
-400
-200
0
200
(E-E F)/mv
Figure 1. Calculated e-dependence ofne) (fully drawn line), lS’f(x,e;q)lz (at q = 0 and x = 6 A, dotted-dashed line), and their product (dashed line) in eq 10. The dependence off( 0 and 0 at x < 0. Both the width of the profile, @-I, and the position of the front, x, are functions of v (cf. below). The exponential form can be expected to apply rather generally sufficiently far outside the metal surface. Unless otherwise stated, we use hereafter atomic units (e = m = h = 1). Substitution of eqs 18 and 19 into eq 15 finally gives
,:;
j,
(AX)’ ( ~ E / A E( ~)1 ’ n(x*,;;,;g)
= N ~ AU (vAD(u))’
x
exp[-F: (;bt,v)/kBT]
( 13)
This form can be, approximately, converted back to the integral form
j, =
A (v,D(a))’ ( w A ~ ) ~
vexp[-F:G;,,v)/kBT1 ) (14)
M(v)=
J dx lQ(x,a)12n(x,;;$?)
(15)
Similarly, for the the anodic current ja
= N6a A
(vA,(a))2
L1 -‘(v)l
exp[-F~(;~n,v)/kB~ (16)
-* -0 where can, unlike cat, is slightly below the Fermi energy. 5. Approximations for the Overlap Integral
We now invoke specific approximations for *(?,a) and n(x,q). The simplest space-normalized representation of the former is the exponential form l*(x,*;a)l’
= (y3/87r)exp(-y[R’
+ ( x - a)’]’/’) (17)
This form is substantiated for long-range ET by molecular orbital calculations.2 By eq 9 IW,a)l2 =
l/4(1
+
- 4))expI-Y(lx - 411 (18)
The simplest approximation for the surface density profile is given by the single-parameter jellium trial function (Figure 3)20
Integration is from --OD to m. Equation 20 frames our numerical calculations in section 6. 6. Variation of the Surface Electronic Profile and Tunael
Factor with the Electrode Charge Overvoltage variation affects the nuclear activation factor by changing the driving force of the reaction. Overvoltage effects on the electronic tunnel factor are associated with modification of the free electron density distribution with the electrode charge. We estimate here the latter effect and compare it with the former in section 7. Electron densities of Ag (8.73 X lC3au) and Hg (12.8 X 10-3 au) are chosen as representatives of two common electrode material^,^^-^^ although Ag is not the best candidate for jellium representation. In addition, we consider the value 1.33 X 10-3 au as the lowest accessible density for common metals.s3vs6 This low density illustrates the electronic profile shifts as the density is lowered toward small values. The excess charge induced density profile shift is controlled by the quantity exp(PX) = exp(Pu/en) in eq 9. B is, however, itself a function of the electrode charge, suitably expressed by a set of charge “lability coefficients”, a, 6,and ds6
+ + bu’ + d d
/3 = bo uu
(21)
which gives a much stronger jellium profilevariation with charge than solely the constant term PO. Analytical expressions for PO and the coefficients a, b, and d were obtained by self-consistent electron density theory for the metal-vacuum interface,56 the results of which are much in line with numerical KohnSham wave function calculations.s7~~8 All the quantities in eq 21 would be affected if a medium is present which would require new selfconsistency schemes reported over the last decade.*~26~32 There is no principal difficulty in combining these schemes, but for our preliminary estimates, we use the free metal surface values. We
Overpotential Effects on Electronic Tunnel Factor
The Journal of Physical Chemistry, Vol. 98, No. 14, 1994 3835
TABLE 1: Electronic Density (a au), Zero-Charge Electronic Profile Decay Factor (BO),and Charge Lability Coefficients (a, b, and d: cf. eq 21) for Hg, Ag, and Cs (au), Calculated According to Corresponding Equations in Ref 56. Densities from Ref 53 ~
metal
Hg Ag
Cs
~
d 12.8X lo-’ 1.215 114.118 -2.885 X 103 -8.885 x lo5 8.73 X 1.22 149.686 -8.578 X lo3 -2.103 X lo6 1.33 X lk3 1.383 289.652 -1.480 X lo5 4.948 X lo5
n
BO
a
b
