Faraday Discuss. Chem. Soc., 1982, 74, 3 1-47 Quantum Theory of Chemical Reactions of the Solvated Electron BY ALEKSANDER M. KUZNETSOV Institute of Electrochemistry of the Academy of Sciences of the U.S.S.R., Leninsky Prospect 31, 117071 Moscow V-71, U.S.S.R. AND JENS ULSTRUP Chemistry Department A, Building 207, The Technical University of Denmark, 2800 Lyngby, Denmark Received 17th May, 1982 The solvated electron represents a more diffuse charge distribution than do electrons in molecules. The electron-exchange integral in rate expressions for reactions of the solvated electron is therefore also more sensitive to a variation in external parameters such as the free energy of the reaction. We have investigated this effect and have applied the Condon approximation, but we have determined the orbital exponent of the electronic wavefunction (of assumed spherical symmetry) by minimizing the total potential energy with respect to the actual polarization, for which we have also incorporated frequency dispersion, rather than only the equilibrium polarization.For low- frequency nuclear modes this causes a more diffuse charge than at equilibrium in the " normal " free-energy region for the electron-transfer reaction, and a more compact charge in the strongly exothermic region. For high-frequency modes the effect is much larger in the strongly exothermic region and of equal importance to other effects which cause a drop in the rate constant in this region. 1. INTRODUCTION A great deal of attention is presently being given to solvated electrons in frozen Much of this interest refers to the nature of the trapping aqueous and organic glasses.sites as revealed by the absorption spectrum in the near-infrared and visible and to long-range electron transfer of the trapped electron to molecular The kinetics of electron transfer display several features which differ from those of " conventional " electron transfer between molecules in liquid solution. First, the decay of trapped electrons follows a logarithmic time dependence, and the rate depends exponentially on the scavenger concentration.8-11 Both these effects are commonly interpreted as the direct tunnelling of the electron to the scavenger molecule, and the rate laws reflect the strong distance dependence of the tunnelling prob- ability.8 -12 Secondly, reactions of the trapped electron are strongly exothermic, the free energy of reaction exceeding the total nuclear reorganization energy.13-15 The reactions are furthermore often followed at low temperatures (< 100 K) where a substantial part of the nuclear modes are quantum mechanically ' frozen.' Both these conditions are favourable for the observation of nuclear tunnelling.It is therefore of considerable interest to study solvated electrons in terms of electron- transfer theory, but in comparison with molecular reactants in liquids the solvated electron exhibits at least the following differences : (1) The disordered medium cannot adequately be represented by a single-mode appro~irnation'~*~~ as is the case for high-temperature processes.18-20 The electrons32 QUANTUM THEORY OF THE SOLVATED ELECTRON are coupled to a broad continuum of nuclear modes, which requires a significant reinterpretation of the Brransted and Arrhenius relations at low temperatures.l8 (2) The perturbation which induces the process cannot solely be identified with the interaction between the electron and the scavenger, as is the case for molecular reactants, since this would invoke conceptual difficulties for the reverse reaction, i.e.the formation of the solvated electron by thermal ionization of the reduced scavenger. (3) The electron represents a more diffuse charge distribution than for molecular reactants. This may be reflected both in the formal kinetics, leading to the importance of diffusion to sites at varying distances from the scavenger, and in the ' elementary ' rate expressions, where the electron-exchange integral may depend on the solvent nuclear coordinates.We have previously analysed several of these effect^.'