Faraday Discuss. Chem. SOC., 1982, 74, 17-29 Quantum Effects on Electron-transfer Processes B Y MORDECHAI BIXON AND JOSHUA JORTNER Department of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel Received 2nd June, 1982 The non-adiabatic multiphonon theory of electron transfer provides a complete description of nuclear tunnelling and final-state excitation phenomena, which modify the classical Marcus theory. Ultrafast electron-transfer processes, where the electronic process competes with medium-induced vibrational relaxation, cannot be handled by the conventional theory, being amenable to description in terms of quantum-mechanical models. 1. INTRODUCTION During the last decade it has been recognized that a unified theoretical framework exists for the description of non-radiative, intermolecular, electronic, relaxation phenomena in condensed phases, which span a broad spectrum of processes in the areas of solid-state physics, physical chemistry and bi0physics.l All these diverse non-radiative phenomena can be described in terms of a relaxation process between nuclear states which correspond to two distinct zero-order electronic configurations and whose energy conservation is ensured by absorption and emission of phonons.For example, electron-transfer (e.t.) processes between ions in ~ o l u t i o n ~ - ~ and in biophysical ~ y s t e m s , ~ ’ ~ which are the subject matter of the present paper, can be envisaged as a non-radiative process within a “ supermolecule ” consisting of the donor (D) and the acceptor (A) pair, together with the entire medium, the relevant electronic states being DA and D+A-.7-13 The following ingredients of the relaxation process are central : (1) The electronic coupling V, which involves the two-centre, transfer integral between D and A.(2) The modification of the nuclear vibrational states of the “ supermolecule ” by the change in the electronic states. These involve the changes in the intramolecular vibrational states specified by the coordinates qc of the D and A centres, as well as the response of the exterior polar medium characterized by the coordinates 4m to the change in the charge distribution. The microscopic rate constant, W,, for e.t. from a single vibronic state xI(Q), where Q = qc,qm, of vibronic states of D+A- characterized by the energies Ej, can usually be described in terms of a non-adiabatic f~rmalism,~-’~ being given in terms of a product of an electronic term 1VI2 and a nuclear Franck- Condon factor The basic implicit assumption is that medium-induced vibrational relaxation and18 ELECTRON TRANSFER vibrational excitation processes in the initial ( / I > } manifold are fast on the timescale of the electronic-vibrational process,14J5 i.e.wZ @ YVR (1 -3) where y V R z 1011-1012 s-l is a typical vibrational relaxation rate.' One can then perform a separation of the timescales for " fast '' vibrational relaxation and the " slow " electronic-vibrational process. The conventional transition probability, W, for e.t. is given by the thermal average of the microscopic rate where F = Zrl 2 exp( -Ez/kT) Fz (1 Sa) I is the thermally averaged Franck-Condon factor and 21 = 2 exp(-Ezlkr) I (1.5b) is the partition function for the initial manifold.Two limiting cases of the non-adiabatic multiphonon rate can be distinguished. (a) The low-iemperature limit is exhibited when the thermal energy kT is low relative to all the characteristic relevant vibrational energies kc^)^). The low- temperature rate is Wz = ,,. This low-temperature limit corresponds to nuclear tunnelling from the vibrationless nuclear state of the initial-state potential surface to the final vibronic states, which are quasidegenerate with it.7 The temperature range where the rate is temperature independent can be specified by the ~ o n d i t i o n ~ ~ J ~ kT < fiW0,/5 for all the relevant vibrational frequencies mK. (b) The high-temperature limit: when the thermal energy kT exceeds all the characteristic frequencies, a classical treatment of P can be utilized which amounts to the replacement of the discrete sums in eqn (1.4) by multidimensional configura- tional integralsI8 where U,(Q) and U,(Q) are the nuclear potential surfaces of the initial and final states, respectively.The high-temperature rate [eqn (1 A)] may be written as The Marcus t h e ~ r y ~ - ~ provides