Electron-transfer properties of quantum molecular wires

Chemical Physics

ELSEVIER Chemical Physics 193 (1995) 237-253

Electron-transfer properties of quantum molecular wires

E.G. Petrov, I.S. Tolokh, A.A. Demidenko, V.V. Gorbach Bogolyubov Institute for Theoretical Physics, Ukrainian National Academy of Sciences, 252143, Metrologichna str., 14-b, Kiev, Ukraine

Received 19 July 1994; in final form 6 December 1994

Abstract

It is shown that interelectrode tunnel current through a quantum molecular wire (QMW) depends essentially on the relation between the dynamic properties of QMW and the relaxation processes within electrodes and donor (acceptor) units. Within the framework of the adiabatic approximation, it is found that image forces from the electrodes change the tunnel current by several orders of magnitude. It depends on: (1) the position of the donor-bridge chain-acceptor (DBA) system with respect to the electrode surfaces; (2) the ratios between the static and high-frequency permittivities of the interelectrode medium; (3) the effective radii of donor (acceptor) and QMW units. The orientation effects connected both with the orientation of DBA units and with mutual spin orientations of neighbouring magnetic QMW units are studied. It is shown that even small changes of the orientation of porphyrin molecules may cause changes of several orders of magnitude of the QMW-mediated tunnel current. In the case of an antiferromagnetically ordered QMW, the interelectrode tunnel current can be regulated by a magnetic field in a wide region up to 7-8 orders of magnitude because of the ability of the magnetic field to influence changes in the spin orientations.

1. Introduct ion

Advances in molecular and biomolecular electron- ics, molecular biophysics, bioenergetics, etc. have given rise to numerous physical problems associated with electron tunneling (ET) in quasi-one-dimen- sional molecular and macromolecuiar systems. That is why the investigations of the tunnel currents through model molecular structures, in particular quantum molecular wires (QMW), are very impor- tant both from experimental and theoretical points of view.

By the term QMW we mean a quasi-one-dimen- sional chain of molecules or molecular groups ended by a special donor (D) and an acceptor (A) unit or attached directly to the surface of the electrodes. The molecular systems (polymers, biopolymers, systems with conjugate w-bonds) and supramolecular sys-

tems that have been recently synthesized [1] can be used as QMW under appropriate conditions.

The direct measurement of the electron tunnel current through a molecular wire is a difficult prob- lem. The conductivity of a single molecular wire (ligand) has been first measured directly in a special experiment [2]. In scanning tunneling microscopy experiments, the resistance of a fat acid molecule was measured in a bilayer [3]. All experimental results show that a model in which a molecular chain plays the role of a bridge structure should be used for the description of the tunnel current through a molecular wire.

The development of a theory of ET through bridged structures has been in the focus of attention for a long time, starting with Marcus' and others pioneering papers [4,5] on the distance ET between redox-centers (the theoretical achievements on ET

0301-0104/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0301-0104(95)00426-9

238 E.G. Petrov et al. / Chemical Physics 193 (1995) 237 -253

Fig. 1. Kinetic scheme of ET process in the Ca-(DBA)-C b system.

through different kinds of spacers can be found, for instance, in the books [6-10], special collected pa- pers [11-13], the reviews [14-16], and the papers [17-38] and references therein). In the present paper, we apply the results of the ET theory in order to analyze the influence of dynamic factors on the interelectrode tunnel current through a QMW. Here, a QMW serves as a bridge chain in the common donor-bridge chain-acceptor (DBA) system embed- ded in a dielectric medium between two electrodes C a and C b (Fig. 1). This medium is organized either by host structures (in this case, each DBA system forms a tunnel electron pathway) or by molecular wires only (mono or bilayer structures from quasi- one-dimensional macromolecules or molecular chains).

Because of the contact between each edge unit of a DBA system and the electrode surface, the edge units play a specific transfer role. Chemically, D and A units differ from the chain units. But nothing prevents the edge chain units to play the role of donor and acceptor groups, even when the regular chain is embedded between electrodes. The last fact is physically due to the action of the electrode surfaces on the energy levels of an electron to be transferred. We consider an interelectrode region which is formed by large organic molecules (like a Langmuir-Blodgett film) below. In this case, the overlap of the electrode and molecule wave func- tions is very weak (see, for instance, the discussion about the nature of the physical contacts between the tip apex and the molecule in Ref. [39]). The electron wave functions of QMW without admixture of the electrode wave functions can be used therefore for the description of an interelectrode ET. This approxi- mation is especially justified for the transferring electron if its affinity to each unit of the D - B - A system is much less than the electrode work func- tion. As a result, the influence of the electrodes on

the energy levels of an extra electron, i.e. an electron transferred through QMW, can be taken phenomeno- logically into account as an external field initiated by image forces. Because the image forces depend on the medium permittivity [40], we have a simple way to determine the efficiency of an interelectrode ET on the polarization properties of an interelectrode medium.

The present paper is structured as follows. In Section 2 we give the expression for an interelec- trode tunnel current at different relations between the times of dynamic and relaxation processes. In Sec- tion 3 we consider the dependence of a tunnel cur- rent through QMW on the number N of chain units, energy parameters of the chain bridge, and dielectric permittivity of the medium where the QMW units are located. In Section 4 the effect of an orientational factor on the tunnel current is examined, and Section 5 is concerned with the role of the magnetic ordering in the ET through a magnetically ordered QMW.

2. Tunnel current through molecular wire

To find the current J through a single DBA system we use the expression

J = eJV a . ( 1 )

Here, e is the electron charge and k~ a is the variation in time of the number Na(t) of free electrons which can be be transferred through a molecular wire from the electrode C a . Below we consider the stationary tunnel current only (A~ a = const). The expression for A} a depends essentially on the relations between the dynamic and relaxation properties of the whole C a- (DBA)-C b system. The dynamic processes are de- termined by the quantum electron jumps between the electrodes and edge units of the DBA system, be- tween the D(A) unit and QMW, and between QMW units. Each separate jump is characterized by the parameter of electronic coupling. The relaxation pro- cesses are connected with the interaction between the electron and phonon subsystems.

In metals, the electron-phonon coupling is rather large, and, hence, the typical time r e of relaxation processes is very small. Therefore, ET occurs on the background of the fast relaxation processes within electrodes. These fast processes support an equilib-

E.G. Petrov et al. / Chemical Physics 193 (1995) 237-253 239

rium electron distribution over different electron wave vectors k and q (and electron energies Eak a n d Ebq) concerning the electrodes C a and C b, respectively.