^^'^^^^'^^ In the present work we investigate the second and third effects in particular and derive expressions for the electron transfer (at a fixed distance) which incorporate these effects. These rate expressions can subsequently be combined with kinetic rate laws for direct tunnellinglo or variable-range diffusion.21-24 2. HAMILTONIANS AND REACTION CHANNELS We shall now introduce zero-order Hamiltonians for the initial and final states corresponding to localization of the electron in a solvent ' trap ' and on the scavenger molecule, respectively. Subsequent application of perturbation theory is then justified in view of the large (20-40 A)"-" electron-transfer distance. The total Hamiltonian of the trapped electron and the molecular scavenger in the medium is fi= Te -+ TN + p e , t r + pe m + p z , t r + p e z + pzm + ptr,m + f i s c + om (2.1) where T, and TN are the kinetic energies of the trapped electron and the nuclei of the medium, V e , t r and Ve, the energies of interaction of the localized electron with the trap (or the nearest solvent molecules) and the scavenger, screened by the inertia-less polarization of the medium, and Ve, the interaction between the electron and the inertial polarization of the medium outside the trap.We have here assumed that the inertia-less part of the polarization is separated adiabatically from the motion of the trapped electron.V z , t r and Vzm are the corresponding interaction energies for the scavenger, Vtr,m the energy of interaction between the trap and the medium, H,, the (vacuum) Hamiltonian of the scavenger, and Urn finally the potential energy of the unperturbed medium. It is now convenient to introduce the reaction channels, corresponding to basis functions of the electron in the trap and on the scavenger m o l e c ~ l e . ~ ~ * ~ ~ However, in contrast to reactions between molecules, the electronic wavefunction of the trapped electron is not determined by its interaction with a molecular core. It is therefore necessary to introduce the reaction channels in the following way, which differs from the procedure for molecular reactants.We assume that the interaction of the electron with the external medium is linear. The total polarization of the medium in either state is then a superposition of the contributions from the scavenger and the trapped electron. For neutral and negatively charged scavengers the spatial distribution of the polarization field furthermore creates two potential wells for the electron, centred in the trap region and at the acceptor. For positively charged scavengers a potential well at the acceptor site is created by superimposed polarization and molecular potentials. This distribution is represented by the equationA. M. KUZNETSOV AND J . ULSTRUP 33 where the total inertial polarization, P(r), is given as a sum of the inertial polarizations in the trap region when the scavenger is absent, P1(r), and in the scavenger region, P2(r), when the solvated electron is absent.This division refers to the polarization field and not to dynamic variables which describe the solvent. Polarization ffuctua- tions of sufficient depth to trap the electron at a given site are thus incorporated in the channel Hamiltonian for localization of the electron at this site, and we assume that possible distortions of electronic wavefunctions at one reactant by the polarization field or molecular potential from the other reactant are sufficiently small to be ignored or handled by perturbation theory. These points and their relation to a formulation in terms of adiabatic basis functions, which include the electronic interactions between the centres, have recently been analy~ed.~’’~’ We are then in a position to introduce the Hamiltonians of the ingoing (Hi) and outgoing (Hf) channels (initial and final states).The total Hamiltonian can be written I? = rii + Pi = Ei, + Vf f i i = p e +,f+N +Ape,tr + “2 + ptr,m + fz,tr + pzm (2.2) where the channel Hamiltonians are + [Vez + V%]dia + urn + LVe,tr + V%ldia + (2.3) (2.4) (2.5) H f = p e +- F N + Fez+ v% f Qtr,m + pz tr + pzm The perturbations which induce the forward and reverse reactions are f i = ivez + p%]c~ff-d; pf = [Ve,tr + v!z]off-d. VLZ and VL2 are the energies of interaction of the electron with the polarization components Pl(r) and P2(r), respectively, and the perturbations Vi and Vf contain only the ‘ off-diagonal ’ matrix elements (off-d) with respect to the electronic wave- functions at the two different centres, whereas the diagonal parts (dia) are