a complete description of the classical nuclear contribu- tion FC to the high-temperature e.t. rate within the framework of the harmonic approximation2 -4*7 Fc(AE;Er) = (4nErkT)-) exp[-(AE + Q2/4E,kT] (1.8) where AE is the electronic energy gap,' while E, is the sum of the classical solvent (Es) and intramolecular inner-sphere (E,) reorganization energies.The only quantum phenomenon surviving in the high-temperature non-adiabatic rate [eqn (1.7)] is the pre-exponential electronic coupling. In order to assess the effects of nuclear contributions to the e.t. rate one has to focus attention on the relevant frequencies { f i ~ ~ ) = (kw,}, {k~,,,}, which incorporate both molecular modes {kcoc> and medium modes {kw,>. For e.t. in solution some ofM. BIXON AND J . JORTNER 19 the medium modes are low ( h m , , , ~ 1 cm-l),19 while for ice hmmz 180 cm-1.20 Molecular-type frequencies are kmC z 200-500 cm-l for the first coordination layer of solvated ionss-13 and hmc z 250-1000 cm-l for intramolecular vibrational modes of molecules D and/or A, e.g.porphyrin rings involved in photosynthe~is.~~~ It is apparent that the low-temperature limit cannot be accomplished for e.t. in solutions, where exterior reorganization provides a dominant contribution. On the other hand, for e.t. in low-temperature aqueous glasses 21 and in low-temperature photo- synthetic reaction centres 5*6 the low-temperature limit is realized. Between these two extremes diverse cases can be encountered. In examining quantum effects on e.t. rates two distinct classes of phenomena will be considered: (A) Quantum effects on conventional e.t. rates occurring from a thermally equilibrated initial manifold. These involve: (1) electron coup1ir.g: for non- adiabatic e.t., W K I YI2, (2) nuclear tunnelling effects: the breakdown of the classical limit for the F factor [eqn (1.8)] provides a signature of nuclear tunnelling effects on the e.t.rate. (B) Quantum effects originating from the breakdown of the separability of time- scales for the slow electronic e.t. process and the fast vibrational relaxation. The conventional e.t. rates, eqn (1.4) and (1.5), are applicable only under some restrictions. For ultrafast e.t. processes competing with vibrational relaxation, the entire conceptual framework of the theory has to be modified. 2. QUANTUM EFFECTS ON CONVENTIONAL ELECTRON The unimolecular reaction probability W(R), eqn (1.4), for a spatially fixed DA pair at a centre-to-centre distance R can be utilized to provide a simple expression for the bimolecular e.t. rate, k. We shall now consider the manifestations of quantum tunnelling on e.t.in solution when the thermal energy kT is sufficiently high to warrant a classical treatment of the solvent, while kT is sufficiently low so that a quantum treatment of the molecular modes is required. Within the framework of the harmonic approximation for all nuclear modes, the Franck-Condon factor is TRANSFER I N SOLUTIONS The parameters involved are: (a) the electronic energy gap AE, (b) the Marcus reorganization energy E, of the exterior solvent and (c) the parameters specifying the first coordination layer, which involve the vibrational frequencies (mK‘> and (mK”) of the first coordination layer in the initial and in the final states, respectively. (d) The changes in the equilibrium configuration {ArK) in the first coordination layer originate from totally symmetric modes, which are expressed for the case of octahedral complexes in reciprocal frequency (cm) units (AdK)‘ = 6(Ar,)’m/h.The calculation of multidimensional Franck-Condon factors can be accomplished by various techniques, such as the direct methods,’* the saddle-point method,8v10-22 the convolution technique,13 the series expansion 23 or the semiclassical m e t h ~ d . ~ Rather than dwell on cumber- some technical details we shall examine the effects of nuclear tunnelling on these e.t. rates in solution, which are manifested by the following phenomena. ENHANCEMENT OF ABSOLUTE ELECTRON-TRANSFER RATES Nuclear tunnelling effects in the first coordination layer are expected to increase the absolute rate of electron transfer (e.t.) relative to the classical In fig.1 we present the results of model calculations for a symmetric e.t. process (AE = 0)20 ELECTRON TRANSFER I 1 I I I 1 : - - - - loo : 200 600 I000 intramolecular frequencylcm- FIG. 1 .