As a rule, the electronic coupling between near- est-neighbor units of a bridge chain is much greater than the electron-phonon coupling 1. This fact shows that electron jumping over chain units must be con- sidered as a dynamic process at interelectrode and donor-acceptor bridge-assisted tunneling. As to the D and A units, these edge units can participate in a tunneling process in the capacity of a pure dynamic or dynamic and relaxation parts of the whole C a- (DBA)-C b system. The result depends on the rela- tion between the relaxation time T o within a separate D(A) unit and the dynamic time ~-a of a tunnel electron jump between this unit and an adjacent electrode. We consider the most important cases,

T d << T O and ~'o << Td"

2.1. Direct interelectrode tunnel current through QMW

Let ~'a << T0" It means that a transferring electron is not practically localized within the DBA system, and its energy states in this system are virtual ones. As a result, the interelectrode ET has an elastic character, and the irreversibility of ET is determined by relaxation processes within the electrodes only (see also Ref. [39]). Using the density matrix method [44] (for the finding of a Jga ), it is possible to show that at a low bias voltage V

2~re J = T E I Tak,bq 1 2 [ W ( E a k ) - W( Ebq -['- eV)]

k,q

X 6(Eak - Ebq ). (2)

The temperature dependence of the tunnel current is determined by the Fermi distribution function,

W(Ejk)= {exp[(Ejk-EjF)/k .T] + l} -1 ,

( j = a, b) , (3)

where Ejv is the Fermi energy, T is the temperature and k B iS the Boltzmann constant. The effective electronic coupling in Eq. (2),

Tak,b q = --LakLbqODBA( E; U), (4)

describes the interelectrode ET between states with energies Eak =Ebq (the &function in Eq. (2) in- cludes this conservation law). The quantity (4) is determined by the electronic couplings L,k a n d Lbq between the electrodes and the D and A units, re- spectively. Following Larsson [23,30], to describe the tunnel properties of a DBA system we introduced in Eq. (4) the ET capability

N-1 IN+2 ODBA(E ; N)=fiDfiA I--I f l . /1--I [ E ( I ) - E ] .

n=l / l=l (5)

This quantity is expressed via the electronic cou- plings riD (/3A) and fin of D(A) to the nearest unit of a QMW and between the nth and (n + 1)th units of a QMW, respectively. E(l) is the energy of an extra electron in the lth eigenstate of a DBA system, N is the total number of chain units. As to the energy E in Eqs. (4), (5), at the tunneling condition, E(1) - E >> [ Lak 1, [ Lbq 1, this quantity coincides with the energy Eak = Ebq = E F of a tunneling elec- tron 2

2.2. Tunnel current through QMW with relaxation on D and A units

Let T O << ~'d' In contrast to the direct interelec- trode tunneling considered above, a transferring elec- tron is capable to occupy now the D and A units. With the density matrix method we obtain the fol- lowing expression for a stationary interelectrode tun- nel current:

kl ,¥a X - b -- k2 Xb X - a J = - e (6)

X - a X - b -[- kl X - b "q-k2 ) ( - a

The physical meaning of the rate constants in Eq. (6)

1 For instance, the electronic coupling for it-electrons of polyenes is about 2 eV [41]; for another chains of organic and biological origin, these couplings lie in a wide range from 1 meV up to 1 eV [12,17,23,25-27,30,34,36,42,43].

2 Here, we consider electrodes with equal Fermi energies, E~F = EbF ~- E F.

240 E.G. Petrov et al. / Chemical Physics 193 (1995) 237-253

is manifested in Fig. 1. In particular, the quantities Xa and Xb where

2w - W(Eak)W(ED.o) Xa h ~ [Lak(/zl /%)12 0

kP4z0

×6(Eak +E°~o-ED~ ), (7)

indicate the electron inputs to a DBA system from the electrodes, while the kinetic constants X-a and X-b, where

27r X - a - h E [Lag(/z[ /z o)[2W(ED~)

k/z/Zo

(8)

determine the electron outputs from a DBA system to the electrodes. In Eqs. (7), (8), ED~ and E°Dg0 are the energies of the/zth and tz0th vibrational states in donor electronic terms when a transferring electron does and does not occupy a donor unit, respectively. The multiplier ( /z ] /~0 ) denotes the overlap integral for these vibrational states. The equilibrium vibra- tional energy distribution function,

W(Ej~) = exp(-Ejv/kBT) E e x p ( - - E j J k B T ) '

U

( j = D , A),

(9)

determines the occupancies of different vibrational states (v = tz, /z 0, u, u0). The expressions for Xb and X-b follow from Eqs. (7); (8) by replacing the indices D, ak and /z with A, bq and u, respectively (u and v 0 are the vibrational states in the acceptor electronic terms).

The donor-acceptor ET rate constants,

2-rr k l - k o - ~ A = h [TDAI2(FC)I'

21T k2 - kA-~ o - h I T°A [ 2(VC)2' (10)

depend on the purely dynamical part, I TDAI 2, and the relaxation Franck-Condon factors, (FC)j. In Eq. (10), the effective electronic coupling D to A has the form

N UlAUNA TOA =/3D/3A E (11) '

where E(A) are the energies of the delocalized bridge eigenstates I h), unh = (n I A) are the elements of the unitary matrix that transforms localized bridge unit states, ] n), to ] A) (for a long regular chain, una are given in Refs. [20,21], for a finite regular chain, the quantities unA coincide with the bridge molecular orbital coefficients CnA [17,23]); and

( F C ) I = ~ ~ I(/x[ /z0)(v 0 I v ) l 2 /z/z o vP 0

×w(EoAw(E° o)

(the expression for (FC) e follows from the above one by the substitutions ED~ ~ EA~ + eV and E°~0 ~E°~0 in the distribution functions W(ED~) and W(E° ~o), respectively).

The quantities (10) coincide in form with the well-known expressions for the rate constants uti- lized in the theory of a long-range bridge-assisted donor-acceptor ET (see, for example Refs. [8,9,14,18,19,23,25,27]). It means that the interelec- trode tunneling includes the donor-acceptor ET pro- cess at the tentative electron localization on D and A units. In particular, a current J is proportional to the rate constants, k 1 and kz, if only ET through QMW is a limit stage for an electron transfer between electrodes C a and C b. Just in the case k I << X-a, ke << X - b , we get

J = - e [ ( X a / / X _ a ) k l - ( X b / / X _ b ) k 2 ] .

