homogeneous and heterogeneous optical and thermal electron transfer

7/30/2019 Homogeneous and Heterogeneous optical and thermal electron transfer

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Electrc&imica Acta. 1968. Vol. 13, pp. 1005 to 1023. Pergamon Press. Printed in Northern Ireland

HOMOGENEOUS AND HETEROGENEOUS OPTICALAND THERMAL ELECTRON TRANSFER*

N. S. HUSHDepartment of Inorganic Chemistry, The University, Bristol 8, England

Abstract-A unified theory of optical and thermal outer-sphere electron-transfer processes is out-lined, in which the equations are obtained as special cases of general expressions for radiative andradiationless transition probabilities. For a particular donor-acceptor ion system, this leads tocorrelations between the rate of homogeneous thermal electron exchange and the frequency andbandwidth of the corresponding optical intervalence transfer absorption. These in turn can becorrelated with thermal and optical transfer at a metal/solution interface. The relationship betweenthe transmission coefficient in adiabatic thermal exchange and the transition probability in thecorresponding optical transfer is discussed, and a number of predictions are made for the opticaltransfer of electrons between a molecule or ion and an electrode, which has so far not been observed.R&m&-Esquisse dune thBorie des processus de transfert des Hectrons des sphkres extemes optiqueet thermique: les Equations sont obtenues comme cas spkiaux dexpressions g&&ales propres auxprobabilitBs de transitions rayonnantes et non rayonnantes. Pour un syst&ne particulier donneur-accepteur dion, ceci conduit d des corr&lations entre la vitesse dkhange thermique homo&ne de18lectron et la frequence ainsi que la largeur de la bande dabsorption propre % intervalence optiquecorrespondante. Ceci & son tour peut ttre coordonne avec le transfert thermique et optique ZL neinterface m&al/solution. Discussion du rapport existant entre le coefficient de transmission duntchange thermique adiabatique et la probabiIit6 de transition du transfert optique correspondant.Des prbdictions sont faites pour le transfert optique des electrons entre une mol&ule ou un ion et uneelectrode, non observe jusquici.Zusammenfassung-Man entwickelt eine Theorie des optischen und thermischen Elektronentibergangsin Aussenschalen. Die erhaltenen Gleichungen stellen Spezialf;ille von allgemeinen Gesetzeny d&Wahrscheinlichkeitsfunktionen fiir Uberglnge mit und ohne Strahlung dar. Fiir ein speziellesDonor-/Akzeptorionensvstem ergeben sich Beziehungen zwischen der Geschwindigkeit des homo-genen therm&hen Elel&ronena%tausches und der Frequenz und Bandbreite der-entsprechendenoptischen Absorption fiir den Intervalenziibergang. Diese ihrerseits kiinnen mit den thermischen undoptischen Uberggngen an einer MetalI/LGsungsgrenzfl&che in Verbindung gebracht werden. DieBeziehung zwischen dem Transmissionskoeffizienden des adiabatischen thermischen Austausches undder tfbergangswahrscheinlichkeit im entsprechenden optischen Vorgang wird diskutiert. Man machteine Reihe von Vorhersagen fiir den optischen Elektroneniibergang zwischen einem Ion oder Molekiilund einer Elektrode, welcher bis heute noch nicht festgestellt werden konnte.

1. INTRODUCTIONELECTRON transitions may in general proceed either by radiative or radiationlessmechanisms. A familiar example of a combination of these processes is found inmolecular phosphorescence. A molecule is photoexcited to the lowest excitedsinglet state l rl * and subsequently undergoes a radiationless conversion to the lowestexcited triplet level 3r1*, after which the system returns to the ground state withemission of a quantum of energy E(31l*) - E(ll?,). In this example the excitedelectron is essentially localized, ie the transition dipole moment is intramolecular.In the solid state, it is also possible not only to excite an impurity hole or electron,such as an electron trapped at an F-centre, but also to ionize it to a free state in theconduction band. This can be accomplished either optically or thermally. The

* Presented at the 18th meeting of CITCE, Elmau, April 1967; manuscript received 23 August,1967.1005


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1006 N. S. HUSHthermal excitation or ionization processes can be regarded as many-phonon jumps,and for many years the mechanism of these radiationless processes was obscure.There was no doubt that (together with the reverse dissipation of light energy onde-excitation) they resulted from coupling of electronic and vibrational motion, butthe detailed reasons for this were unknown. According to an early theory of Frenkel,slight differences in the lattice vibrations in electronically excited states from thosein the ground state are the cause of the many-phonon emission when the electronreturns to the ground state. Subsequent work made it clear that differences of latticeequilibrium positions in the two states were often more important. There is anextensive literature on this subject, both for processes in insulators on semiconductorsand in ionic crystal~.~Chemical or electrochemical electron-exchange processes can also be consideredfrom this point of view. In a thermally-activated electron transfer between two ionsin a polar medium at the high-temperature limit, an electron can be considered to hopfrom one nearly localized site to an adjacent one (or to the conduction band of ametal) by an adiabatic mechanism. There is usually a considerable dispersion ofequilibriumconfigurations and to a lesser extent of vibration frequencies accompanyingsuch a process. These can be interpreted, in the high-temperature limit, in terms ofabsolute reaction-rate theory. Little attention has yet been paid to the correspondingoptical transfer processes *. Here we shall firstly briefly outline the theory of theconnexion between homogeneous optical and thermal electron transfers. A numberof aspects of these will be discussed, and finally the types of optical transfer whichcan be predicted to occur between an interfacial ion or molecule and a metal will beoutlined. It is possible to make reasonably definite predictions about the frequenciesand characteristics of the absorption envelope of a variety of heterogeneous transitionsof this kind, none of which have yet been observed.We shall assume localized (or near-localized) electronic functions in the subsequentdiscussion. For adiabatic processes, there are many advantages in the use of de-localized (M.O.) functions (c$ ref. 14). However, this would complicate the develop-ment of a theory not restricted to the high-temperature limit.