-Ratio of the quantum-mechanical Franck-Condon factors to their high-temperature classical limit plotted against intramolecular frequency in symmetric electron-exchange reactions at room temperatures (kT = 200 cm-') (I) (Ad)' = 0.1 cm, (2) (Ad)2 = 0.025 cm and (3) (Ad)' = 0.0025 cm. at room temperature characterized by a single mode, co, which suffers a configurational change, (Ad)2, while all the other frequencies are not modified. We have calculated the ratio F/FC, where FC(AE; E, + E,) with E, = (Ad)2tico2 is the classical vibrational overlap, eqn (1.8).From these results it is apparent that the role of nuclear tunnelling is pronounced for high frequencies and for large configurational changes. For the R u ( N H ~ ) ~ ~ + - R u ( N H ~ ) ~ ~ i- exchange, where inner-shell reorganization is very 10w,22*25 i.e. ( A q 2 z 5 x cm and tzw z 400 cm-', the quantum enhancement effect of the rate is only lo%, being exceedingly small. For the CO(NH,)~~+-CO(NH~)~~+ exchange reaction, which is characterized by large inner-shell r e ~ r g a n i z a t i o n , ~ ~ ~ ~ ~ i.e. (Ad)2 z 0.1 cm and tzw z 400 cm-l, the quantum effects enhance the e.t. rate by about one order of magnitude. Detailed calculations 22*25 for this system indicate a huge (lo-*) Franck-Condon reduction of the e.t. rate. However, most of this dramatic nuclear effect is accounted for by the classical result, the quantum effect being just F/FC = 5-7.22*24 Nuclear tunnelling effects on symmetric e.t.reactions at room temperature are modest. TEMPERATURE-DEPENDENT ACTIVATION ENERGY The contribution of high-frequency modes results in the increase of the activation For symmetrical e.t. the activation energy is 8*25 encrgy with increasing temperature. 8lnF E, (Ad,J2tzwL2(cosech2v: + cosech2vz) E = - - a( l / k T ) = 4 -k ? 2(coicothvL -t o.&cothv:)2M. BIXON AND J . JORTNER 21 For the symmetric CO(NH3)62+-cO(NH3)63+ exchange reaction in solution EJ4 = 7.0 kcal mol-' and E,/4 = 11.5 kcal rnol-', so that the " high-temperature '' limit of the activation energy is E,(T + a) = 18.5 kcal mol-I.The activation energy is calculated to vary in the range E, = 15.7 kcal mo1-1 at 273 K to E, = 16.6 kcal mo1-I at 343 K. KINETIC ISOTOPE EFFECTS Isotopic substitution in the first coordination layer of ions, or in a molecular donor or acceptor, is expected to modify the e.t. rate. This kinetic isotope effect originates from frequency changes and from distortion of the equilibrium configurations of totally symmetric stretching modes.26 On the basis of detailed calculations 26 for the C O ( N H ~ ) ~ ~ + / ~ + and C O ( N D ~ ) ~ ~ + / ~ + exchange reactions, we have concluded that the effects of frequency changes are minor and the major contribution to the isotope effect arises from the configurational changes of the totally symmetric mode. The isotope effect for symmetric e.t.is then given by26 where kH/kD = exp(YD - yH) (2.4) Y, = (E,/hco,) tanh - a = H or D (2) ; where E,, the internal reorganization energy, is invariant to isotope substitution. From the temperature dependence of kH/kD for symmetric e.t. (fig. 2), it is apparent that (i) the highest low-temperature value is reached at hco/kT >, 10, (ii) at high temperatures (kT/hw w 1) one can expand eqn (2.5) to get h(kH/kD) cc T3, (iii) in the high-temperature classical limit (kH/kD + 1) quantum effects erode, and (iv) the isotope effect is modest at room temperature, e.g. for the CO(NH&~+/~+ exchange kH/kD varies from 1.26 at 25 "C to 1.12 at 70 "C. The isotope effect is normal, i.e. kH/kD >, 1. FIG. 2.-Temperature dependence of deuterium kinetic isotope effect for symmetric electron- exchange reactions (AE = 0) between hexa-ammine complexes.22 ELECTRON TRANSFER The effect of the change of the electronic energy gap AE on k,/kD (fig.3) reveals the following features. (a) The maximum value of kH/kD is exhibited for symmetric reactions. (b) The AE dependence of h(kH/kD) in the normal region is described by a bell-shaped curve. (c) kH/kD reaches its minimum value for activationless and barrier- less reactions at AE = F(Es + Ec). (d) For highly exothermic and highly endo- W E , FIG. 3-Deuterium isotope effect for e.t. between ammine complexes as a function of the reduced electronic energy gap AEIE,. E, is the total reorganization energy E, = E, + E,. The parameters are as follows: AoH/kT = 2.0 and EJE, = 0 (- .