If Xa/X-a = Xb/X-b = K (identical electrodes, and identical D and A units), this equation will reduce to the expression

J = --eK(k 1 - k 2 ) , (12)

which corresponds to Ohm's law at a small bias voltage V.

In order to examine the general expressions (2) and (6) for the QMW-assisted tunnel current, it is necessary to analyze the dependencies of different kinds of electronic couplings on the dynamic factors of a DBA system, and, also, the dependencies of the Franck-Condon factors on the vibrational structure of the wire and the interelectrode medium 3. Below

3 The Franck-Condon factors are calculated analytically if one utilizes the harmonic approximation [8,14,15,18,19].

E.G. Petrov et al. / Chemical Physics 193 (1995) 237-253 241

we consider the role of the most important dynamic factors (the electron energies and electronic cou- plings) on the interelectrode current.

3. Role of image forces from electrodes

As was mentioned in the Introduction, in the case of a macromolecular interelectrode medium, when the contact between the electrode and the macro- molecule (here, the DBA system) is very weak, one can introduce the image forces from the electrodes. The physics of the influence of the image force on the interelectrode tunnel current is connected with the shifts of the energies of localized DBA system states for an extra electron. Our purpose is to make qualitative estimations of such an influence. Here, we cannot use the results from solid state electronics (see for instance Ref. [45]) because of the specific structure of DBA systems. The previous investiga- tions on the role of image forces in transfer pro- cesses in macromolecular systems were only con- cerned with the ion transport [46], the formation of a polaron state [47] and electron pathways [48], and the calculation of the reorganization free energies [49].

We consider a regular bridge (QMW consists of repetitive units) without too much loss of generality in the physical picture. It allows us to obtain the analytic expressions for tunnel currents, and hence, to make a detailed discussion of the physical results.

3.1. Analytical expressions for effective electronic couplings

The effective electronic couplings Tak;b q (Eqs. (4), (5)) and TDA (Eq. (11)), which determine the contribution from dynamic processes to the currents (2) and (6), have much in common in a physical sense. Both of them depend on the structural and energetic properties of the DBA system. This fact was manifested by numerous estimations and quan- tum-mechanical calculations of TDA [17,20--38].

McConnel [17] was apparently the first who ob- tained an approximate expression for TDA in the form (11). In his version, the energy E coincides with the bottom of a donor term, E D. More correct investigations, performed in Refs. [20,21,23] within the framework of the Lifshitz [50] and L6wdin [51]

methods, respectively, had shown that Eq. (11) is valid not only at the rigid condition [23] E ( A ) - E >> ] /31 1, I /30 ], (E = ED), but under the weaker condition [20,21] E ( A ) > E , (E = E (~), where E (~) is the lowest electron energy of the whole DBA system). One can find E (~) solving the set of equa- tions (A.1) and (A.2) from Ref. [20] or Eq. (27) from Ref. [21]. In the present work, we analyze these equations for a regular bridge chain, when /3 n =/3 and

E(A) = E 0 - 2 1 /3lcos[~rA/(N+l)], (13)

where E 0 is the center of the bridge energy band. The analysis shows that a quantity E (~) is close to E o if the last one is situated under the band bound- ary, E 0 - 2 [ fi [cos ] nv/(N + 1)]. At the D - A tun- neling, this condition is usually satisfied, and, hence, we may apply the form (11) in a wide energy interval,

AEo> 21 filcos['rr/(U+ 1)1 ,

(A E 0 = E 0 - E = E o - E D ) . (14)

For further analysis, let us rewrite Eq. (11) as

TDA = --~O/3AON(E), (15)

using Larsson's ET capability of the bridge in the form

N Ou( e ) = ~x -1 / 1--I [E(A) - E ] . (16)

The product in Eq. (16) can be put in the simple analytic form

sinh ~0 ~)N(E) = O)~eg(~0) = [ /~ [-1

sinh[(N + 1) ~0] "

(17) Because o~eg(~o)~ e x p [ - ~ o ( N - 1 ) ] at large N, the quantity

~0 = ln{2 I / 3 [ / [ A E o - ( A E 2 - - 4 1 G 2 ) I / 2 ] } (18)

can be considered as a decay parameter. Earlier, a similar parameter was used to explain the exponen- tial decrease of the tunnel rate at the donor-acceptor ET through polypeptide [20,21] and fat acid [24] chains, and for an estimation of the electronic cou- pling D to A along primary and secondary protein structures [22] (see also Ref. [26]).

242 E.G. Petrov et aL/ Chemical Physics 193 (1995) 237-253

Under the condition of deep tunneling, AE 0 >> 2[ /3 l, the decay parameter gets the form ~o = ln(AEo/I /31), and the ET capability (17) is simpli- fied to the expression

1 ( I / 3 1 ) N-1 O N ( S ) = offg(~o) = ~ o ~ S ~ 0 (19)

Analogously, the ET capability (5) of the DBA system, with allowance of the inequality (14), takes the form

ODBA(E ; N ) = [ j~DJ~A/(ED--EF)2]ofveg( ~0)"

(20)

But it is necessary to take into account that the energy gap in ~0 is determined now by the energy difference AE 0 = E o - E v. The expressions (17)- (20) form the physical basis for an analytic descrip- tion of the current dependence on the dynamic fac- tors of QMW.

3.2. Polarizing shifts

Under the influence of the image forces from the electrodes, the energetic regularity of the QMW units is broken. This fact must be displayed in the effi- ciency of ET and, hence, the interelectrode tunnel current J. Let E D, E A and E o be the bottoms of the electronic terms corresponding to the location of a transferring electron on D, A, and the nth unit of QMW, respectively (in the notation of Section 2, E D --- ED0 q- E°o, E A --- EA0 q- E°o, E 0 - EOo q- EA00 + I where I is the affinity to the selected bridge unit). In the semiphenomenological physical model we use, the image forces are considered as an addi- tional external electric field initiated by a charge particle (transferring electron) being in an interelec- trode medium (assuming that the rest of the interac- tions are included into E D, E A and E 0 quantities). Therefore, the electron energies are presented in the form

ff~D = ED "~ E~ ), EA = EA q- E(i), En = So q- E(i). (21)

For a calculation of the energy shifts, E~ ), E~A i) and E~ i), let us note that the electrodes C~ and C b are made of conducting materials (e.g., metal or graphite), and the interelectrode region is an insula- tor. As a result, the strong inequality, G >> s, is

satisfied between the dielectric permittivities e c and s of the electrodes and the interelectrode medium, respectively. Note also that e may have both the static ( s = s 0) and high-frequency ( s = s~) value. The situation depends on the ratio between the time of electron stay (location) on D(A) or QMW, ~'L, and the time of establishing of the stationary polar- ization near the charge (i.e. the solvatation time), ~-p. If ~'L >> ~'P, then the permittivity s has the static value. In the opposite limiting case, ~'L << ~'P, a permittivity of s= should be taken. We must obvi- ously put s = e~ for virtual states of a transferring electron. The value s = s 0 is used in the case of a tentative electron localization on D and A units.