also disregard pseudopotential corrections inside the metal. With these reservations we have collected the appropriate parameters in eq 21 in Table 1 (calculated numerically slightly more accurately than in ref 56). Figure 4 shows calculated electronic tunnel factors using the formalism above and the data in Table 1. Two distances, viz. a = 4.5 au (2.4 A, Figure 4, column B) and 20 au (10.6 A, Figure 4, column A), are used. The former represents small molecules (e.g. H3O+)in contact with the surface, and the latter, a “screened” discharging group such as an immobilized redox group inside a metalloprotein or insulating film. Figure 4 reveals the following: (a) For short ET distances, charge induced electron density effects are small for “normal” metals (Hg, Ag) and undetectable in the background of the activation Gibbs energy variation (Figure 4, column B). The effect amounts to 1 order of magnitude over a 10 pC cm-2 charge range when molecular electron localization is strong (large y). For a 30 pF cm-2 interfacial capacitance, this range corresponds to 0.3 V. (b) The charge profile effects for strong and weak electron localizations on the discharging molecule in short-range E T cross at a certain, quite large negative charge. Strong localization is more favorable; i.e. the overlap is larger a t larger negative charges. This can be understood in terms of the center-of-mass motion of the excess electron charge. For small charges the center-of-mass is close to the metal surface and the overlap more favorable for the shallower molecular charge distribution. As the charge cloud expands and the center-of-mass reaches the discharging molecule, the overlap order is opposite and more favorable for stronger electronic trapping at the molecule. (c) Strong charge cloud expansion effects are exhibited for long-range ET (Figure 4, column A). The effect is more pronounced the stronger the molecular electron localization (the larger y). The effect reaches no less than 7 orders of magnitude for the tighter and 5 orders of magnitude for the shallower molecular electron distribution for low-density metals over a 10 pC cm-2 charge range (0.3 V potential range for 30 pF cm-2 capacitance) but is of course smaller for realistic higher-density metals. They are still very notable for Ag, namely 5 and 2 orders of magnitude, respectively, while they amount to 1 order of magnitude and 50%, respectively, for Hg. The nuclear factor varies by 5-6 orders of magnitude over a 0.34.4-Vpotential range (cf. section 7). In conclusion, therefore, electronic profile effects might be detectable for normal electrode metals a t longer-range ET, representative of metalloprotein and other immobilized redox group ET. The effect is small and probably undetectable for small molecules in contact with the electrode surface unless the bulk electron density is abnormally small.
7. Charge-Induced Electronic Profile Effects as Corrections to the Electrochemical Transfer Coemcient From Figure 4, the a-variation of the electronic factor is approximately exponential log ~ ( q=)const + const’u(q) = const
+ consiU(cp/kBT) (22)
where the constants depend weakly on u and 7,and interfacial potential drop. More specifically,
cp
is the
const’ = constb(k,T/e)K = constbe/4?rLCH
(23)
where K is the integral compact layer capacitance, related to CH by (4nQ-l = CH and L = e2/kBT the vacuum “Bjerrum length”. For Hg a t the pzc, CH = 0.3 A = 0.6au (1 p F cm-2 = 4.77 X au) while L = 560 A = 1057 au at room temperature. In the absence of charge lability, constb has the simple form consib = p/n = j3o/n, or log W
v)
-
const + (80/4*LCHn)(cp/kBT)
(24)
Equation 24 can becompared with the cpdependence of the current expressed by the Tafel law lOgj, = B
+ CYe(cp/kBT)
(25)
where B is another constant and the transfer coefficient CY frequently close to 0.5 a t small overvoltages. These features are broadly accounted for by the nuclear activation factor. In comparison, insertion of the appropriate parameters into eqs 2224 gives cons( = 0.01,0.02, and 0.10 for Hg, Ag, and the lowdensity metal in Table 1, respectively, as @ 00. Close to the pzc, charge-induced electron density effects can thus be disregarded compared with the u- or cpdependence of the nuclear activation factor. In contrast, as the electrode charge increases to significant values, the lability coefficients dominate and 0 is now much smaller. The M(r])-variation is then comparable to the variation of the nuclear factor but cannot be given the simple form in eq 22 since not only 0 but also CH (or K ) vary with u in these ranges.