2. ENERGY-DISTRIBUTION FUNCTIONSThe essential features of the radiative and non-radiative transition processes willnow be briefly outlined. The discussion is not restricted to electron-transfer processes;the solutions of the equations for these involve only a particular choice of parametervalues.In recent years, the properties of electrons in polar media have been discussedmainly in terms of polaron theory,2 which originated with the work of Landaus andFr61ich.6 This provides a general theoretical framework applicable to weak, inter-mediate and strong coupling of the motion of the electron with that of the lattice.The situation in which we are interested is that in which electrons are essentially

localized in the fields of individual ions, so that the system electron + accompanyinglattice vibration constitutes a small polaron. For our purposes here, the theoreticalapproach of Kubo et a17*8 cf also Huang and Rhyse) is particularly appropriate,even though it is less general in scope. In particular, the use of co-ordinate andl The existing literature on the colours and absorption spectra of mixed-valence compoundsand solutions has been reviewed..


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Homogeneousand heterogeneousoptical and thermal electron transfer 1007momentum variables rather than creation and annihilation operators for the polariza-tion field results in an intuitively simpler representation of the physical problem.We consider the Hamiltonian H of a system comprising an electron localizedinitially in a discrete bound state at a site a in a medium L. The medium may be polaror non-polar. In general this can be written asH=H,+He+Hrwhere H L and H e are the sum of kinetic and potential energy operators for L and forthe electron, respectively, and H I represents the interaction energy of the electron withL. The analytical form of these operators naturally depends critically on the nature ofL and in particular on whether L is an insulating or a conducting medium. Writingnuclear and electronic co-ordinates as Q and r respectively, the energy El(Q) forthe lfh electronic state with fixed Q is defined by

[He + Hr(r, Q)ld,(r, Q>= E,(QM,(r, Q). (2)The wavefunction of the system L + electron can then be written asyro(r, Q) = +,(r, QKv(Qh (3)where I and u are electronic and vibrational quantum numbers. This expresses theBorn-Oppenheimer condition, which is usually a good approximation for stationarystates. In adiabatic theories of thermal electron-transfer reactions it is assumed thatdepartures from electronic/vibrational separability are so small that they will makelittle difference to the activation free energy, but are large enough to permit the trans-mission coefficient to be high.According to (2) and (3), the nuclear vibrational energy E,, in the electronic state1 is given by

[HL + E~(QL(Q) = J%,,(Q). (4)However, it is easily shown that HYJr, Q) is not exactly equal to E,,Y,, if theinteraction potential H I is non-vanishing. In fact,

mlc(r, Q) = GY,, + [HL~,,+, d&,L,l= E,Y,, + HY,,. (5)With time-dependent perturbation theory, the last term in (5) leads to a finite value forthe transition probability Wlr,.. or the thermal excitation 1 + I. Summed over ailvibrational levels, this is

with E + AE > AE > E. The Boltzmann factor of the initial state (i) is M-~. orthe corresponding optical transition I -+ l the optical absorption constant k(v) forfrequency v isk(v) = (8&Vi/3nc) eEz 2 ~~l(l~(MI1u j2.v* v* (7)

In (7), Ni is the concentration of electrons per unit volume, c and n are the velocityof light in vacua and the refraction index of L. M is the electric dipole momentoperator eR which may be a function of Q. This is assumed to dominate the opticalprobability.


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1008 N. S. HUSHIn the high-temperature limit, the general features of the distribution functions are

as follows.,s For an arbitrary adiabatic potential U(Q), such that the kinetic energycan be expressed in unit quadratic form but not otherwise specified, the thermaldistribution function for M #f(Q) is

:= IWWv)12 Jexp - {BU(x)}G{(E - U(x)) + U(x)} dxJ exp - {@U(x)) dx (84where x = *(Q + a> and p = (kT)-l.

This shows that the most probable optical transition is a vertical transitionobeying the Franck-Condon principle. On the other hand, the corresponding distribu-tion function for the thermal process is proportional to W, where

w, = .f exp @u(x)}~{~(x) u(x)>dx.Sexp @Wx)} dx @b)From this it is clear that the radiationless path (at high temperature) proceeds throughthe point of intersection of the two adiabatic surfaces. This agrees with the expressionobtained by Marcus lo for the thermal electron-transfer case.It is also to be noted that there is a close connexion between the distributionfunctions for the thermal transfer and for the optical transfer in the zero-frequencylimit (E = 0). This in fact forms the basis of the detailed interconnexion of theoptical and thermal excitations.The connexion between homogeneous and heterogeneous thermal electron-transfer reactions and other types of radiationless processes was first pointed out byLevich and co-workers.lr Their results will be discussed after the general approachis outlined. 3. THERMAL ELECTRON TRANSFER

The considerations outlined above apply generally to electron-excitation processes.We now obtain more detailed expressions, and consider the special case of electrontransfer between ions in an ionizing medium or between an ion and an electrodeand discuss the thermally-activated process in the high-temperature limit.The adiabatic theories of MarcusI and Hush, although different in approach, arein agreement* (apart from points of detail) for outer-sphere transfers, which are theonly types to be discussed here. The non-equilibrium statistical mechanics of thesesystems has been explored in considerable detail by Marcus. We can reproduce themain features of these results by considering the general equations for the probabilityof a radiationless transition. In this way it is possible to see clearly the connexion

* This is true for homogeneous reactions. For heterogeneous reactions, the role of the imageforce is not entirely clear in solutions of finite concentration. A usual experimental condition isthat in which the electrolyte concentration is sufficiently high for the diffuse double layer potentialto be very nearly suppressed. If the reactant ion is located at the outer Helmholtz plane in such asolution, the inner potential at the reactant plane is the same as that in the interior of the solution,and hence the ion is not subject to an image force as it would be in an infinitely dilute solution.For this reason, the image contribution is neglected in calculating an activation free energy by theauthors method, whereas Marcus includes it. Marcus predicted activation energies are thereforelower. It is probable that these are limiting approximations, and a detailed analysis of the effect ofa discrete double layer is needed.