To find the energy shift we choose a model where an interelectrode medium is approximated by the plane-parallel plate of thickness d = 2(r D + Nr o + r A) (the radii rD, r A and r 0 characterize the effec- tive sizes of D, A and bridge units, respectively). Then taking into account the general expression from Refs. [40,46], we obtain a shift (in electron-volts, eV) of the virtual energy level for each nth bridge unit in the form

13.5 E(~ i) --~ - - - F ( N + 6 D + 6A; n + 6D), (22)

pea

where po= ro/rB, 6 D = r v / r o , 6 A = rA/ro, and r B = 0.53 A is the Bohr radius;

F ( M ; m)

1

2 m - 1

+M I l 2 - [ ( 2 m - 1 ) / 2 / ] 2 l " (23)

Since D and A units are near the electrode surfaces, it follows with a high accuracy that

13.5 13.5 E~ )-.~ - - - - , E(A i)..~ - - - - , ( s = oo0, o%¢),

ePD SPA

(24)

where PD = rD/rB, PA = rA/rB" In Eqs. (22), (24) and further, the permittivities are given in units of the vacuum permittivity, Sv, c.

Eqs. (22) and (24) show that the energy shifts depend essentially on the polarization ability of the

E.G. Petrov et aL / Chemical Physics 193 (1995) 237-253 243

interelectrode medium (through the permittivities e o and e=) and the geometric sizes of the structural units of the DBA system. Particularly, because of the inequality e o/> e=, the D(A) energy shift, connected with the electron localization on the D(A) unit, exceeds (by the factor e0/e~) the energy shift achieved at the condition of direct interelectrode tunneling.

Eqs. (21)-(24) together with Eqs. (5), (16) can be used for numerical estimations of the image force effect on long-range ET. Below we give quite simple analytic expressions which are sufficient both for numerical estimations and physical analyses.

Since all QMW units form the inner part of the DBA system, the energy shifts E~ (° are much less than those of the D(A) units. For instance, at e = s~ = 2 (nonpolar medium), p = 4, 6 o = 6 A = 1, d = 40 A, we have E~ ) = - 1 . 7 5 eV, E~ 0 = - 0 . 5 2 eV, E2 (° = -0 .35 eV. Therefore we introduce the quan- tity

1 N if(i) = -N E E(i), (25)

n=l which characterizes the average lowering of electron bridge levels. The introduction of the quantity ff;0 (° keeps the energy regularity of the bridge units, and, hence, we may use the quantity

E0 = Eo + E0 (i) (26)

for determination of the center of the bridge band. As a result, we may estimate the tunnel currents (2) and (6) (or (12)) with the help of expressions (4), (20) and (15), (17), respectively, changing in Eq. (17) the quantity C0 on the decay parameter

~:= ln{2 1 f i I / [AE--(AE2--41~2) l /21} . (27)

In Eq. (27) the energy gap, AE, depends on the kind of interelectrode tunneling. In the case of a direct tunneling,

z X E = E o - E F = A E o + u ~ ), ( a e o = u o - e F ) . (28)

If ET is connected with a tentative electron localiza- tion on D and A units, we must take

AE =/77o - /~v = AEo + ff2o(i) -- E~ ),

( AE o = E o - ED). (29)

3.3. Analytic expressions for estimation of tunnel currents

In order to simplify the analysis of the image force effect from the electrodes on the tunnel current J, we restrict ourselves to the case when Ohm's law holds true. We estimate the image force effect by the relation

77 = j / j o , (30)

where j o is the expected tunnel current when one disregards the image forces from the electrodes in the calculation of the tunnel current.

At the condition of direct interelectrode ET through a linear DBA bridge system (Fig. 1), in accordance with Eqs. (2), (4), (20) and (21), (26) we have

I ODB A 12 ~7 = I OOB A 12

E D _ EF ) 4, - A ( N , Co, ~) ED~)~F+E~)

(31)

s inh[ (N+ 1)~o] sinh__~ t2

A(N, C0, ~) = ( s i n h [ ( N + 1)~1 sinh Co ] "

(32)

Here, because of the virtual nature for all states of a transferring electron, E~ ) is given by Eq. (24) with e = e~. The decay parameters, ~ and ~0, are deter- mined by Eqs. (18), (27) and (28) where AE 0 = E 0 - E F. Because /T (i) < 0, then AE < AE0, and the image forces facilitate ET, i.e. tunnel current. This fact is manifested in Fig. 2.

If the electron has a tentative localization on D and A units, then with the help of Eqs. (12), (10), (15) and (18), (27) we find 4.

= K I TDA 12/Ko [To°A 12

= exp[ -E~)/kBT] A'(N, Co, ~). (33)

The quantity A' coincides in form with Eq. (32) but the decay parameters (18) and (27) are determined by the energy gaps A E o = E 0 - E D and Eq. (29),

4 Approximate ratio •/K ° ~ exp[-(ffS D - ED)/kBT ] = exp[-E~)/kBT] follows from definitions (7)-(9) of the rate constants X~, X-a, and energy conservation law at ET.

244 E.G. Petrov et al. / Chemical Physics 193 (1995) 237-253

8.0

6.0

CY~ 4.0

2.0

0.0 I , I 6.0 8.'0 10.0 12.0

Fig. 2. Image force effect from electrodes on direct interelectrode ET. The numbers 1, 2, 3, 4 for both upper (8 D = 8A= 1) and lower (8 D = 8A=2) sets of the curves correspond to N= 4, 10, 16, 22, respectively; calculations in allowance for Eqs. (31), (32) with AE o = E o - E v = 3 eV, E D - E F = 2.2 eV, I /3 ] = 0.25 eV.

respectively. In Eq. (29) the quantity E~ ) is calcu- lated with Eq. (24) where e = s 0. Provided that lowering of a donor level, ED, is more considerable than that of the center of the bridge band, E0, the difference ff~0 (/) - E ~ ) in (29) is positive, and, hence, A E > A E 0. It means that the tunnel parameter A ' ( N , ~o, ~ ) falls when Q M W length increases. However, this fall is compensated by the thermal factor e x p ( - E ~ ) / k s T ) in the ratio (33). As a result, this ratio can be 7} > 1 or rl < 1 (it depends on the number of chain units, Fig. 3).