-), E,/E, = 1 (-), E,/E,= 2 (- - -). thermic e.t., i.e. IAEl > E, + E,, the isotope effect increases with increasing the electronic energy gap. It is important to emphasize that the deuterium isotope effects on outer-sphere e.t. reactions originate, in our opinion, from the quantum motion of the entire ligand with respect to the ion. Thus the isotope effect originating from the replacement of an H2160 molecule by a D2160 molecule in the first coordination layer will practically be identical with the effect induced by substituting H2l60 by H2180, extending the scope of isotope effects. FINAL-STATE VIBRATIONAL A N D ELECTRONIC EXCITATION Marcus predicted a decline of the rate in the " inverted region " for highly endo- ergic reaction^.^,^ Excitation of internal quantum states of the donor-acceptor system results in a substantial moderation of the decline of the rate in the inverted region, exhibiting large quantum corrections on the rates of exoergic reactions.For e.t. processes involving only the ground electronic states of DA and of D+A-, the e.t. process results in the vibrational excitation of the internal vibrational modes, which have to be treated quantum me~hanically.~'-~~ The e.t. process involves contributions from parallel decay channels (2.6) D($) A(&$) + D+(E;) A-(E$) where EEL?) are the energies of the vibrational quantum states of the donor (acceptor) in its initial (final) state. The total e.t. rate can be expressed as a sum over allM. BIXON AND J . JORTNER 23 vibrational excitation channels.EP = EP = 0, the rate isl2?l3 For an idially internally " frozen " system, i.e. where SD(O,&F) and SA(O,&) are the Franck-Condon factors for the donor and the acceptor, respectively, each initially being in its ground vibrational state. The effective energy gag is FC(AE - &; - &$), so that a reduction of the electronic gap by the vibrational excitation is exhibited. The contribution of vibrational excitations adds new channels effective at high IAE], thereby reducing the inversion effect. It has been recognized that excitation of final quantum states can also involve electronic excitation when the exoergic e.t. process results in an electronic excitation of the A- and/or the D+ centres.27 The e.t. rate now incorporates additional additive decay channels to final states, characterized by electronic energy gaps A&, AE2, etc.which are lower than AE. The e.t. rate incorporating both electronic and vibrational excitations can be expressed in terms of the sum where Wis given by eqn (2.7), and W(AEj) is given by the same equation with AEj replacing AE. The contribution of the electronic excitations yields a substantial contribution at large IAEl resulting in the moderation of the decrease of k. From the point of view of the experimentalist, there is a qualitative difference between the role of electronic and vibrational excitations in the sense that Vibrational excitations will exhibit a unimodal " distribution " of Ink against AE, while the presence of an additional channel involving an electronic-excited final state will result in a bimodal distribution.An interesting analogy can be drawn in this context with line shapes in optical emis~i0n.l~ The interrogation of free-energy e.t. relations can be considered as an experiment in " chemical type " spectroscopy. The general appearance of Ink plotted against AE bears a close analogy to the line shape L ( h ) in emission of a photon hv, where the dependence cr; r(hv) on (AE - hv) for the optical process is analogous to the dependence of W(AE) on AE for the thermal e.t. process. When emission occurs to a single final electronic state, the line shape will be unimodal, while when emission involves two lower-lying electronic configurations, a bimodal intensity distribution will be exhibited. The examination of nuclear tunnelling phenomena and the effects of excitation of internal quantum states results in some quantitative modification of classical relations and correlations.The following conceptual and technical extensions of the theory will be of interest. (1) Electronic energy gap and the free-energy change. The multiphonon rate expressions incorporate the electronic energy gap AE as the relevant energy parameter, while the Marcus classical r e l a t i ~ n , ~ - ~ eqn (1.8), with AG replacing AE, contains the free-energy change. In this context, one should note that the general multiphonon rate W, eqn (1.4), for the DA -+ D+A- reaction, which will be denoted by W(DA + D+A-), and the rate W(D+A- -+ DA) for the inverse process, satisfy the condition of detailed balance W(DA -+ D+A-)/W(D+A+ --f DA) = exp(-AG/kT) AG = AE - TAS A S = kln(Z&) (2.9) with the entropy change being determined by the ratio of the partition functions for24 ELECTRON TRANSFER the initial DA, eqn (1.5), and for the final D+A- states.Practical calculations, which rest on the harmonic model for nuclear potential surfaces, result only in a minor contribution to A S from frequency changes,19 while the major contribution to A S due to rotational reorganization of the solvent and to electrostriction is missing. (2) The electronic coupling, Y. Three general coupling schemes can be dis- tinguished : (i) spin-allowed direct e x ~ h a n g e , ~ * ~ ~ ' ~ * ~ ~ (ii) spin-forbidden direct exchange,25 and (iii) indirect spin-allowed s~perexchange.~~ We are currently ignorant concerning several basic points.First, the effects of dielectric screening on direct exchange have not been treated in a self-consistent manner. Secondly, for e.t. between molecular D and A, e.g. for e.t. between large aromatic molecules, V has to be calcul- ated using many-electron molecular wavefunctions, as utilized for N a ~ - N a p . ~ * The dependence of Yon distance R provides a grossly over-simplified description and one has to consider orientational dependence. Thirdly, a realistic estimate of super- exchange interactions, which may be important for biophysical processes, will be desirable. The criteria for the validity of the non-adiabatic formalism, which is based on the Landau-Zener are applicable at best to the classical limit of e.t.dominated by the contribution of solvent modes. No satisfactory non-adiabaticity criteria for the multidimensional e.t. problem involving both classical and quantum modes have been provided. In the low-temperature limit only the non-adiabatic kinetic situation is meaningful, while for the limit of large V the zero-order states are heavily mixed (see section 3). The adiabatic rate provides the highest value of the pre-exponential factor for the high-temperature unimolecular rate, which for a single-mode system with a frequency co is given by3' (3) Applicability of the non-adiabatic formalism. (4) An upper limit for a thermal e.t. rate. W = LL) exp[-(A€ + Es)2/4EskT]. (2.10) Accordingly, the upper limit for the rate of an activationless adiabatic process is W M co z 1012-1013 s-'. The corresponding ultrafast bimolecular rate calculated according to eqn (2.10) is k M 1 O I 2 dm3 moI-l s-'.This rate is considerably higher than that of the diffusion controlled rate, kDiff = 109-1010 dm3 mol-' s-l. The observed rate of ultrafast bimolecular e.t. processes is limited by the diffusion process and therefore reveals nothing about the e.t. mechanism. On the other hand, fast unimolecular e.t. is not masked by diffusion and may occur on the picosecond time- scale. However, for such ultrafast e.t. the conventional theory should be modified due to competition between the electronic process and vibrational relaxation. 3. COMPETITION BETWEEN ELECTRON TRANSFER A N D VIBRATIONAL RELAXATION The advent of mode-locked lasers made ultrafast e.t.processes, occurring on the picosecond (and subpicosecond) timescale, amenable to experimental interrogation. Several experiments on ultrafast e.t. which come to mind are: (1) e.t. from electronically excited states in solution and in glasses; (2) e.t. to the ground state of a cation produced by photoionization in a dense medium; (3) geminate recombination of ions and of solvated (or partially solvated) electrons ; (4) intramolecular e.t. from excited states of DBA chemical and biophysical model compounds, where B is a bridge; (5) the ultrafast primary charge-separation process in bacterial photo~ynthesis.'