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Homogeneous and heterogeneousoptical and thermal electron transfer 1009with the corresponding optical process. To simplify the algebra at this point, weassume throughout this section that all vibration frequencies are the same in initialand final states of the system.The frequency tensor associated with the vibrations (assumed to be harmonic)of all atoms of the medium L is designed Q. The displacement vector B = Qof -Qt is the vector difference of equilibrium positions of atoms in the final and initialstates of the system.? Then, in the high-temperature limit, the activation energyE* of the thermal transition (between adjacent sites for electron transfer) with over-allenergy E,, is (with mass normalized to unity)

E* = (UQA + Eo)~2AR2A

For a solution electron-transfer process, the total activation energy is E* plusthe energy APE of bringing the ions together in solution from the initial state (i) toform the close-contact precursor state (P).~On the continuum approximation, in which the medium L is regarded as a con-tinuous dielectric with static and optical dielectric constants K and ~~ the generalexpression for E* is

where8E* = (%+ EoY

4x (10)

In (lo), E,. and EIn are the electric fields due to the charge distributions & and 4&srespectively. An improvement to this approximation can be made by regarding themedium as continuous in an outer volume V and as consisting of a discrete set ofoscillators in the complementary inner volume V, with no correlation between thenuclear motions in the two sub-volumes. Writingand Xinner

= +A&PA,

Xo,,ter =; (; - ~)~& - &12dv,we have

E* = [xinner + Xouter + Eo124[Xinner + Xouterl

(12)This can also be written

E* = I:[x + E,,],where

1: = *[x + Eollx (13)and x is now Xouter+ xtnner.

We are considering here a process which in the outer-sphere electron-transfercase corresponds to transfer between ions which have been brought together to thet An early and illuminating discussion of the orientation of normal coordinates in an elec-tronic transition was given by Duschinskyls. I am indebted to Professor M. Kasha for providing atranslation of this paper.


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1010 N. S. HUSHclose-contact internuclear separation. The over-all activation energy, starting withions at infinite separation, is that given by (13) plus the work of bringing the ionstogether, and E,, is the over-all energy change at the close-contact separation. As weare assuming here that the frequency tensor Q and IR are identical, one can writethe total free energy of activation for the outer-sphere electron transfer as

A@+ = ~APG + (x+4yG)2 UWwhere i, p and q refer to the initial, precursor and successor states. The formalsimilarity of this equation to previous expressions for adiabatic electron transfer isapparent ; in particular, it is formally identical with the expression of Marcus:i2in Marcus terminology, APG = w, x = 3, and PAqG = AF,). The expressionof Levich and co-worker@ is based on the continuum approximation (neglectingxi,& and is formally similar to (13a) with the appropriate approximation of (10)for xouter. These authors also obtain a low-temperature limiting equation which isessentially identical with that of Kubo and Toyozawa* for the general low-temperaturelimit for a radiationless transition. The main point we wish to stress here is that allequations up to (13) are general equations for radiationless transitions of the typespecified earlier in this section and contain no special features arising from the factthat they are applicable to the electron-transfer process. These special features comein through the particular approximations subsequently introduced in interpretingthe x terms.

We have discussed the leading features of the high-temperature approximation,in which the radiationless transition is regarded as proceeding at the intersectionpoint? of two adiabatic surfaces. If the temperature is lowered, or the vibrationfrequencies become very high, the activation energy will decrease until it is finallyequal to the over-all energy E,,. At this point, the reaction proceeds entirely byvibrational tunnelling to the ground state of the final state system (or successor state inthe case of electron transfer) (cfFig. 1). This is closely connected with the narrowingof the absorption band width of the corresponding optical process when the tempera-ture is lowered ($4). For a thermoneutral process (E, = 0) we can regard E* asarising from the dispersion of equilibrium distances if, as here, the difference offrequency tensors is neglected. This would formally be regarded as a contributionto the free energy of activation (PA%) in analysis of a rate constant. This will varyconsiderably with temperature at lower temperatures, in spite of the assumption ofequality of the frequency tensors. Considering the maximum value of E in theLaplace transform of the energy dispersion exp (-_BE) due to d*, one obtains for theabove conditions

E = kTA tanh2tiand (14a)

7 Strictly speaking, at the lowest intersection point. There may be more than one such point ifthe frequency tensors are different in initial and final states, in which case the equation for B, (13),becomes a quadratic (cf (146)).


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Homogeneous and heterogeneous optical and thermal electron transfer 1011Both are identical (setting E = E*) with (9) for E, = 0 at high temperature, but notat lower temperatures. This points to the complexity of the temperature-dependenceof electron-transfer rates if the conditions are such that either the temperature islow or some vibration frequencies are very high. This may be called the intermediateregion, in contrast to the high and very low temperature regions, Investigation of theintermediate region would be fruitful for the theory of both homogeneous andheterogeneous reactions. It is worth noting that Huang and Rhyse have obtaineda general expression for the rate of a radiationless transition that covers the completetemperature range, for the special case of a single frequency, as is assumed, forexample, in the pure continuum approximation. This gives a useful guide to thebehaviour of the rate constant in the intermediate region, for the electron-transfercase.It has so far been assumed in this section that R = U. If this restriction isrelaxed, it is found that the free energy of activation for the radiationless transitionis given by the maximum value of the expression

@(l - 2)AE2{(1 - @X2 + izQ2}-Q2A+@Ztr In {P2((1 - A)S22 KI2)) + 1E,,. (14b)

This is obtained from the Laplace transform of the transition probability,8 with 1as a dimensionless variable. With a number of simplifications, whose importancecannot be discussed here, this can be brought to the approximate form of (13a)with the term APG added for the electron-transfer case. The term xinner is now givenbY

Xinner = A$Z(Q2 + c-J2)-R2~, W)This last expression is identical with that of Marcus l2 for the electron-transfer case.In actual calculations of inner-shell terms ,12*14t is assumed for further simplicitythat the non-diagonal frequency elements are zero.

For metal/solution electron exchange, a similar approach yields(xinner) electrode = +(xinner) homogeneous,(xouter) electrode = &out,& homogeneous

(image potential included, with ion-metal distance equal to half ion-ion distance inhomogeneous case),

(xouter) electrode = (~~~3 homogeneous(image potential neglected).If the image potential term is included, we have simply

Xelectrode =+ Xhomogeneous.If the image potential is neglected, we obtain

%electrode = khomogeneous + 3(XouterJ homogeneous,giving rise to a larger activation energy.