included in the reaction channels.The matrix elements have the form = <ilPez + PL2jf) - <ilPez + P!zli><ilf> (2.6) with an analogous expression for (Vi)fi, where (i) and If) are the electronic wave- functions in the initial and final states. Eqn (2.3)-(2.6) define the zero-order channel Hamiltonians and the perturbations. In particular, we see that if there are no pre-existing traps so that V,,,, = 0, the perturbation for the reverse reaction is the interaction with the polarization field in the absence of the scavenger, i.e. Pl(r), and an analogous contribution appears in the perturbation for the forward reaction. We conclude this section by introducing the Born-Oppenheimer channel wave- functions.The total electronic-nuclear wavefunctions are written Ysb; !z) = lys(x; 4)xs((l) (s = i,f> (2.7) where lys(x; q ) are the electronic wavefunctions and solutions to the stationary electronic Schrodinger equation ( H s - TN)Vs(x; q)l= & S ( ~ ) W S ( ~ ; q)- (2 - 8) x is a set of electronic coordinates, q the total set of all local and “ collective” medium nuclear coordinates, and ~ ~ ( 4 ) the electronic energy in the state s. The nuclear wavefunctions, xs(q), are subsequently found from the equation [ T N + &s(dlxs(!?) = Esxs(4). (2.9)34 QUANTUM THEORY OF THE SOLVATED ELECTRON The polarization contributions to the perturbations are then v g = - J P ( r)&(x; r) dr (2.10) v g = - SP( r)&(x; r) dr (2.11) where &(x; r) is the vacuum field in the point r caused by an electron at x.3. RATE PROBABILITY 3.1. THE CONDON APPROXIMATION REVIEWED We shall assume that the transition probability per unit time, Wif, can be ade- quately calculated by first-order perturbation theory. For large electron-transfer distances a second-order mechanism involving high-energy intermediate electronic states might compete. Such a mechanism can be described by second-order perturba- tion t h e ~ r y , ~ ~ . ~ ~ but we shall not discuss it further in the present work. A quite general form of Wif, averaged over the initial and summed over the final vibrational states, is then 26-31 c + im wif = ( ~ / i h ) do expvFi)Tr(Lipi(l - o)~;pf(9)) (3.1) c - icn where p = (kT)-’, k is Boltzmann’s constant and T is the absolute temperature.pi(l - 8) and pf(8) are the nuclear parts of the statistical operators32 in the initial and final state, at the temperatures T/(1 - 0) and T/8, i.e. pr(l - 0) = exp[-P(l - €J)Z?~]; pf(9) = exp(-P8Ei,) (3.2) where RS = TN + E,(q) (s = i,f). Fi is the free energy of the initial state, i.e. exp@Fi) = {Tr[exp(-pEJ])-l Li an abbreviated notation for the two-centre electronic resonance integral, and L’: its hermitian conjugate. As an operator Li can formally be written where [ Vi(q)lif is given by eqn (2.5)-(2.8). Finally, c is a real number which determines the integration contour in the complex plane. We shall further assume that Li depends sufficiently weakly on the nuclear co- ordinates that it can be replaced by its value at q = q*, for which the trace in the coordinate representation takes its maximum value.We thus formally invoke the Condon approximation, but we do account for the variation of [Vi(g*)IIP with q*. q* generally differs strongly from the equilibrium values of 4 in the initial (4io) and final (&o) states and is subject to significant changes when parameters such as the free energy of reaction vary. For this reason [Vi(*q))lif also differs from both [Vi(qi0)lif and [Vi(qf0)lif and varies through such a series of reactions, and the effect is more important for the trapped electron than for molecular reactants. The transition probability thus takes the form c i - i m wif = (p/ih) [vi(q*)lif12 exp(pFi)Tr{pi(l - e)Pf(e>> (3.4) c - i m and we can proceed to calculate the trace using effective Hamiltonians 26*31933 or the correlation-function Here it is convenient to consider the low-frequency classical and high-frequency ‘ quantum ’ nuclear modes separately.A .M. KUZNETSOV A N D J . ULSTRUP 35 3.2. LIMIT OF CLASSICAL NUCLEAR MODES In this limit the trace in eqn (3.4) can be written as an integral of a functional of the polarization P(r). P(r) represents the dynamic variables of the medium, and the trace could be recast as an integral with respect to a set of harmonic normal c ~ o r d i n a t e s . ~ ~ * ~ ~ * ~ ~ Using the first formulation we can write Wif in the form - W J f l . The potential energies in the initial ( Ui) and final ( Uf) states are (3.5) = um[p(Y)l+ EP - / dr[~i(r) + ~2(r)1/ dxIV/i(x\,12&(x; r) (3 4 Uf = Um[P(r)] + efO - / P(r)Bf(x; r)dr = um[~(r)l+ E ~ o - \dr[~,(*) + ~,(r)l/dxlV4x; r)12&(X; r).U,,,[P(r)] is the potential energy of the unperturbed dielectric medium, determined by the polarizatio'n components P(r) of the different kinds of molecular motion (orienta- tional, atomic etc.) by the relation33 Um[P(r)] = 2 2 / [PV(r)l2dr CV where cV = EL; 1 - 8;' and e, and cV+ 1 are the dielectric constants on each side of the absorption band for the vth polarization mode. For a single (relaxational) mode c = e0-l - E ; ~ , where E, is the optical and E, the static dielectric constant. ey and &Of are finally the electronic energies excluding the interaction with the polarization field. The integrals with respect to 6 and P(r) can be calculated by the saddle-point method for rather general potential surfaces.The saddle points 6" and P(r) are determined by the equations 3 1 9 3 7 9 3 8 (1 - e)(mi/6p) + e(suf/sp) = o (3.7) where the variations must be extended to any discrete molecular or short-range solvent modes, when the appropriate potential-energy contributions have been added to eqn (3.6). We shall assume that the explicit variation of the electric field b(x; r ) with changes in the electronic charge distribution caused by the polarization fluctuations is small compared with the dependence of the electronic resonance integral and Ui and Uf on P(r). This assumption is a reasonable first approximation, but the occasional rather notable dependence of the electronic wavefunctions on P*(r) (section 3.3) does require explicit consideration of the variation of &(x; Y) in these cases.28 Following previous procedures 26*38 P*(r) is then found to be Pv*(r) = (1 - 6*)P,lb(r) + 6*P,o(r); P*(r) = 2 P"*(r) (3.8) V where Pr0(r) and P,'o(r) are the equilibrium inertial polarizations of the initial and final state, respectively.For spherically symmetric charge distributions or for36 QUANTUM THEORY OF THE SOLVATED ELECTRON infinite homogeneous media39 P:o(r) and P,"o(r) are related to the induction fields, D, by the equations c C Pro(r) = 2 Dio(r); Pfo(r) = Dfo(r - R); Pso(r) = 2 P(r,Yo), (r) s = i,f (3.9) which we shall exploit in the following. We thus assume that the electron-transfer distance is sufficiently large, and Ve,tr z 0. Insertion of eqn (3.8) in eqn (3.7) then gives an equation for the saddle point of the Lagrange multiplier O*.Further application of the semiclassical procedure 38 gives the rate probability. If the electron is only coupled to the medium, Wif is (3.10) where E," is the solvent reorganization energy. At high temperatures this quantity is well approximated by the limiting formula for a structureless medium 4.n 47E V Wif = (n/h2kTE;)*1 [ Vi(AFO)]ifl2 exp[-D(Es + Af',-J2/4E:] E: =- [DiO(r) - Dfo(r - R)I2dr (3.1 1) 87~ "I where R is the distance between the trap and the scavenger. At low temperatures (e.g. at 77 K) and for strongly exothermic processes frequency dispersion of the nuclear modes is, however, important,18*20 and E," takes a more general form which we shall discuss in section (3.4).Finally, AFO = (&; - EP) - 1 dr[p,(r) + ~2(r)1\ dxg(x; r)[lvf(x; pfo)12 - Ivi(x; pio)121 is the free energy of reaction. electronic factor on AFo explicitly expressed through P*(r) Eqn (3.10) differs from the usual form of this equation by the dependence of the [~i(p*)lif = / v / ~ ( x ; p*)[vez - / ~2*(r)g(x; r)drlv;(x; ~ ) d x (3.12) The rate probability takes a different form if frequency shift or anharmonicity in discrete modes is important. Such modes could represent the motion of the nearest solvent molecules around the localized electron, for which strong anharmonicity in a low-frequency (50- 100 cm-l) breathing mode has been ~uggested.