Faraday Discuss. Chem. SOC., 1982,74, 49-56 Quantum Theory of the Processes of Charge Transfer Recent Advances BY ALEXANDER 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. Received 10th May, 1982 A new aspect of the interaction of an electron with the polarization of the surrounding medium is discussed. Effects caused by the modulation of the electron wavefunctions by fluctuations in the polarization are considered. New methods of calculating the transition probability are presented. They permit one to take into account the effects mentioned above and to go beyond the Condon approximation. Many new results have been obtained in the quantum-mechanical theory of elementary charge-transfer processes in polar media during recent years.Some of these are related to the fundamentals of the theory itself, others concern the application of the theory to various phenomena, including chemical and electrochemical reactions of the electron and proton transfer, adsorption, processes involving solvated and trapped electrons in solids and in liquids, biochemical reactions etc. In this paper we focus attention on those results which are of principal importance in developing the fundamentals of the theory. They will be presented for homo- geneous electron-transfer reactions as an example. A review of earlier results may be found in ref. (1)-(5). INTERACTION OF THE ELECTRON WITH THE POLARIZATION OF THE MEDIUM (A NEW ASPECT) It is understood that fluctuations in the polarization of the medium play an important role in the processes of charge transfer.The microscopic mechanism of a non-adiabatic electron transfer between two centres A"1 and BZ2 located at a distance R from each other in a polar medium may be described briefly as Owing to the fluctuation of the inertial polarization of the medium from the initial equilibrium value Poi to the transitional configuration P* the equalization of the electron energies E , and e, required by the Franck-Condon principle occurs. In this configuration the electron transfer from the centre A'I to the centre Bz2 takes place followed by the relaxation of the polarization to the final equilibrium value. The behaviour of the degrees of freedom describing local vibrations in the medium or intramolecular vibrations in A and B depends on the values of their vibrational frequencies and on the potential-energy profiles, and may be either classical or quantum-mechanical in nature.It is of no major importance in the context of the problems discussed below. The work required to change the polarization from Poi to P* determines the50 A QUANTUM THEORY OF CHARGE TRANSFER activation free energy, Fa. The transmission coefficient K is largely determined by the value of the electron-resonance integral where qA and tpB are the zeroth-order electron wavefunctions describing the initial and final electron states, Prd the off-diagonal parts of the interaction operator pi leading to the transition of the electron from A to B. To calculate Fa we must final states Ui and U,. In reference model (RM), the approximation are described where c = I/&, - I/cS and is REFERENCE MODEL know the free-energy surfaces (FES) of the initial and the simplest model, which will be referred to as the free-energy surfaces of the solvent in the continuum by the formula U = (27~1~) P2(r)d3r the polaron-theory parameter.The electrostatic interaction of a charge ez with the solvent is chosen in the RM in the form where cfz(t) is the vacuum electric field created by the charge ez. Due to this interaction a polarization Po(r), which depends on the charge distribu- tion in the system, arises in the medium. It is assumed in the RM that the fluctuations of the polarization do not change the charge distribution (i.e. the electron-density distribution) in the initial and final states.Then the free-energy surfaces Ui and U, are quadratic in the deviation of the polarization from the initial (Poi) and final (Po,) equilibrium values. This results in the known expression for the activation free energy F," = (Er + AF)2/4E, where E, is the reorganization energy of the medium and AF is the free-energy of the transition. To calculate the electron-resonance integral we must know the zeroth-order wave- functions and the interaction Pfd. In the RM the interaction of the electron with the second centre (B), Vl;, is usually chosen as the interaction causing the electron transfer from A to B. The superscript od denotes the off-diagonal part of this interaction. The interaction of the electron with the polarization, yep, which is usually strong, is involved in the zeroth-order Hamiltonians determining the electron wavefunctions VIA and VBZ where T, is the kinetic energy of the electron, and the superscript d denotes a diagonal part of the interaction which does not lead to the electron transition and only distorts the electronic states.Note that in the RM the entire interaction of the electron with the polarizationA . M. KUZNETSOV 51 Ve, is involved in eqn (4) in determining the electronic states. Furthermore, the dependence of qA and pB on the polarization is neglected when calculating the transition probability, as mentioned above. A NEW APPROACH Recently the role of the interaction of the electron with the polarization has been analysed in detai1.7-13 It was suggested that the zeroth-order states should be determined using the equations of the type iTe + VeA + Ve$ + VgB + (V&>dlpA = &AvA [Te + VeB + vz + v,dA + ( V & ) d b B = EBvB ( 5 ) where VeAp is that part of the interaction of the electron with the polarization which creates (together with VeA) the potential well for the electron near centre A.I/: is defined in a similar way. Then the perturbation operators pyd and PFd have the form Ppd = (veB)od + (v&)od = VeB - (VeB)d + V$ - <V$>d (6) = ( VeA)od + (v$)od = VeA - ( veA)d + - (V&)d* The above definitions of the zeroth-order states and of the perturbation operators are more reasonable from a physical point of view than those used in the RM. The zeroth-order states determined in the RM, taking into account all interactions with the polarization at various values of the polarization, may not describe the electron localized near a given centre.In our new approach eqn (5) involve only that part of the interaction of the electron with the polarization which retains its location near a given centre at any value of the polarization. This new approach enables us to consider all the physical effects which are due to the interaction of the electron with the polarization of the medium and to take them into account in calculating the transition probability. These effects are: (i) The eflect of diagonal dynamic disorder. The fluctuations of the polarization change the position of the electronic energy levels E~(P) and cB(P), enabling us to satisfy the requirements of the Franck-Condon principle.