4. HOMOGENEOUS OPTICAL ELECTRON TRANSFER(INTERVALENCE TRANSFER)

The corresponding optical electron-transfer process is symbolized by the verticaltransition in Fig. 1. For the special case of zero frequency difference between initial


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1012 N. S. HUSHand final states, it is evident that the maximum absorption occurs at a frequency

kn,X = E,, + &AQ2A (15)There is thus a simple relationship between the optical frequency maximum and theactivation energy E* of the thermal process, viz,

React icnco-ordinateFIG. 1. Schematic representation of intersection of adiabatic surfaces for electronicstates I and I, with adiabatic potentials U and U and over-all energy difference Eo .Vertical transition occurs with most probable frequence hvmax. Thermal activationenergy in high-temperature limit is E*. At low temperature, the probability of nucleartunnelling from point A increases for the thermal transfer.

Electron transfer between pairs of adjacent ions in which the electron is nearlylocalized on one ion or the other in initial and final states (intervalence transfer)exhibits some unusual features. Let us denote by p an electron-phonon couplingconstant, defined as

p = ~AC12A/Eo.With this parameter, the frequency maximum is

(17)

&n,, = E,(l + P>. (18)For many transitions, p is small; this is particularly so in ionization of trappedelectrons to the conduction band of a semiconductor, as the nearly free electron willhave little effect on the lattice equilibrium configurations. It is larger in F-centreexcitations, where it may be of the order of 0.5. In intervalence transfer absorption,p is usually very large. For the special case of thermoneutral transfers in which E,,(arising from splitting of the levels of adjacent ions) will normally be a very smallfraction of an eV, p is nearly infinite. Intervalence transfer is thus a many-phononjump with an unusually large electron-phonon coupling constant. If it were not for


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Homogeneousand heterogeneousoptical and thermal electron transfer 1013this effect, an optical transition such as

Fez+ + Few -+ Fe3+ + Fez+would occur at almost zero frequency. Experimentally, it is found that intervalencetransfer absorption usually occurs in the visible region of the spectrum, althoughexamples of ir and uv bands are also known. In Table 1, calculated frequencies arelisted for outer-sphere transfer between pairs of adjacent aquo transition ions at theclose-contact separation. These are obtained from the calculated values of E*.14 In

TABLE 1. CALCULATEDAC~VATIONENERGIES(E*)ORTHERMALOUTER-SPHERRELECXXONTRANSFERBETWEEN IONS IN AQUEOUS SOLUTION" AND CALCULATED PEAK FRBQUENCIES (h,,,) AND BANDHALF-WIDTHS (Al,*, 300K) FOR THE CORRESPONDING HOMOGENEOUS OPTICAL INTBRVALENCE

TRANSFER.

Ion systemTja+_Tiz+va+-ve+Cr*+Xr+Mns+-Mnz+Fes+-Fee+Co*+-Co~+Pu4+-Pu*+TI*+-TI+

E* h l , * A1/rKcal/mol cm-l cm-l9.5 13,300 55009.0 12,300 530019.5 27,300 so0013.4 18,800 660012.3 17,200 630011.5 16,100 61001@7 15,ooo 590017.6* 29,400 7500

* This is the experimental value; the value of E. for the process Tl*+ + Tl+ -+ 2T18+ has pre-viously been calculated~ from this to 13.7 Kcal. This value of E, , s used in calculating hv,,,.

practice it is difficult to obtain sufficiently high concentrations of adjacent outer-sphere ion pairs of these elements for these predictions to be directly verified.However, outer-sphere transitions for a number of non-transitional ions are known,3*4such as Sb(III)C1,3--Sb(V)Cl,- and Pb(IV)Cl,4--Pb(II)Cl,2- in the solid state.Most intervalence transfer bands for transition elements have been observed ininner-sphere environments.3*4However, the frequencies are not far from those calculated for the simpler case;Fe(III)-Fe(H) transfers, for example, usually give rise to absorption in the region16-18 Kk in a number of dissimilar environments. 3*4 Further details of the transfercan be obtained from the shape of the absorption envelope.

The band envelope will usually have a somewhat more complicated shape than issuggested by the simple expression of (15a). This is because of differences in frequencytensors and the neglect of higher moments than the second of the energy distributionfunction.If F(E) is the distribution function, the moments are defined by

r E,F,(E) dEpFCn = J-,"s ,(E) dE--m (19)

3


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1014 N. S. HUSH

The first three moments for the optical distribution function are*,ul = aAlR2A + t tr(CF coth (@XY/2)r) + E,,,

pa = ;b*ap coth (j?tiCI/2))a2A+ T tr ({W coth (~KY/2)I}2) ,coth (/%Q/2) IX-Y-r coth (/3KY/2)) SY2A

+ $ tr (U-l coth (/3AQ/2)I2)+ p tr ({a-l coth (@KY/2)r}3). (20)

These are measures of the peak frequency, the band width and the skewness ofthe absorption envelope, respectively. Higher moments cannot be ignored in anexact treatment-for example, the fourth moment measures the kurtosis (flatteningof the peak and increase of intensity in the wings relative to a pure Gaussian distribu-tion), which is clearly visible on a number of experimental bands. However, in firstapproximation it is adequate to consider only the second moment. In this approxima-tion, with neglect of variation of frequency tensors (ie, I = 0) the band width canbe calculated from (20) if the frequency maximum is known. In the high-temperaturelimit, this is proportional to T- 2. Estimates of the band half-widths obtained in thisway for typical transfers are also listed in Table 1. These agree qualitatively with theobserved fact that homogeneous intervalence transfer bands are characteristicallybroad owing to the large dispersion of equilibrium distances.