4. Orientational effects

This section is concerned with the study of orien- tation ET in C a - ( D B A ) - C b systems which occurs due to the dependencies of the electronic couplings /3D, /3A, /3, Lak and Lbq on the mutual orientations of the adjacent DBA units and their orientations relative to the surface of electrodes. The important steps in this area have been made in the theoretical works of Larsson and co-workers [23,30,36] and

20.0

1

16.0

~.12.0 i

8.0 2

4.0 4

0.0

i i i i i i p i i i i

¢°o lO 20 3o N

Fig. 3. As for Fig. 2, but for ET with tentative electron localiza- tion on D and A units; the curves 1 (8 D = 8 A = 1, pe~ = 6), 2 ( 8 D = 8A=l , p ~ = 8), 3 (8 D = 8/,=2, p~ = 8), 4 (8 D = 8A= 2, p~ = 10) are calculated in allowance for Eq. (33) at knT = 0.025 eV, e 0 = 2 e~.

Newton [54] where it was shown that the geometry of the bridge affects significantly the effective elec- tronic coupling TDA. Several models for the consid- eration of orientation effects on electronic coupling between neighbouring molecules have been proposed by Brocklehurst [52], Marcus and co-workers [53] and others [60,61]. It has been found that the orienta- tion effects depend crucially on the symmetry and extended properties of the transferring electron wave functions connected with the electronic structure and geometry of molecules, and, hence, they are essen- tially manifested in the electron transfer systems including plane molecular structures. Among the lat- ter, the porphyrins and metalloporphyrins play an important role in the realization of ET. For example, these molecules are integrated into electron pathways of photosynthetic systems [55,56] and they also can form artificial electron transfer systems [1,57] or arranged aggregates in a binary solvent mixture [58]. For this reason, examining orientation effects on ET in bridged systems, we consider iron porphyrin (FeP) molecules as the bridge units of QMW.

4.1. Orientat ional e lectron tunnel ing be tween bridge

units

Here, we consider orientation effects connected with the couplings t = / 3 D, /3 A, /3. We restrict our-

E.G. Petrov et al. / Chemical Physics 193 (1995) 237-253 245

selves to the case when adjacent units of an electron transfer chain are not in close contact (such disposi- tion is often found in real bridge systems [57]). As a result, the overlap of electron wave functions of adjacent units is specified by the asymptotic tails [56]. Besides, in this case the polarization properties of the intervening medium should be taken into account properly [59]. We carry out the calculations of couplings, mentioned above, within the frame- work of the one-electron semiphenomenological model where one-electron orbitals are composed from effective hydrogen-like atomic wave functions with an effective decay parameter c~ [52,56,60,61]. In accordance with the experimental studies of ET in condensed matter [10], theoParameter c~ can be taken in the range a = 0.5-1.0 A -1. For simplicity, in our estimations we take a = 0.7 ~ -1 for all kinds of atomic orbitals (AO).

Considering ET between FeP molecules, we as- sume that the effective one-electron orbitals gt corre- spond to one of the lowest unoccupied FeP molecu- lar orbitals (MO) which (in the MOLCAO approxi- mation) include the 3d AO of central Fe ion [62,63]. These mixed MOs can be written in the form

Ig t ) = Yl ~M) + ( 1 - y2) l /21qt~) , (34)

where I q~M} is the 13dx2_y2} or 13dz2} AO of the iron atom, I qti~} is the symmetry-adapted linear combination of the 2s and 2p AOs of the porphyrin ring, and y is the mixing coefficient. Because the degree of MO delocalization between metal and porphyrin ring AOs depends on the calculation meth- ods [62-67], we shall put below y as an arbitrary parameter for a qualitative analysis of the orientation effects.

To calculate the electronic couplings, one should know the explicit form of the effective potential energies of a transferring electron in the FeP com- plexes in condensed matter. But it is impossible to evaluate these energies exactly without carrying out rather complicated quantum chemical calculations. Therefore, we estimate the orientation effects by calculating only the intermolecular overlap integrals S = (1/f 1 I1/P 2 ) o f the relevant MOs 1//1 and 1/f 2 .

In the present, we study the orientation effect for a configuration-like face-to-face type (Fig. 4). Then the electronic coupling is a function of the angle 0,

YY 13 r-n tl I tr

- - I ' 1 ( I - -

u ~ ' ~ u

Fig. 4. Relative orientations of two FeP molecules of QMW.

t = t(0). It is convenient to introduce an orientation parameter o-(0) = t(O)/t(O* ), where 0* is the an- gle corresponding to one of the fixed configurations of the DBA system. In accordance with our approxi- mation, we may put

0) -- s( o ) / s ( o* ). (35)

As to the whole DBA system, we shall estimate the orientation effect by Eq. (30) where the tunnel cur- rent j 0 corresponds now to the configuration with fixed angles 0~, 0 A and 0* for D , A and QMW units, respectively. For the case of tentative electron localization on D and A units and at deep tunneling Eqs. (10), (14), (15), (19) yields

7 = (0D, 0A, 0)

= [O'D(OD)o'A(OA)]20-~(O) 2(N-l), (36)

where o-D, o- a and o- B are the relevant orientation parameters (35) connected with the overlap integrals

S D = (1/ t D 1 ~ 1 ) , S A = (1F u I~/PA) and S B =

(~ , Igr,+l}, where gt D, gt A and ~, are the relevant MOs of D, A and the nth bridge unit, respectively. In Fig. 5 it is seen that the quantities (35) depend essentially on molecule mutual orientation (angle 0), type of MOs and also mixing coefficient y. There- fore, in accordance with Eq. (36), the orientation effect (here, the changes in an interelectrode tunnel current) can reach many orders of magnitude.