~O*~~ In such cases the activation energy is still obtained by solution of eqn (3.7), but the pre-exponential factor depends on the detailed topology of the potential surfaces.38 3.3.VARIATION OF THE ELECTRONIC FACTOR We shall now proceed to the effect of the transition configuration on the electronic factor. We shall analyse the effect for the isolated solvated electron, i.e. for sufficiently large electron-transfer distances, and represent the electronic wavefunction by a Mike wavefunction of the form vls(x> = (A3/7E>+ exp(-Alxl) (3.13) where we determine the orbital exponent by the variational calculus introduced byA . M . KUZNETSOV AND J . ULSTRUP 37 Pekar.38 The electronic wavefunction at the initial-state equilibrium polarization is found by minimizing the functional where p is the mass of the electron. Pio(r), Dio(r) and yls(x) are related by the equations where e is the electronic charge, and v/is(x) is obtained from yls(x) by taking the orbital exponent, Lo, corresponding to the equilibrium polarization Pl0(r).Fbo can thus be written (3.16) In comparison, in the transition configuration where P(r) = P*(r) where y f S is the wavefunction for the polarization PT(r) P*(r) = -2 [(l - 6*)D:o(r) + O*D&(r - R) + Dfzo(r - R)]. (3.18) and D the induction fields created by the electron (e) and the scavenger (z) in the initial (i) and final (f) states. By inserting eqn (3.18) in eqn (3.16) and ignoring the field contributions from the acceptor site, the following simplified equation is obtained Fb,(ryl*,) = -\ IVyTs(x)I2dx - & (1 - 6*) ,/ D:o(r)Dr(r)dr. (3.19) The functionals P'! and F i thus only coincide for activationless processes where 8* -+ 0 and Plo(r) = P:(r). The equation for 3, is (3.20) h2 - 1 61Vyys(x)12dx - & D:o(r)dD;o(r)dr = 0 2P whereas the equation takes the following form in the ' normal ' (0 > 6* > 1) and strongly exothermic (6* < 0) freeAenergy regions k2 - ,/ 61Vy,*,(x)12dx - f (1 - O*)/ Dieo(r)dDf(r)dr = 0.2P 4x h2 2P (3.21) With the particular form of yls given by eqn (3.13) this equation becomes (3.22) h2 - 3, - (1 - 6*)I(A) = 0 2P where38 QUANTUM THEORY OF THE SOLVATED ELECTRON This quantity is calculated in the appendix. The result is (3.24) and in combination with eqn (3.22) and the known value of A. for 8" = 0 this equation provides (numerical) values for A. Before doing so we shall, however, derive a simplified expression for A valid for small 8*.For IS*] < 1 the solution of eqn (3.22) can be written A(8*) M A,{ 1 - 8*/[1 - p(1 - 8*)(dZ/dA)lA = ~~/h']> = &(l - @*) < = [l - p(dl/dA)IA = ~Jh~1-l. (3.25) Since dI/dA < 0, it follows that 0 < 5 < 1. From eqn (3.24) its value is found to be 5/11, and A thus decreases with increasing 8". Eqn (3.25) has several implications. Since the main contribution to the two- centre resonance integral in eqn (3.12) is provided by a small region, Ax*, around a point x*, where the integrand is maximum, [Vi(P*)lir is approximately where the superscripts refer to an electron at the trap or at the acceptor. If the explicit interactions for large distances are introduced, eqn (3.26) can be written [ Vi(P*)lif M Ax*&(x* ; P*)ylfs(x*; '*){ [- ~~1.r: RI where ze is the charge of the acceptor.For small 8* the perturbation is dominated by the screened electrostatic interaction with the acceptor. On the other hand, for finite lo* I or large electrostatic screening the perturbation is dominated by the second term, i.e. the interaction of the electron with the inertial polarization field. The major effect is, however, on the overlap integral, and the resonance integral has the approximate form [Vi(P*)Iif = [ ~ i < ~ i o > J i ~ e x ~ ( ~ o ~ ~ ' I x * l ) . (3.28) The electronic transmission coefficient, Kif(P*), would then be 'Cif(P*) = 'C*r(Pio)exP(2Ao@* Ix* I). (3.29) An important effect is thus that in the " normal " free-energy region the radius of the solvated electron is larger than at equilibrium.For Ix*J M 5 A (an electron-transfer distance of ca. 10 A), A x 0.5-0.8 A-';l and O* = 0.1 5, i.e. a fairly exothermic process, the effect amounts to a factor of 1.2-1.3 in the radius (1-l) and 1.4-1.7 in the trans- mission coefficient, but may arise to a factor of 20-30 for larger distance (Ix*l = 15 A) and larger l8*I (0.3), where A must, however, be calculated numerically. The effect vanishes for activationless processes, while the electron is more strongly localized in the transition region than at equilibrium for strongly exothermic processes, where 8 isA . M. KUZNETSOV A N D J . ULSTRUP 39 negative. The effect is here enlarged by the fact that the first two terms on the right- hand side of eqn (3.27) partly cancel each other.These effects should be reflected in the activation volumes being significant and positive in the normal, but negative in the strongly exothermic region. 