It is useful to know, for the purpose of practical band analysis, the conditionsunder which it is permissible to make the approximation coth x M x-l in the momentexpressions. In Table 2, values of the quantity

gr = (Z)coth(&)for a single frequency u) at 293K are listed. Also, the ratio of square of band half-widths at 293 and 90, {A1,2(293)/A1,2(90)}2,or a transition involving only a singlefrequency arelisted. (This conforms to the continuum model with no localized modes).TABLE 2. RATIOOF HALF-BANDWJDTHSAT 293K AND 90K~s FUNCXIONOF~U$%T~(T~ = 293K)

AND g-FACTOR AT 293K, ON SINGLE-FREQUENCYMODEL(SEETEXT)

hw/2kT, coth (hw/2kT,)10.90.80.70.60.50.40.30.20.1

1.3130 1.313 1.3131.3961 1.387 1.257l-5059 1.490 I.2051.6546 1.620 1.1581.8620 1.795 1.1172.1640 2.005 1.0822.6319 2.260 1.0533.4327 2.575 1.0305.0665 2.830 1.01510.0333 3.140* 1@03t* limiting value 3.256t limiting value 1


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Homogeneous and heterogeneous optical and thermal electron transfer 1015A rough idea of the average frequency involved can be obtained from experimentalmeasurements of the half-widths at these two temperatures. Then, if x is calculatedfrom the band width using the high-temperature expression for the moment pz, afirst-order corrected value is

XCOlT(T) X(T)/&PThis expression is of use in the absence of detailed information about the frequencytensors.Useful correlations can be made between the activation energy of the thermalprocess and the frequency maximum and band-width of a corresponding opticalprocess. For identical conditions, these correlations are at present limited to solid-state transfers, where the concentration of pair sites is high. In this case the activa-tion energy of the thermal exchange is in principle obtainable from the temperaturecoefficient of semiconduction, assuming this to proceed by a phonon-assisted hoppingmechanism.

Direct evidence for the operation of this mechanism in a particular case is reallynecessary before definite conclusions can be drawn, however. It should be noted thatrecent work on the Hall effecP and Seebeck coefficient9 of alkali-metal-doped NiOhave cast some doubt on the interpretation of the semiconduction process as NP+-Niw thermal hopping, although the situation is not completely resolved. A furtherpoint of interest here is that for an isolated two-site system, thermal hopping will beexpected to lead to a Debye loss, which could be obtained by measurement of thecomplex dielectric constant. There is some evidence for such an effect in a-Fe,O, atlow temperatures from the work of Volger.lt5. THERMAL TRANSMISSION COEFFICIENT AND OPTICAL

ABSORPTION PROBABILITYWe shall also expect to find a correlation between the transmission coefficient of athermal process and the dipole strength of the corresponding optical transition. Thismay be useful in estimating transmission coefficients, as the optical intensities are inprinciple easily measurable.The transitions we discuss here are those in which the degeneracy of the two statesat the intersection point of the adiabatic surfaces is removed by resonance interaction,so that the separation of upper and lower surfaces at this point is of the form

&I = 2((cl H lb) - (a I b) @l-H la>>1-(alb)2 P-9

where u, b represent orbitals centered on adjacent sites and H is an appropriateone-electron perturbation.In the high-temperature limit, the transmission coefficient of the thermal transitionis

K = 1 - exp 4rr2&2- Ihv Igrad (U - V),lj (21)where v is the average thermal velocity and the quantity grad (U - U), is measuredat the intersection point. In order to estimate this, the reaction co-ordinate must

t See note added in proof p. 1023.


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1016 N. S. Huskbe known. This can either be (i) a displacement in the dielectric, or (ii) an inner-shell normal co-ordinate. Estimates of both can be obtained.

On the continuum model, with mass normalixed to unity, we have-lgrad (u - u,I = ~42~~~ = 2/2x. (22)If an inner shell mode is taken as the transfer co-ordinate, the frequency and thegradient of (U - U), can be calculated* from the vibrational potential functions.lan14The problem common to either choice is calculation of the energy splitting 2s.It is useful to have the magnitude of this for typical cases. Consider an outer-spheretransfer between two six-co-ordinated aquo-ions, with a transitional metal ion at thecentre of each octahedron. It is assumed that the splitting arises through overlapof the oxygen 2p, orbitals of neighbouring ions in close contact. Two cases can bedistinguished. If the transferring orbital is of type dy*(ie, eB*) in Oh symmetry(eg C!?+-Cr2+), the most favourable configuration for overlap is apical contact,Fig. 2. On the other hand, for d&* (ie t,,*) transfer (eg Fe3+-Fes+), a skew config-uration is more favourable.

Y

1_ x

(a) (b)FIG. 2. u-overlap of octahedrally co-ordinated transition ions in apical contact (seetext for orbital notation).

This arises because the wave-functions of the transferring electrons are of the formIy* > = a d,,_,, - &(l - ~G)l~(c~4 o, + 0, + u4),[.s*> = /3 d,,_,, - 6(1 - B2Y2(r1 + 2 + 773 + n44), (23)

where a(1 - a2>, a(1 - /3) measure the delocalization of d-electron density on toeach of the appropriate oxygen (Tor r oxygen orbitals, respectively. Estimates of theseamplitudes are available from electron-resonance experiments. In each case, thecentral problem is calculation of the interaction energy a of two oxygen p-functions

* If we consider a volume integrated coefficient, ie K = J K(r)4r(r)L dr,we obtain, on expanding the exponential in (22),K =$q~e+ ;;j$: _ E/)2 du)lllR1dr

This is the expression given originally by Levich and Dogonadzel However, this procedure isinvalid, as an average integral total rate should be considered. An expression similar to (21), withexpanded exponential and a slightly different numerical factor has been derived8 for the generalradiationless transition with resonance interaction of the degenerate states.


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Homogeneous and heterogeneous optical and thermal electron transferI .

1017

-4-

-5 0 2 4 iid %mr

FIG. 3. Separation of energy surfaces (2~) as a function of distance d,, of adjacentoxygen atoms for a-overlap of octahedrally co-ordinated transition ions with a * = @7.This value is appropriate to the Cr*+-CP+ couple. The pointp indicates the close-con-tact separation, at which the distance d m s equal to the diameter of a water molecule.at a distance A,, overlapping as in Fig. 2. The energy E is then

E = &(I - c?)& (dy * transfer).E = $(l - @2)& (d&* transfer).In a one-electron approach, E is given by

eN = Z,,,((u jr,-l lb) - (~~b)(alr,-~~a))(1 - (~(b)~)-~, (24)where Z,,, is the effective nuclear charge of oxygen. Taking Z,,, = 4.55 and usingSlater functions, E is easily computed as a function of the distance d,. In Fig. 3,a plot of log (2.5) (ie the energy separation of the surfaces at the intersection point)is shown for the particular case of a = 0.7, which is that appropriate to Cr(I-I,0),3f.The close contact O-O separation can be taken to be twice the radius of a watermolecule, ie 2.8 A. At this separation 2~ is approximately O-046 eV. This is in factof the order of magnitude that has been implicitly assumed in previous work. Theenergy separation varies very sharply with distance, and is less dependent on theionic nature of the bond (as measured by (1 - a2) or (1 - p>l).This connects with the oscillator strength of the corresponding optical process. Thefirst-order electronic wave-functions for the states &, and & are?

t The interaction energy E is assumed to be identical in precursor (P., and transition states in thisestimate.