4.2. Orientational electron tunneling between ring molecule and crystal surface

Let us consider orientation effects connected with a direct interelectrode ET and which arise from the dependencies of the electronic couplings La~ and Lbq (see Eqs. (2), (4)) on the orientation of D and A units relative to the surfaces of the electrodes. Within the framework of the one-electron model used above

246 E.G. Petrov et al. / Chemical Physics 193 (1995) 237-253

these couplings are expected to vary as the overlap integrals S = (~IXA)[~akCoq)) with orientation, where ~akCoq) is the wave function of the electron state in electrode C a (Cb).

Let the D(A) unit be a plane ring molecule. To define ~ we use a cycle polyene model [68] accord- ing to which the nr-MOs may be expressed as

qt m = ~7. Cjm [ (2Pz) j ) , J

(Cym=N~ 1/2 exp(2"rrijm/ND) ), (37)

where m is the quantum number of the cycle polyene eigenstate, N D is the number of atoms with [(2pz)) AOs. As for FeP molecules for evaluating the asymptotic tails of 2pz AOs of the cycle polyene, we use the corresponding hydrogen-like AO with the decay parameter a .

The crystal surface of the electrode is modeled by a plane at which there is a step in the electron potential energy under the transition from the crystal to the interelectrode dielectric region. In such a surface region we restrict our consideration to the surface states of the electron [69] only, which can be written as

l/-rkll(r) = BUk,l(r) exp(ikll • r ) exp[ - ( ( z + h ) ] ,

(38)

where B is a normalization constant, Ukl(r) is the Bloch amplitude function, kll is the surface Bloch wave vector of the state, ( is the decay parameter of the surface state (the magnitude and sign of ( are different in the crystal and interelectrode dielectric regions), the coordinate system axis z is normal to the surface, and the coordinate center is situated in the center of the molecule at a distance h from the surface.

The overlap integral we need can be rewritten in the following way:

No S = (1/? m I~k,,) = E C~;Sj, S j = ((2pz)j lqek, ,) .

j = l

(39)

Because the functions ~,, and ~k decay exponen- tially and simultaneously in the bulk electrode re- gion, the contribution of this region to Sj can be neglected and the integration is carried out only in

b

1.0

0.8

0.6

0.4

0.2

0 . 0 ~ i

0 15 30 ( d e g r e e )

(o)

45

5.0

5.0

~9 ~.o b

1.0

3.0 15 30 65

( d e g r e e )

Fig. 5. Dependencies of ET orientation parameter o- on angle 0 and mixing parameter y for the cases: (a) blgdx2_y2 FeP MOs and (b) algdzz FeP MOs. The centre-to-centre separation R = 12 A; curves numbers 1, 2, 3, 4, 5 and 6 correspond to the magni- tudes of ,y2 = 1.0, 0.8, 0.6, 0.4, 0.2 and 0.0, respectively.

the interelectrode dielectric region ( z > - h ) . In this region we suppose the function ukll(r) to depend weakly on r and it can be taken as a constant.

Let us consider the specific case of the orientation dependence of S=S(O) only, when the molecule turns around the y axis which is perpendicular to the vector kll. In the initial position (turning angle 0 = 0) the molecular plane is parallel to the surface.

E.G. Petrov et al. / Chemical Physics 193 (1995) 237-253 247

Replacing the summation over j in Eq. (39) by the integration over angle coordinate q) and calculat- ing the overlap integrals we obtain a rather compli- cated analytical expression for S(O). Below we pre- sent only the formula for the case ktl = 0,

S = S ( 0 ; kl~=O)

=NK cos O(exp(-~h)Al lm(pl )

+ exp( - ah){azlm(P2)

+s in o a3[Im_~(p2 ) +Im+a(P2)]

+ s i n e 0 a4[Im_z(p2 ) + 2 Im(p2 )

+Im+z(p2)]}). (40)

Here I m are the modified Bessel functions of the arguments

Pa = - ffP sin 0, P2 = - 0/P sin 0, (41)

where p is the radius of cycle molecule, and

3 2 a 2 ( A 1 - (;2__ 0/2) 3,

2 ( ( - 30/) [ 1 - h ( ( - a ) ] 2h2 A e - -

30/- ~ 2h ) A3=p 0~i-i--a) 2 ( - a '

p2 A4

2 ( ¢ - a ) ' NK = 8uk,,( ,UD 0/3)1/2.

(42)

The value o-(0), connected with S(O; kll = 0) is shown in Fig. 6. The behaviour of o-(0) is deter- mined by two main factors. The first one is the rise of the function In(p) connected with the approach of the ring atoms to the surface with increasing 0. The second factor is the decrease of the magnitude of the function cos 0 connected with the inclination of 2pz AOs relative to the crystal surface. This factor causes an abrupt decrease in S(O; kll = 0) at 0 ~ 90 °.

In the case of small kll (I kll[ p << 1), the main difference in the qualitative behavior of IS(0) ] as compared with IS(0; k l l = 0 ) l is the nonvanishing (at 0 = 90 °) magnitude of [S(0) I which is propor-

1.0

Io 0.5 ~ / @.@ ,

@ 3O (degree)

Fig. 6. Plot of ET orientation parameter cr versus angle 0 of molecule orientation relative to the crystal surface (0 = 0 corre- sponds to parallel position of the molecule).

60 90

tional to I klll. Hence, the decrease of IS(0) [ at 0 ~ 90 ° is not so sharp.

5. ET through magnetically ordered chain

We intend to clear up how the magnetic ordering in a QMW affects the interelectrode ET in the present section. The research in this field is impor- tant both for the problems of biomolecular electron- ics (since there exist ferrosulphur-, molybdenium- and copper-containing clusters [70]) and for the prob- lems of molecular electronics (due to a synthesis of various types of magnetically ordered chains [71]). Here, we observe a regular DBA system with non- magnetic D and A units. The image forces from the electrodes are taken into account by introducing an average energy (Eq. (25)).

It is obvious that not only the magnetic ordering influences the efficiency of the quantum electron jump between the magnetic ions belonging to QMW units but also every electron jump acts on spin ordering. In order to find the states of an extra electron in the magnetically ordered chain, we note that an extra electron is an electron that is transferred from D to A. It has been shown [72] that the combined action produced by an external magnetic field, inner exchange and resonant interaction, gener- ating the quantum electron jump between magnetic

248 E.G. Petrov et al. / Chemical Physics 193 (1995) 237-253

ions, results in an inhomogeneous spin distribution, "spin cloud", formed around the electron. The elec- tron band width 41 /3 [ is, hence, essentially depen- dent on the magnetic field. The above mentioned results were obtained in the quasi-classical approxi- mation [73]. Below we use the same quasi-classical approximation in order to characterize the states of a spin system when the rate of donor-acceptor ET through a finite antiferromagnetically ordered chain is found (we consider the cases described by Eqs. (2) and (12)).