~ For such ultrafast processes, the microscopic rates W,, eqn (1.1), may be compar- able to, or even exceed, the medium-induced vibrational relaxation (v.r.) rates within the initial { / I ) } vibronic manifold.Thus, the basic assumption, eqn (1.3), pertainingM. BIXON AND J . JORTNER 25 to the separation of the timescales for fast v.r. and slow e.t. may no longer be applicable, and the conventional e.t. theory breaks down. We are then concerned with a new and intriguing physical situation, when e.t. occurs during v.r., or even prior to it. One can then assert that for ultrafast e.t. the thermally averaged rate is meaningless, and the microscopic rates, W, [eqn (1. l)] have to be considered explicitly. Obviously, an ultrafast e.t. will be exhibited provided that two conditions are satisfied. This requires: (1) spatial proximity of D and A resulting in large V (ca. 10-100 cm-l) ; (2) large vibrational overlap is exhibited in the vicinity of the crossing point of the potential surfaces.(B) Accessibility of the states with large W,. The system is excited so that the vicinity of the crossing point is accessible during the relaxation process. The relevant v.r. processes during ultrafast e.t. span a broad spectrum of medium- induced phenomena. Some examples are : (a) Rotational relaxation of the solvent during ultrafast e.t. Such an e.t. process occurs on the timescale of the dielectric relaxation time z,, or rather the transverse constant-field dielectric relaxation time 32 7:: = (Dm/Ds) z,. An activationless e.t. characterized by a large Y (10-100 cm-l), where the major nuclear contribution originates from solvent reorganization and which takes place in a system characterized by a long z, (or T:), is expected to occur from a non-equilibrium solvent configuration, which relaxes during the e.t.(b) Con- (A) Some of the W, are large. FIG. 4.-Schematic description of initial excitation followed by vibrational relaxation, which is designated by arrows. (2) When E, > Ex the system can undergo efficient e.t. at the crossing point x before equilibration has been accomplished. (1) When E, < Ex the system usually equilibrated thermally.26 ELECTRON TRANSFER ventional intramolecular v.r. induced by the medium in D and in A, which consist of large polyatomic ions or molecules. At present the treatment of the interesting implications of solvent relaxation and of configurational medium relaxation will be avoided, and we shall consider an ultra- fast e.t.between molecular D and A, where the coupling with exterior-medium modes is small and the major nuclear contribution to each W, originates from intramolecular reorganization. Such a situation of e.t. accompanied by intramolecular reorganiza- tion (e.t.a.i.r.) may prevail in glasses with a low dielectric constant or in biophysical “ rigid ” systems, such as photosynthetic reaction centres, where we expect conven- tional medium relaxation effects of type (b) to prevail. Considering explicitly the conditions (A) and (B) for ultrafast e.t.a.i.r., two relevant physical situations are of interest. First, activationless e.t.a.i.r. involves an exoergic process with A23 = -&, so that the potential surfaces cross at the minimum of the initial-state potential.Second, e.t.a.i.r. during v.r. When W, is large at the crossing point of two potential surfaces and the system is initially excited at an energy E, above the crossing point (Ex), efficient e.t.a.i.r. is expected when the system relaxes vibrationally through the crossing point 1 4 9 1 5 (fig. 4). Let us consider first the problem of ultrafast e.t.a.i.r. occurring from the vibration- less 11 = 0) level, as may be the case for some activationless e.t.a.i.r. The 11 = 0) = 11) level interacts significantly with a small number of final intramolecular states, which will be represented by a single final level IF). The IF) level undergoes medium- induced vibrational relaxation, which may be envisioned as weak interaction with a continuum of medium states superimposed on lower lying intramolecular levels in the final manifold*(fig.5). The residual v.r. coupling can be accounted for by assigning a vibrational relaxation width, I?, to the final state (fig. 5), whose energy is EF. On the other hand, the 11) state, whose energy is E,, is not subjected to v.r. The quantum- ( b ) FIG. 5.