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1018 N . S . HUSHTak ing hvm~ = (U ~ ' - U ' )p , a nd neg l ec ti ng t he ove r l ap i n t eg ra l be tween ff~ .and 4~z., the t rans i t ion m om en t o f the o pt ica l in te rva lence t ransfer i s thus

whe re R i s t he i n t e r ion i c s ep a ra t i on an d g i s t he nu m ber o f equ iva l en t t rans f e r r inge l ec t rons pe r c en t r e [g- -- - 1 fo r bo th FeS+ -Fe~+ and CrS +-C r~] . Th e osc i l la tors t r e n g t h f i sf = 4 .32 10 -9J r dv

= 1.085 10-Sh~,m~x IMI~ /eL (27)wh ere h*'ma i s expressed in wa ve-n um bers an d ]Ml/e s i n A n g s t r o m s 3 F o r a b a n dof G auss i an shape , t he exc i t a t ion coe f f ic i en t o f the pea k m ax im um i s

f 1 0 9emax -- 4.6Al/2 , (28)whe re emax i s the molar decadic ex t inc t ion coef f ic ien t , wi th the band ha l f -width At /2in wavenum ber s . Thu s , f o r t he Cra+ -Cr a+, aquo - ion op t i c a l t r ans f e r , u s ing t hetheo re t i c a l va lue s in T ab l e 1 , we ca l cu l a t e

em~x = 11.The expe r imen ta l va lue s so f a r o b t a ined fo r t he o sc i l la to r s t reng ths o f in t e rva lence

t r ans f e r band s sugges t t ha t t he ex t en t o f e l ec tron de loca l iz a t i on i n the g ro und s t a t ei s u sua l l y qu i t e sma l l, and com parab l e w i th t ha t c a l cu l a t ed abov e fo r the C r (H~O)es+ -C r ( H 2 0 ) 6~" c lo se - con t ac t pa i r . Fo r exam ple , t he f -va lue fo r t he T P+ (C1- ) e -T i~ - (CI - )xin te rva lence t ransfe r (20"41 KK ) in conc ent ra ted HC1 i s es t im ated as 0 .0026, co r re -spon d ing t o a va lue o f ab ou t 0 .11 h fo r the d ipo l e s t r eng th [Ml / e ./ .i s Th i s imp l i e s t ha t{ e ' / ( U ~ - - U')}2 , (25) , i s 0"5 10 8 for th i s ch lor ine-b r idged inner sphere TiS+-Ti ~-c o m p l e x , i e tha t the e lec t ro n i s 99 .95 ~o loca l ized on Ti ~- orb i ta l s and 0 .05 ~o de loca l -i z e d o n t o T ia+ i n t he g roun d s t a te . A s imi l ar c a l cu l a t ion f rom the i n t ens i ty o f t he

Fe(III)t2gaeg~(6S) + Fe(II) t2g6 (xS) -,.F e ( I I ) t z o 4 e ~ 2 ( ~ D ) + Fe(IlI)tzg6(2I)

t rans i t ion in Pruss ian Blue (14 .1 kk) shows tha t the Fe( I I ) t~g e lec t rons a re 99 .03 percen t a s soc i a t ed w i th the d on or Fe ( I I ) i on i n t he c ry s ta l . 4a9 In t h i s c a se , e ach do no r i oni s su r ro und ed by s ix equ id i s t an t a ccep to r s , r e su lt i ng i n a r el a ti ve ly l a rge t o t a l de loca l i z a -t ion . Als o the fac tor g i s 6 for the low -spin Fe( II )t2g 6 conf ig ura t ion , so tha t the f ina li n t ens it y o f the b and ( sum med ove r a l l exc i t on s t a te s ) is comp ara t i ve ly h igh .

In t he t heo re t i c a l e s tima te o f op t ic a l t r ans f e r o sc i l la to r s t reng th w e have no tcons ide red t he po s s ib l e Q-dep endence o f t he t r ans i ti on m om en t . Th i s w i ll beimp or t an t f o r t r ans f e r t o som e exc i t ed s t a te s , i n wh ich t he s t a t i c m om en t van i shes bysymm et ry . A l so , o the r sou rce s o f i n t ensi t y ( e g i n t ens it y bo r ro win g f rom ne ighbour ingcha rge - t r ans f e r s t at e s ) m ay be p r e sen t i n pa r t i cu l a r sy s t ems . Ho we ve r , i t i s c l e a rtha t f o r sy s t ems in wh ich i t is pe rmi s s ib le t o a s sum e tha t t he a bov e mod e l is adequa t e ,a c lo se r e la t i onsh ip ex i st s be tween t he t r ansmis s ion coe f fi c ien t o f t he r ad i a ti on l e s st r ans f e r and t he o sc i l l a to r st r eng th o f t he co r r e spon d ing op t i c a l tr ans f e r.


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Homogeneous and heterogeneous optical and thermal electron transfer 10196. OPTICAL ELECTRON TRANSFER BETWEEN AN INTERFACIALION OR MOLECULE AND AN ELECTRODE

These considerations are easily extended to the case of optical transfer betweena species present in the electrode interface and an electrode. The type of experimentenvisaged here is one in which a transparent electrode (eg a stannic oxide Glm) isused, preferably mounted on the face of a prism in a multiple-path attenuated totalreflexion cell.To begin with, we shall assume that the centre of the ion is located at the outerHelmholtz plane, so that there is no specific adsorption and that the ionic strengthof the solution is sufficiently high for the diffuse doublelayer potential to be suppressed.It is also assumed that the difference r is zero. Under these conditions it is easilydeduced that optical transfer at zero overpotential (denoted by the zero right-handsuperscript, below) will occur at a maximum frequency

hv:,, M 2E*M $(hv,,J homogeneous, (29)where E* is the activation energy PAE) for the symmetrical homogeneous thermalprocess, & + a=1 -+ al-1 + &.This is predicted to be independent of the nature of the metal.