-
8. Concluding Remarks
We have shown that excess surface charge variation on simple metal electrodes can significantly affect the current-voltage relation in electrochemical E T involving suitable electrodemolecule separation. The effects are larger the stronger the electron localization on the discharging molecule and the smaller the metal electron density. We have used approximate integral expressions for the current on the basis of the electron density. This is analytically simple and holds prospects for the nature of the effects. The exponential density forms used are also supported by molecular quantum chemical calculations,lJ particularly for long-range ET when dominating contributions to the tunnel factor are from spatial regions well outside the electrode surface. A more rigorous approach must rest on metallic electronic wave functions (eqs 7 and 8,in some cases available by the Kohn-Sham scheme,26.57-5* but this would give entirely numerical solutions. Another option would be to use analytical trial wave functions for charged j e l l i ~ m . Our ~ ~ .analysis ~~ suggests that such investigations would be worthwhile. Individual wave function approaches would also be important for d-metal electrodes where charge modulation needs much better substantiation than the jellium model, local molecular orbital calculations being appropriate. Electronic structure calculations for electrochemical systems based on wave functions or density profiles should include the In view of unsettled discussions of the solvent in interfacial density functional theory,@ this problem is challenging. A first glance at Figure 3 in ref 32 suggests that the solvent diminishes the surface electronic profile lability, weakening the effects on the tunnel factor compared to those in a vacuum. The reason is simply that the dielectric screens the fields near the surface for a given charge density. This should, however, be investigated by incorporating screening effects also into exchange and correlation energies. The latter may increase the profile lability. Charge modulation of the metal electron density can be compared with other recently discovered molecular electronic
3836
Kornyshev et al.
The Journal of Physical Chemistry, Vol. 98, No. 14, 1994
(A)
-0.003
-0.002
-
10-6 10-8
-0.001
!0 002
-
0
0
-0.003
-
-
I
10-10
I
Q.001
!o 002
-0
-
-0.002
0.001
iop4
0
-
I
L
0
I
low density metal
10-5 io7 10-9 -
0.001
10-11
10.002
I
I
I
I
0.0 1
-0
-
!0.002 0.001
I
-
-
I 0
.\\ I
I
Figure 4. Electronic tunnel factors for diabatic electrochemical ET at simple metal electrodes as a function of electrode charge (atomic units, 1 pC cm-2 = 1.756 X 1 Pau), calculated by eq 20 with parameters characterizing the surface electronicprofile in Table 1. The surface density range shown, 2 X 10-3 au (1 1.4 pC cm-2), is representative of accessible electrochemical surface charges. The three rows correspond to Hg, Ag, and a hypothetical low-density metal with the density of Cs. Solid and dotted line correspond to weak (y = 0.5 au) and strong (y = 1 au) electron localization on the discharging molecule, respectively. (A): short ET distance, a = 4.5 au (2.24 A). (B): long ET distance, a = 20 au (10.6 A).
modulation effe~ts.6'4~The molecular wave functions in longrange E T protrude into environmental space; they can be strongly modulated by configurational fluctuations in the latter and much more delocalized at the moment of ET than at equilibrium. Together with the modulation effects considered above this calls for careful assessment of the different E T factors in specific system applications. We finally note two issues upon observationof electrode charge modulation. As noted, the effects are small for "normal" electrochemical systems involving contact E T but increase strongly with decreasing metal density and increasing distance from the electrode. Bismuth is a suitable low-density electrode metal. Jellium certainly cannot represent Bi, but detailed quantum mechanical treatment of the Bi surface could reveal interesting charge modulation. Distance effects could be exploited using "normal" metal electrodes such as Ag if the molecular redox groupcan be fixed at suitabledistances from thesurface. Reports
of such systems have appeared.65 Other possible systems in the realms of metalloprotein ET and surface films have been noted above. Acknowledgment. We acknowledge useful discussions with Ansgar Liebsch and Michael Partenskii. This work was supported by the Danish Natural Science Research Council. References and Notes (1) German, E. D.; Kuznetsov, A. M. Mod. Aspects Electrochem. 1993, 24, 139. (2) Newton, M.D. Chem. Reo. 1991, 91, 767. (3) Meral Ions in BiologicalSystems; Sigel, H., Sigel, A., Eds.; Marcel Dekker: New York, 1991; Vol. 27. (4) Bowler, B. E.; Raphael, A. L.; Gray, H. B. Prog. Inorg. Chem. 1990, 38, 259. (5) Photochemical Processes in Organized Molecular Systems; Honda, K., Ed.; North-Holland: Amsterdam, 1991.
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