3.4. EFFECT OF HIGH-FREQUENCY NUCLEAR MODES In this section we consider the variation of the electronic factor when a continuum of high-frequency modes is present. Such modcs are important at low temperatures and even at room temperature for processes in aqueous solution where the vibrational frequencies of a notable fraction of the solvent modes are higher than kT/h.41943 At low temperatures the number of such modes which fluctuate by nuclear tunnelling rather than by thermal activation further depends much more sensitively on the temperature than at high temperatures.18-20 In the following we consider a continuous set of " linear," and either classical or quantum-mechanical solvent modes.We could incorporate local modes of rather general form as well. If they have " low" vibrational frequencies such as the an- harmonic breathing modes around the solvated electron 40741 we can follow a modified version (see below) of the procedure represented by eqn (3.7) and (3.8). If they are of a quantum-mechanical nature their Franck-Condon overlap factors and vibrational energies appear in the expression for Wif such as shown e l ~ e w h e r e . ~ ~ ~ ' * * ~ ~ It is convenient to follow the procedure based on effective Hamiltonians 26931*33 but slightly modified from previous formulations. The Hamiltonian of the solvent corresponds to a set of harmonic oscillators (3.30) The vibrational spectrum is dztermined by the spectrum for dielectric absorption, and the coordinates related to the Fourier components of the polarization vector, P k , by a linear transformation (3.31) where &v are coefficients determined by the dielectric permittivity function.between the field and the solvent polarization is In the presence of electric charges giving the field cf(r), the energy of interaction (3.32) where 8 k are the Fourier components of the field. The corresponding equilibrium co- ordinate shifts, qkv0, are (3.33) The transition probability per unit time is40 QUANTUM THEORY OF THE SOLVATED ELECTRON The integrals can be calculated by the saddle-point method and the saddle points {q&) and 6* determined by the equations This corresponds to the polarization components Pz in the saddle point phmkv(1 - e*) 2 tanh P ; = z & ( V phUkv(1 - 0*) phmkv0* &’ + tanh- 2 tanh phmkv6* tanh- 2 phWkv@* .;i.+ tanh- phmkv(1 - 0.) + 2 2 tanh Using the sum rules for the polarization we can write eqn (3.38) as phm(1 - 0*) 2 tanh phmS* & + tanh - phm(1 - 0*) 2 2 tanh pmo* tanhT (3.38) L + tanh- + ptim(1 - 8”) 2 2 tanh where G(k,m) are the space and time Fourier components of the retarded Green’s function for the solvent polarization, and Im denotes its imaginary part, If spatial dispersion of the medium can be ignored, the k-dependence of G vanishes and eqn (3.39) has the form pm(1 - 0”) 2 + tanh- tanh phcug * ‘i ( r ) p*(r) =2[*mG(m)( ?c tanh pkm(1 - 2 - S*) 2 PhC06* tanh If we can also ignore spherical asymmetry in the charge distribution caused by a non- spherical trap or by the field of the scavenger, G(m) is related to the frequency-depen- dent dielectric permittivity, &(a), by the equation G(m) = L( 1 - &&).4n (3.41)A . M. KUZNETSOV AND J . ULSTRUP 41 The equation for the saddle point @* is then pkw(28" - 1) (3.42) 2 sinh - 2 sinh J and the rate probability [ Y,(P,,)]~~~~ exp[-p@*A& - @(@*)I (3.43) ptiw(1 - 0) pti~08 Pha sinh - . (3.44) 2 2 sinh @(O) = /dr[gi0(r) - gfo(r)l2 :/% f ImG(w) sinh - 2 We shall now modify this formalism by including the variation of the electronic resonance integral with 8". The procedure is again to minimize the function given by eqn (3.17) but now the polarization must be replaced by eqn (3.40), which incorporates the frequency dispersion.P is approximately determined solely by the field of the trapped electron, Bio(r). phw(1 - @*) tanh Q&). (3.45) phm(1 - $8) + tanh- ptiwo* P*(r) z ImG(co) K c i , 2 2 tanh Further calculation of P*(r) now requires specification of ImG(w). ImG(w) is determined by the absorption spectrum of the medium. For liquid water this is well represented by a broad " dissipative " band with maximum at 1-10 cm-I and several narrower but still relatively wide infrared