The approximate equality sign in (29) indicates the uncertainty about the magnitudeof the electrical image term, a complication which will also arise in a slightly differentform (as the electrode dielectric constant is no longer infinite) for semiconductorelectrodes. However, this uncertainty will usually be about 0.1 eV, or less than 1 KK.It is therefore assumed here that the equality sign holds. Calculated values of hv;,,and band-widths for ion metal transitions for the aquo ions discussed earlier are shownin Table 3. Again, calculated values of the quantity x(=4E*) have been used.For typical transition-ion transfers, the optical bands at zero overpotential willlie in the ir, often at very low frequencies. (It would be necessary to use D,O assolvent for many of these systems). The bands will be narrower than those observedin homogeneous transfer in the ratioAl,z(homogeneous)/A1,z(heterogeneous) = 2/2.Perhaps the most interesting feature of these predicted transitions is that the peakfrequencies will vary linearly with overpotential, q, according to

hv;,, = hv;,, + FT. (30)TABLE 3. CALCULATZD FREQUENCY MA~~AANDBANDI-MLF-WIDTHS (300K) FOR

ION sr. ELECTRODE OPTICALELE~TRONTRANSFERATZEROOVERPOTENTIALJ dmxt A11sIon system* cm-l cm-l -Ti*+-Tj*+ 6,600 3,900va+-vs+ 6,100 3,800Crsf-Cra+ 13,600 5,700Mn*+-Mnx+ 9,400 4,700Feat-Fe*+ 8,600 4,500co*+-Coe+ 8,m 4,300Pu+-pus+ 7,500 4,200TP+-TI+ 14,700 5,300

* Aquo-ions in aqueous solution. t Source of theoretical values for x as in Table 1.


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1020 N. S. HUSHThis will also aid in the identification of the band, and it will also make it possible

to shift it away from absorption regions of the individual ions. Thus, an Fe(H,0)69+-Fe(H,0)2+ band would be expected at 8.6 KK at zero overpotential and to shift to16.67 KK at a positive overpotential of 1 V. On the other hand, the band half-widthshould be invariant with potential. This behaviour is illustrated schematically inFig. 4 for Few-Few and Tlw-Tl+ optical transfers at the interface with an aqueoussolution.

$v): 0 +I Fe 2t- Fe 3c

AA ,

r)(v): -I 0 ti Tl+-T13+

3KK

FIG. 4. Predicted frequencies and approximate band envelopes for Fes+-Fe*+ andTP+-T1+ optical transfer at a metal-aqueous solution interface as a function of over-potential. The relative intensities are arbitrary.The band intensities (and hence the extinction coefficients) are expected to be

sensitive to changes of overpotential. The relative intensity 8, of a metal-to-iontransfer at overpotential 11 o the total intensity at 7 = 0 is approximatelyvq (M,12eq= (1 + exp (-F#T))-lTV ( 1max lwll

= (1 + exp (-F~/RT))-~$+.nlaxIn this expression, the first factor expresses the dependence of the acceptor-ionconcentration on overpotential. If x is of the order of 2 eV, 8, is 413 at 0.5 Vpositive overpotential and 0 at O-5 V negative overpotential. For the reverse ion-to-metal transfer, the same relationships hold with the sign of r reversed.For the ions discussed, the absolute intensities will be very low. With a value of100 as an estimate of a typical molar extinction coefficient, by analogy with the


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Homogeneous and heterogeneousoptical and thermal electron transfer 1021homogeneous case and with a probable path length of ca 5 A , the optical densityfor a 1 M ion solution is only 5 x IO-6. There is thus a problem analogous to thatencountered in the study of homogeneous transfers. It is clear that in order to obtainbands of moderate intensity, closer interaction with the metal is desirable. This maybe effected in a number of ways, such as(i) the use of large organic ligands which have extensively delocalized n-systems

(eg dipyridyl, o-phenanthroline)(ii) the use of bridging ligands such as N3-, or others in which there is a possibilityof forming a bond to the metal surface through specific adsorption.In organic systems, it is often likely that specific adsorption will occur, which

will greatly enhance the intensity. The dependence on overpotential will now bemore complex, however, owing to the finite potential gradient in the inner Helmholtzlayer.At finite overpotential, the forward and reverse transfers

az+ + e(M) -+ a(-l)+az-l)f -+ aa+ + e(M) (11;will occur with different energy maxima.In the high-temperature limit, the difference is given by

/W;,,(I) - hv;,,(2) = 2Fy. (31)Thus, at small overpotentials it may be possible to distinguish two peaks, formetal-to-ion and ion-to-metal transfer, respectively. As the concentration ratio[az+]/[a(z-l)+] is given by exp (-Fq/lU), the intensity of (ii) will rapidly diminishat positive overpotentials while that of (i) will rapidly diminish at negative over-

potentials. Thus, at positive overpotentials, the transfer will be almost entirely(i), ie transfer to the ion, while at negative overpotentials the reverse ion-to-metaltransfer will predominate. The actual intensities can be calculated by an extensionof the method outlined above, taking into account the relative concentrations ofoxidized and reduced ions as a function of overpotential.

The experimental observation of optical transfer at electrodes would be of greatinterest, as we would then have systems in which the optical and thermal rates couldsimultaneously be studied with continuous variation of the over-all energy difference.In addition, photocatalysis of the thermal electron-transfer step, analogous to thatobserved in some homogeneous electron-transfer systems, could also be examined.7. CONCLUSIONIn outlining a unified theoretical treatment for optical and thermal transferprocesses, it has been necessary to restrict the treatment to the general features of

the phenomena. There are many detailed aspects which have not been discussed-for example, the operation of spin conservation rules. However, the essential featuresof the correlations seem clear. Further experimental work is necessary before it willbe profitable to pursue them in greater detail.REFERENCES

1. J. FRENKEL, Phys. Rev. 37, 17, 1276 (1931).2. Pohrons and Exci tons, ed. C. G. Kuper and G. D. Wbitfield. Oliver and Boyd, Edinburgh andLondon (1963).