5.1. Physical model

The magnetic ordering in the bridge chain makes it necessary to take into account the spin degrees of freedom. Let us consider a simple situation when an extra (transferring) electron on the magnetic ion of QMW unit reduces necessarily the ion spin S by 1/2. If the chain has N magnetic ions, the ( 2 S - 1)(2S + 1) ~v- 1 spin states of the chain should be taken into consideration in the presence of an extra electron. Besides, it must be kept in mind that each of these states is an N-multiply degenerated state because of the possible location of an electron on each of the N ions. In the general case, it is impossi- ble to take rigorously into account all r = N ( 2 S - 1)(2S + 1) N- 1 states. For instance, the number of states equals already 48 even for the case of a bridge dimer (N = 2) at S = 5 /2 . Although it is still possi- ble to carry out the quantum mechanical calculations based on symmetry considerations [74] in this situa- tion, for N > 2 it is necessary to employ approximate calculation techniques. The most convenient one proves to be the quasi-classical approach. Within the framework of the latter the state of an electron in the chain is characterized by degenerated states (in view of allowance for an electron to have a localized state on each magnetic ion). The spin projection Mj of each magnetic ion j on its own spin quantization axis (SQA) is regarded as a spin quantum number. In the ground state, the value Mj is minimal and coin- cides either with - S or with - S ' = - S ÷ 1 / 2 (when the transferring electron occupies virtually a magnetic ion). All spin-excited states are character- ized, hence, by a deviation of SQA (or spins) from the direction where the ground state energy of the chain is minimal.

The difference in the parameters of exchange interactions I and I ~ between two QMW nearest- neighbour magnetic ions with ( I e) and without ( I ) an extra electron causes the inhomogeneous spin distribution near the ion with an extra electron. The electron coupling interaction resulting in the quan- tum jump of an electron between ions tends to make this spin distribution homogeneous. The analysis re- veals [72] that if the electronic coupling, L, is much larger than I and I e, the size of the spin cloud around an extra electron is about ten units of mag- netic lattice, and the spin distribution itself is close to the homogeneous one. Below we assume the spin cloud size to exceed the length of the bridge chain. Therefore, the SQA directions for the magnetic ions are identical SQA direction for separate magnetic sublattices. In application to a two-sublattice antifer- romagnetically ordered chain, this assumption means that SQA of the both magnetic sublattices are de- clined by an angle O symmetrically to the direction of an external field h 0. Such a homogeneous distri- bution of a SQA corresponds to the de Gennes model [75] used to describe the ground state of a magnetically ordered crystal in the presence of free electrons. The main contributions to the ion spin-de- pendent energy are given in this model by the Zee- man spin energy in a magnetic field, the exchange interaction between magnetic ions and the interior resonant interaction energy. We can show that in the case of a bridged chain, the de Gennes model is valid if the inequality

INS >> I~(S - 1 /2 ) (43)

is satisfied, and when the boundary exchange effect is neglected. The latter implies the edge magnetic ion of QMW to interact with a single nearest-neighbour ion only, and the internal magnetic ions with two. Since in our model, the exchange interaction is much smaller than the resonant one (I, I ¢ << L), a consid- eration of the exchange boundary effect is inessen- tial.

5.2. Analytic expression for tunnel current

To find the interelectrode currents, we restrict ourselves to the case of deep tunneling when the general formulae (2), (4), (20) or (12), (10), (15), (19) are valid if one only substitutes the energy gap

E.G. Petrov et al. / Chemical Physics 193 (1995) 237-253 249

AE 0 for AE (see Eqs. (28), (29)) and the energy difference E D - E F for / ~ n - Ev. Because of the magnetic ordering of QMW, the position of the center of a bridge band is now determined by the expression

Eo((9) =/~0 - IxsgSNho cos (9+ 2NS2I cos 20,

(44)

where ffS 0 is given by Eq. (26). The additional terms in Eq. (44), which depend on the declination angle (9, do not play any essential role in the energy gap AE but they are very important for finding the SQA directions. It is sufficient to determine the angle (9 as the angle corresponding to the minimum of the QMW band bottom. With a line of Ref. [72] we have

/3 =Ld~2/2_,/2(2(9) =L cos (9, (45)

where dSM,((9) is the Wigner function (overlap integral of spin functions). The parameter L coin- cides with the electronic coupling /3 when SQAs are parallel.

Minimizing the energy Ex (Eq. (13)) at A = 1 with respect to the angle (9 with account of Eqs. (44) and (45) shows that

~sghoSN + 2L c o s [ w / ( N + 1)] = ( 4 6 ) cos (9 8IS2 N

Below we choose the case of nonmagnetic D and A units when

( c o s ( ( 9 / 2 ) , m~= + 1 / 2 ,

=L j× l--~sin((9/2), m s = - 1 / 2 . (47)

Here Lj ( j = D, A) is the electronic coupling for the ferromagnetic phase ((9 = 0), and m s is the electron spin projection.

With the help of Eqs. (10), (12), (15), (19), (45) and (47) we obtain

L ]2(N-1) d 1/2 t'(9 "~14 J = C --~o ] [ +l/2msk ,] COs2(N-1)((9)'

(48)

where the value of C does not depend on the spin orientation. A similar expression (but with another value C) is valid for direct interelectrode tunneling.

The form of Eq. (48) is very convenient for a study of noncollinearity effects. Here, we discuss one of them.

5.3. Blocking of ET by magnetic field

Let j 0 be the current in the absence of a mag- netic field. The expression for j 0 follows from Eq. (48) at O = O 0 where the angle O 0 is given by Eq. (46) at h 0 = 0. The magnetic field influences the tunnel current by the change of the inclination angle O. We determine the magnetic field effect by the relation (30) which, in accordance with Eqs. (45), (47) and (48), takes the form

( ) , l + 2 m ~ c o s O 2[ cos (9 I (49)

~7= l + 2 m s c o s O 0 tco- -o-s~o] •

In Eq. (49), the quantities cos(9 and cos(9 0 are found from Eq. (46).