resonance^,^^ while the spectrum is likely to be dominated by infrared absorption bands for amorphous solids. We can, however, obtain some insight into the behaviour of P*(r) without specification of ImG(w) by dividing all the modes into two broad classes separated by a particular frequency 6.43 For w < G the hyperbolic tangent functions are replaced by their arguments (the classical limit) and for w > i3 by unity (the quantum limit).i3 is therefore ca. kT/h, but more precise values can be determined from a fit of the approximate expressions to eqn (3.43). Note that since@* is typically numerically small for reactions of the solvated electron, validity of the replacement of tanh@hwO/*2) by unity does not follow solely from the presence of high frequencies. We can then write P*(r) in the form p*(r) z x w z p E I m G ( w ) ( l - 8*)B:o(r) roo 1 c?:o(r). (3.46) +iJ $Im(Gw) 1 + tanh pfic00* - 2 w If O* is sufficiently small that ptiw8*/2 < 1 in spite of large frequencies, eqn (3.46) takes the simple form P*(r) z (1 - o*)P;;(r) + Py;(r) = Pc*I(r) + P&(r) (3.47)42 QUANTUM THEORY OF THE SOLVATED ELECTRON where the classical [P$(r)] and " quantum " [P$(r)] components refer to parts of P* (r) where the integration with respect to w is from zero to 5, and from 6 to infinity, respectively.If the inverse inequality, phw8*/2 + 1, is valid, then P*(r) FS (1 - t?*)P;;(r) + +Pyl(r) = PC*1(r) + P&(r). (3.48) For the 1s-like wavefunctions 3, is now determined by the equation (3.49) h2 P - a -f(e*)~(;l) = o where I@) is given by eqn (3.24), and the functionf(6)* is phw(i - e*) (3.50) 2 + tanh- tanh ptiwe* * f(S*) = f $Ie w ImG(w) pti@(i - e*) 2 2 tanh This equation is numerically tractable, but for small 18* I the following simple solution emerges. Expansion off(8*) to first order gives h2 lu f(8*) NN f(8) + O*df(B*)/dO*lp = 0 = 1 + df(8*)/d8*le* = 0 = -a/I(a).(3.51) By means of eqn (3.50) and by following the same procedure as for the classical limit we obtain From eqn (3.52) it is seen that p = 1 if ImG(w) # 0 only for classical modes. Other- wise p > 0 since for all values of Btiw/2. Insertion of eqn (3.53) in eqn (3.43) gives Eqn (3.54) can be further converted to the following form valid close to the activation- less region where is small. In this region 's 8* E (EP' + AFo)/pA2 (3.55) where the total nuclear reorganization energy, E,!Ot, is (3.56) E:O' = z/[&'io(r) 1 - d'fo(r)]2dr/$ ImG(w) and A2 = ,/dr[Bio(r) - tiTfo(r)l2 $paImG(w)coth P a -. 2 (3.57)A . M. KUZNETSOV AND J . ULSTRUP 43 The rate probability then takes a Gaussian dependence of AF,, + AFo)2/A2].(3.58) If EPt and A are unaffected by the variation of the electronic wavefunction, Wit takes the following form for small l8*1 Since p > 1, eqn (3.59) shows that the variation of the electronic wavefunction with O* has a stronger effect for high-frequency modes than for the classical limit, caused by a stronger dependence of the high-frequency part of the polarization on O*. The effect is therefore expected to increase with decreasing temperature. For larger 8 we must use numerical solutions of eqn (3.42), (3.45) and (3.50). An analytical rate expression can, however, still be obtained for the particular case where ImG(co) vanishes in a frequency region around kT/h, so that ImG(co) consists of separated low- and high-frequency parts, provided that the latter constitutes a narrow resonance centred at a frequency co,,.For large negative AFo where O* < 0 the equation for O* is -AFo = E;'(l - 20*) + E$"exp(-Phw,,B*) (3.60) where E;' and E$" are the reorganization energies of the low- and high-frequency modes, respectively. O* is then approximated by 8" ---@hc~~,,)-~ ln(/AF,J/E:") (3.61) and the transition probability x exP[-2~ol~*lp(p~co,.)-' ~n(lAFol/E,9") (3.62) where y = ln(lAFoI/EI,Qu) - 1. Eqn (3.62) is the energy-gap law known from the theory of radiationless p r o c e s ~ e s , ~ ~ ~ ~ ~ but it incorporates here the variation of the electronic wavefunction with the energy gap. 4. NUMERICAL CALCULATION A N D CONCLUDING REMARKS Eqn (3.42) is the basis for numerical calculation of 8*. Taking E:"' as a parameter this equation can be rewritten as (AFo/EPt) = 2 Phm(20* - 1) ImG(co)/[@ co2 ImG(co).sinh - 2 Similarly, eqn (3.44) can be given the form (4.1) Phco(1 - 0) phcoo sinh - 2 Phw ImG(co)/[*ImG(co). co (4.2) sinh - 244 QUANTUM THEORY OF THE SOLVATED ELECTRON I I I - - e* -0.5- \. - \. I 1 1. 