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1022 N. S. Husrr3. G. C. ALLEN and N. S. HUSH, in Progress in Inorganic Chemistry, Vol. 8, p. 357, ed. F. A.Cotton. Interscience, New York (1967).4. N. S. HUSH, in Progress in I norgani c Chemistry, Vol. 8, p. 391, ed. F. A. Cotton. Interscience;New York (1967).5. L. D. LANDAU, Phys. Z. Sowjetunion 3,644 (1933).6. H. FR~LICH, Proc. roy. Sot. A160,230 (1937).7. R. Kuno, Phys. Rev. 86,929 (1952).8. FL Kuno and Y. TOYOZAWA,Prog. theor. Phys. 13,160 (1955).9. K. HUANG and A. Rws, Proc. Roy. Sot. A204,406 (1950).10. R. A. MARCUS, J. them. Phys. 24,966 (1956).11. V. G. LEVICH and R. R. DOGONADZE, Dok. Akad. Nauk SSSR 124, 123 (1959); Proc. Acad.Sci. USSR Phys. Chem Sect. 124,9 (1959).12. R. A. Mucus, Ann. Reo. Phys. Chem. 15,155 (1964).13. F. DUSCHINSKY,Acta Physiochim. USSR 7,551 (1937).14. N. S. HUSH, Tr ans. Faraday Sot. 57,557 (1961).15. V. P. ZHUZE and A. I. SHELYKH,Sou.Phys. Solid State 5, 1278 (1963).16. A. J. BOShiAN and C. Cmmcmxm, Phys. Reu. 144,763 (1966).17. G. VOLGER, Disc. Faraday Sot. 23,63 (1957).18. C. K. JORGENSEN, cta Chem. Scand. 9,405 (1955).19. M. ROBM, Znorg. Chem. 1,337 (1962).

DISCUSSIONZ. R. Grabowski .-You speak of an optical transition of an electron initially localized on one atomto another orbital localized on the second atom. In the case of an intervalence transfer between likeatoms, cannot the upper state correspond to a delocalized electron, common to both atoms?N. S. Hush.-The term intervalence transfer is used for transitions in which the electron is verynearly localized on the donor ion in the ground state. Thus, the electronic wave-function of theground state can be written as IO> = $d + ~da+a,where +d and $B are the donor and acceptor wave-functions, and Ada is small. The wave-functionfor the excited state is correspondingly

Provided that the orbital overlap is small (as will always be the case in intervalence transfer) we canset &a = Aad. This shows that the excited state contains only a small admixture of the donorfunction. The amplitude I2 is often of the order of 0.0005 for typical intervalence transfers, whichusually have comparatively small oscillator strengths per electron.These expressions are indeed quite good approximations in the high-temperature limit.R. A. Marcus.-The comparison of the energies involved in thermal and light-induced electrontransfers and made by Dr. Hush is an interesting one. By examining the polar solvent contributionto these energies, one can show that the simple relation is obtained between the two, with or withoutdielectric continum theory, if a dielectric unsaturation approximation is used and specitic interactionsare neglected. From (68) and (14) with appropriate substitutions, one finds

(1)humax = A+ A, (2)where* A = AFO + wp - wr, vmBXs the frequency for maximum absorption and the other symbolshave usual meaning.* When vibrational motions of the reactants are included and treated as harmonicthe same results obtain, but now rZ s lo + i(r, detined elsewhere, instead of lo.

* If an electronically excited state of the product is found in (1) or (2) the AFO in that equationis the standard free energy of formation of that state from the reactants in the prevailing mediumand at the prevailing temperature.1. R. A. MARCUS, J. them. Phys. 43,679 (1965).2. R. A. MARCUS, J. them. Phys. 43, 1261 (1965).3. R. A. MARCUS,Electrochim. Acta 13, 995 (1968).4. R. A. MARCUS,Ann. Rev. phys. Chem. 15, 155 (1965).


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Homogeneous and heterogeneous optical and thermal electron transfer 1023Note added in proof: (30 April 1968)

There is now definite evidence for electron-transfer Debye loss in a number of inorganic solids.S. Van Houton and A. J. Bosman (I nformaI Proc. Buhl I nt. Conf. Materi als, Pittsburgh 1963,p. 123. ed. E. R. SCHATZ. Gordon and Breach, New York, 1964) have studied Li-doped NiO andhave observed dielectric loss attributed to the thermoneutral exchangeNi*+ + Ni+ -+ Nis+ + Nix+

between ions adjacent to the substitutional Li+ ion. The similar transferco*+ + co*+ + con+ + co*+

has been studied in alkali-metal doped COO.[BOWAN and C. CRE~ECXXEUR,. Phys. Chem. Soli & 29,109 (1968)). In the authors laboratory[N. S. HUSH and R. C. &APPELL, to be published], similar results have been obtained for Li-dopedNiO; in addition, Debye losses in Nb-doped TiO, and in non-stoichiometric ferric ferricyanide havebeen observed, and are attributed respectively to the exchanges

Tia+ -C Ti4+ + Ti4+ + TiafFe8+ + Fe*+ -+ FeB+ + FeS+

An interesting feature of this effect is that the method is the analogue for homogeneous systems ofthe determination of heterogeneous electron transfer rates by measurement of faradaic admittance.Electron-transfer Debye loss will be less easily observed in stoichiometric compounds. Here thetransfer is not thermoneutral (EO # 0). The dielectric loss E at the Debye peak will be smaller thanthat for a corresponding thermoneutral transfer by the factor sech* (&/XT). This is about l/100 forEO = 6kT, so that high sensitivity will be generally required even for systems in which E,, is com-paratively small.The possibility of resonance absorption at low temperatures as a result of transition to a non-adiabatic vibrational tunnelling mechanism [cf. Fig. 11 is also being explored.

homogeneous and heterogeneous optical and thermal electron transfer

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