-Energy-level scheme for ultrafast e.t. from the vibrationless II> level. (a) Coupling scheme, ( 6 ) model system. The 11) level is coupled by V to IF), which in turn is subjected to v.r. to medium- phonon states superimposed on the IF’) state.M. BIXON AND J . JORTNER 27 mechanical model for the e.t.a.i.r. (fig. 5 ) corresponds to interstate coupling 73 = ITIcF), where FI(Fl is the ]I) - IF) Franck-Condon overlap, with the IF) state being characterized by the width r.This relaxation problem can be handled by solving the secular equation hich has the solutions E = +[EI + EF - iT] &- +[(EI - EF + ir)2 + 4P2]*. (3 -2) Several limiting cases are of interest. AEIF = EI - EF, one obtains to second order the solutions For weak p, i.e. IPI < IAE(IF + i r l , when The energy shifts are of no interest. state has acquired, according to eqn (3.3), a width It is important to note that the coupled initial which is proportional to P2. yI can be, identified with a decay rate, which for a large AEIF, i.e. IAEIFI $ r, is given by yI = V2r/(AE)2, while for near resonance IAEIFI $ r the highest value of yI = P2/r is obtained. These weak coupling situations are analogous to the non-adiabatic limit. The results are drastically different for the strong coupling situation, i.e. P 9 IAEIFI.Strong scrambling between the two zero-order states is exhibited. The resulting states are with the corresponding energies which are split by 2P and have equal decay widths. As the initial state is prepared by optical excitation, the dynamics depends on the coherence properties of the excitation. If Il} and 12) can be excited coherently, which means that the initial state is \I), then the electron is transferred in a decaying oscil- latory manner with an oscillation period of h/2 P and a decay rate of T/h. In real life it would be very difficult to achieve a coherent excitation and more probably the excitation would either be to 11) or to 12). In such a case, the electron is already “ half transferred ” in the preparation act of the initial state, which involves excitation of a charge-transfer state.The e.t. is continued due to v.r. at a rate of r/h, which is then independent of the magnitude of V. This simple quantum mechanical model for the28 ELECTRON TRANSFER activationless ultrafast e.t.a.i.r. is illuminating as it retains all information about coherence effects. In view of inhomogeneous broadening effects and dynamic phase- erosion processes, which prevail even in low-temperature solids, it will be extremely difficult to observe the manifestation of such coherence phenomena in ultrafast e.t. At the other extreme, when coherence effects are insignificant, the stochastic model l5 is applicable. The stochastic model for e.t.a.i.r.I5 is very useful as it adheres to chemical intuition.This model (fig. 6) treats the populations pau(t) of the states in the initial manifold STATE a ”p; a 4 wa 3 a 3 STATE b FIG. 6.-Stochastic model for competition between e.t. and v.r. The microscopic e.t. rates are W,,, while yUju’ denotes the v.r. rates. 11) = lav) of DA in terms of kinetic equations involving medium-induced vibrational relaxation with the rates yu+”*, and e.t. is characterized by the microscopic rates Wa,, which are given by eqn (3.5): . dpau(t) - 2 yu-+u’ pau(t> + 2 yu’-+u pao,(t> - ~ a c p a u ( t > (3.8) dt u‘#u U ’ Z V with appropriate initial conditions. The time-dependent e.t. rate under non- equilibrium conditions is while the population NDA(t) of the D-A pair is NDA(t) = N D A ( 0 ) exp (3.10) exhibiting a non-exponential decay.between e.t. and electronic energy transfer, one can assert that, while the time depen- Pursuing the analogy advanced by HopfieldM . BIXON AND J . JORTNER 29 dence of an ultrafast energy transfer which competes with v.r. is determined by a time-dependent donor-emission/acceptor-absorption spectral overlap the time-dependent e.t.a.i.r. rate [eqn (3.9)] is expressed in terms of a time-dependent transition probability, which in the non-adiabatic limit reduces to a time-dependent Franck-Condon factor. J. Jortner and R. D. Levine, Adu. Chem. Phys., 1981,47, 1 . R. A. Marcus, J. Chem. Phys., 1956,24, 966; 1965, 24, 979; 1957,26,867, 872; Trans. N. Y. Acad. Sci., 1957, 19, 423. R. A. Marcus, Discuss. Faraday Soc., 1960, 29, 21. R.A. Marcus, Annu. Rev. Phys. Chem., 1964, 15, 155. J. J. Hopfield, Proc. Natl Acad. Sci. USA, 1974,71, 3640. J. Jortner, J . Chem. Phys., 1976, 64, 4860. W. Jost (Academic Press, New York, 1970), vol. 9B. N. R. Kestner, J. Logan and J. Jortner, J . Phys. Chem., 1974, 78, 2148. R. R, Dogonadze and A. M. Kuznetsov, Elektrokhimiya, 1967, 3, 1324. 125; 425. ’ V. G. 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