'a,. - 1 - 2 - 3 -0.4 When 8* is inserted in eqn (3.43) and (3.44) the rate probability can be calculated numerically for the continuous distribution of nuclear modes represented by ImG(co). These equations must subsequently be combined with eqn (3.49) and (3.50) in order to incorporate the variation of the electronic factor. For disordered media representative forms of ImG(m) are combinations of Debye and resonance absorption bands ImG(co) = where RDi and QRj are the *frequencies of maximum absorption for the Debye and resonance bands, respectively, rj the widths of the latter, and Ggi and Gij constants determined by the dielectric constants in the same way as for the c constants above. For solid glasses the Debye functions must be considered as empirical distribution func- tions since relaxational motion is here of minor importance.Fig. 1 shows plots of 8* against AFo for two temperatures representative of experi- mentally investigated reactions of the solvated electron. We have represented the medium by a single Debye band of maximum frequency 10 cm-l for T = 298 K (corresponding to a liquid soiution) and 150 cm-l for T = 77 K where the medium is a solid. Fig. 2 shows the corresponding dependence of the nuclear part of Wi, where against AF,.These curves exhibit characteristic asymmetric maxima which reflect tunnelling of the nuclear modes in the high-frequency branches of the distribution. For large values of E:O* and high temperatures the numerical results are only insig- nificantly modified if, for the same ImG(w) is represented by one Debye band with the same maximum frequency and one narrow resonance at 50-100 cm-l corres- ponding to the " breathing " mode of the solvent molecules nearest to the electron.A . M. KUZNETSOV A N D J . ULSTRUP 45 - 1 - 2 -3 AFoleV FIG. 2.-lnA plotted against AFo, where A is the " nuclear part " of the rate probability. The symbols The difference is, however, more pronounced at lower temperatures and values of E:"'.Fig. 3 shows the variation of 1 with O* calculated from eqn (3.22)-(3.25) for the classical limit and from eqn (3.49) and (3.50) for the general case covering also high- frequency modes. In the classical limit there is no significant numerical difference from the limiting form for small lO*l [eqn (3.25)] up to values of 1O*I z 0.3. For larger 16.1, are the same as in fig. 1. Energy quantities in eV. 1.5 2 1 - --- 0 - 5 I 1 -0.5 0 0.5 8* FIG. 3.-A/A0 calculated from eqn (3.22), (3.24), (3.49) and (3.50), plotted against 8*. (-) The classical limit [eqn (3.22) and (3.24)]. (- - - -) The limiting classical case for small lo* I [eqn (3.25)]. (. - . - .) A/A, calculated from eqn (3.49) and (3.50). The ordinate scale for the latter curve should be multiplied by two.46 QUANTUM THEORY OF THE SOLVATED ELECTRON A varies more slowly in the strongly exothermic region but more strongly in the " normal " region.For high-frequency modes A also depends rather slowly on 8" in the normal region. However, for 8" < 0 there is a very significant rise in A/& to values which imply that the rate constant could drop by several orders of magnitude as a consequence of this effect. The cause of this difference is the much stronger dependence of P*(r) on 8* in this region as reflected in the high- and low-frequency parts of this quantity [eqn (3.45) and (3.46)]. The effect of the variation of the electronic factor on the drop in the rate constant with increasingly negative AF' for strongly exothermic processes is thus expected to be no less important than the drop caused by increasing vibrational excitation of the nuclear modes such as revealed by fig. 2.This effect should therefore be incorporated in analysis of experimental data for electron transfer in this free-energy region. APPENDIX We derive here the integral in eqn (3.24) from eqn (3.21) and (3.23). Since I(1) arises from the second term in eqn (3.21) we calculate at first the integral 9 = Dfo (Y) D; (Y) dv. (Al) 9 can be rewritten in terms of the Fourier components of Dfo ( r ) and Df(r), i.e. Dfo(k) and Dt(k), respectively I where Df* is the complex conjugate of D:. 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