According to Eq. (46) the noncollinearity effect is especially pronounced at small values of the elec- tronic coupling L when the Zeeman energy contribu- tion exceeds that of the resonant interaction. Fig. 7 shows a growth of the ET rate by many orders when the magnetic field h 0 increases by several times. Tunneling for m s = - 1 / / 2 is blocked for fields h 0

8.0

6.0

O~ 4.0

2.0

o 0 ~ ! 0 20 50 40 50

/ ~sgSho / I

Fig. 7. Magnetic field effect at interelectrode ET for m E = + 1/2 and m s = - 1/2 (curves 1 and 2, respectively). Calculations in allowance for Eqs. (46), (48) with L = 0.01 eV, I = 0.5 X 10 .3 eV, S = 5/2, N = 5.

250 E.G. Petrov et aL / Chemical Physics 193 (1995) 237-253

> 4 0 I / t % g and the ET rate for m s = + 1 / 2 attains its constant value (corresponding to the tunneling rate in the ferromagnetic phase). Since an energy of about 1 cm - I ~ 1 0 - 4 e g corresponds to a magnetic field of 1 T, for chains where the exchange interac- tion between the nearest ions is of the order of 0.1-1 cm -1 (i.e. I ~ 10 -3 eV), the blocking of electron tunneling for m s = - 1 / 2 can be induced by a mag- netic field of the order of 1-10 T.

6. Concluding remarks

We have examined the role of some dynamic factors (energy shifts and electronic couplings) in interelectrode tunnel currents through a molecular wire. The results show that image forces from elec- trodes shifting electron energies in a DBA system can significantly decrease or increase the tunnel cur- rent through the molecular chain (Figs. 2, 3). It depends on the combination of several factors. The main of them are as follows: the kind of ET (direct ET or ET with a tentative electron localization within D and A units), the number of bridge units and the ratio between stationary and high-frequency permit- tivities E 0 and ~ . Interelectrode ET is very sensitive to the size of D(A) units (compare the upper and lower sets of curves in Fig. 2, and, also curves 1 and 2 with curves 3 and 4 in Fig. 3). This fact is completely explained by the proximity of these units to the electrode surfaces and, thus, a large image force effect.

The choice of energy difference, A E 0 = E 0 - E F ,

and electronic coupling between chain units, /3, is the principal question at providing numerical esti- mates. Physically, AE 0 is determined by the differ- ence of the electron affinity to the separate QMW unit and to the electrode, i.e. AE 0 = ~ b - K , where q~ = 4 -5 eV there is a tunnel barrier for the electron (vacuum work function), K is the electron affinity to the insulator that is formed by the interelectrode region. Experiment reveals [76] that, for instance, in the case of packed fat acid layers, AE 0 = 2 -3 eV. As to the electronic coupling, /3, its absolute value lies in the range of 0.01-1 eV (see footnote 1). In our estimates we take AE 0 = 3 eV and [ 13 I = 0.25 eV. Correspondingly, with allowance for Eq. (18) the energy position E D is restricted by the inequality

(14), and we have E D - E F = 2.2 eV (or E 0 - E D = 0.8 eV).

It is seen in Figs. 2, 3 that, similar to solid state electronics, in molecular electronics the image forces affect significantly ET if a bridged chain is embed- ded between the electrodes. In other words, at any quantum-mechanical calculation of a tunnel current mediated by a bridged chain, one must take image forces into account. If the image forces from the electrodes are ignored, the error in the theoretical estimation of the tunnel current may achieve several orders of magnitude.

The interelectrode tunnel current depends essen- tially on the mutual orientations and the type of MOs of adjacent QMW units. In this case, the current orientation effect is caused due to the change of electronic couplings. It is seen in Figs. 4, 5 that even a small change of porphyrin molecule orientations may cause large changes of overlap integrals. As a result, because of the participation of all QMW units in the ET process, we can have a giant integral orientation effect. If, for example, the bridge config- uration is changed by external factors (electric field, stretching or compression of the bridge) so that 0 D = 0 D, 0 A = 0 ; (or S(0 D)=S(0D) , S(0 A)~- S(O~ )) and the value orB(0) is in the range from 0.5 to 3.0, then, in accordance with Eq. (36), the tunnel current J , even at N = 6 , varies by a factor of 10-3-5.9 × 104. It is also seen (Fig. 6) that the plane edge units of a DBA system influence appre- ciably the boundary orientation effect in the vicinity of 90 ° .

Another type of orientation effect is connected with antiferromagnetic ordering in QMW. Considera- tion of the ET process from D to A through such a QMW has revealed the importance of noncollinear spin structure of a bridge in tunneling. Physically, the noncollinearity can be attained either due to delocalization of the transferring electron in a chain (delocalization is determined by the parameter L) or due to an external magnetic field ho, or because of a combined action of both factors (Eq. (46)). The role of the magnetic field is especially important since using the latter one can, in fact, control the ET rate within several orders of magnitude. Besides, the magnetic field induces the blocking of ET from D to A if the electron spin projection m s onto the direc- tion of the magnetic field is negative, and spin

E.G. Petrov et al. / Chemical Physics 193 (1995) 237-253 251

magnet ic ordering becomes ferromagnet ic ordering.

In other words, the Q M W is capable to transfer electrons selectively at a definite value of the mag- netic field. Note here that the selectivity effect in ET through a fer romagnet ic barr ier in the m e t a l - b a r - r i e r - v a c u u m system was discovered exper imental ly about 15 years ago [77]. Here, we have shown that

the selectivity of ET through an ant i ferromagnet i- cally ordered Q M W is manifes ted in the presence of

an external magnet ic field. The above men t ioned or ientat ion effects con-

nected with the br idge properties of Q M W can serve as a physical way for regulat ion of ET in different

kinds of molecular devices. (Some discussions on interelectrode tunnel conduct iv i ty of a s ingle D B A system can be found in Ref. [78] and references

therein). The results obta ined in the present work are based

on a phenomenolog ica l model of an interelectrode m e d i u m (by int roduct ion of permit t ivi t ies e 0 and ~ )

and semiphenomenolog ica l models of different quan- tum characteristics (energy posit ions, electronic cou-

pl ings, and wave funct ions of D B A units and elec- trodes). Such an approach is the s implest way to obta in reasonable analyt ic expressions for the ET rate constants and, hence, to compare the inf luence

of different dynamic factors on the interelectrode current.

Acknowledgement

This research was made possible in part by Grant No. U 4 U 0 0 0 from the Internat ional Science Founda- t ion and by Grant No. 2 . 3 / 6 5 7 from the Ukra in ian

Fund of Fundamen ta l Invest igat ions.

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