(ii) The eflect of of-diagonal dynamic disorder. The interaction of the electron with the polarization fluctuations near the other centre leads to the appearance of additional terms V$ and V$ in the perturbation operators for forward (pi) and backward ( vf) transitions in eqn ( 5 ) . These terms depend on the polarization of the medium. In some cases they may considerably exceed the terms describing a direct interaction with the centres B (VeB) and A (VeA). The interaction of the electron with the polarization fluctuations plays an important role in processes involving the solvated, trapped or weakly bound electrons, and in particular in the electrochemical generation of solvated electrons. (iii) The eflect of diagonal-ofl-diagonal dynamic disorder.The fluctuations of the polarization produce a variation in the distributions of the electron densities near the donor A and the acceptor B (i.e. they cause a modulation of the electronic wave- functions qA and vB). This leads to a modulation of the overlapping of the electron clouds of the donor and the acceptor and hence to a change in the transmission coefficient as compared to the value calculated in the approximation of constant electron density (ACED) (table 1). The variation in the (iv) An additional efect of the diagonal dynamic disorder.52 A QUANTUM THEORY OF CHARGE TRANSFER TABLE 1 .-KINETIC PARAMETERS OF ELECTRON TRANSFER FOR THE SYMMETRICAL SYSTEM IN THE CONDON APPROXIMATION TAKING DUE ACCOUNT OF THE MODULATION OF THE ELECTRON WAVEFUNCTIONS BY FLUCTUATIONS IN THE POLARIZATIONu PISb FJF; 2a,R = 4" 2a,R = 8" KlKo WlWo KIKO WlWO 0 0.811 4.70 56.99 22.11 268.0 0.5 0.864 5.21 76.92 27.11 400.5 1 0.893 5.54 93.39 30.69 517.4 2 0.925 6.05 117.92 36.60 713.4 -0.1 0.795 5.11 58.34 26.09 298.0 a Calculated neglecting the mutual influence of the donor and the acceptor on the zeroth-order distribution of the electron densities near the centres A and B due to fluctuations in the polarization leads to a change in the interaction of the electron with the polariza- tion of the medium and hence a change in the shape of the free-energy surfaces iYi and U,.This effect leads to a change in the activation free energy Fa with respect to the value Fi calculated in the ACED (table 1). A detailed discussion of these effects and numerical calculations are given in ref.(7)-(13). It is shown that the electron densities in the transitional configuration and the activation free energy calculated with due account for these effects and beyond the Condon approximation may differ significantly from those calculated in ACED.7-13 characteristics; PIS = 16z/5c~,; up = 5mce2/16h2. DEVELOPMENT OF THE METHODS OF CALCULATION Below we shall discuss the methods of calculating the probability per unit time, W, of the elementary act of a non-adiabatic charge transfer and the approximations used. One of the main methods of calculation for the non-adiabatic reaction is quantum-mechanical perturbation theory. It is shown that the approximate expression for W may be written as follows:i4 = pe[q*, nA(q*), nB(q*)l pn(Pi,Pf> (7) (8) (9) where Pe and P, are the electronic and nuclear factors, given by pe [q,nA(x * 4) ,nB (x * 4) 1 = A2 1 (x * 4) I nA(x * q>nB (x * 4) Pn(Pi,pf> = (l/ihkT) S do exp(PFi - PoAJ)Tr[pi(l - e)p,(e)l where pi is the off-diagonal part of the interaction leading to the transition, n A and nB are the electron densities of the transferable electron near the donor A and the acceptor B, respectively, Fi is the free energy of the initial state, AJ is the free energy of transition, p = l/kT, pi and p f are the statistical operators of heavy particles in the initial and final states calculated at the temperatures T/(1 - 0) and 778, q is the set of coordinates of the heavy particles, x* the point of the maximum overlapping of the electron " clouds " of the donor and the acceptor, A the effective region giving the major contribution to the electron resonance integral, and q* the transitional configura- tion.If the dependence of pi, n A and nB on q is neglected, we obtain the approximation uf constant eZectron densities (ACED). The results obtained in this approximation are those obtained in the reference model. The usual Condon approximation (CA) is obtained if the dependence of the electronic factor P , on q in eqn (7) is taken into account but the nuclear factor P, is calculated in the same manner as in the ACED.A . M. KUZNETSOV 53 tf in calculating pi and pf involved in P, we take into account the modulation of the electron densities nA and n B by the vibrations of heavy particles, we obtain an improved Condon approximation (ICA).If a classical approximation is used for the calculation of pi and pf we may consider all the cases of Condon approximation (ACED, CA, ICA) and in certain models we may go beyond the Condon approximation (BCA).l0-l4 ZEROTH-ORDER ELECTRON STATES The calculation of the zeroth-order electron states at arbitrary values of the nuclear coordinates is a very complicated problem. It becomes a little simpler if the vibrational subsystem is a classical one. To calculate the transition probability in the CA we have to find the electron wavefunctions only in the equilibrium configura- tion and in the transitional configuration. In the classical limit the transitional configuration is determined as follows 4" = (1 - Q*)qoi + 6"qof (1 1) where 8" is a symmetry factor.If the interaction of the electron with the vibrational subsystem is linear, the terms V& and V$ involved in eqn (5) in the transitional configuration will differ from those in the equilibrium configuration only by the factors (1 - 8*) and 1 9 * . ~ This fact simplifies significantly the calculation and enables us to use the methods developed for the equilibrium configurations. In particular a direct variational method has been The wavefunctions of the initial and final states are chosen in the form qi z exp(-air), qf z exp(-EfIr - RI). (12) The dependence of the orbital exponents ai and orf on the symmetry factor were calculated. It was shown that for 0 < 8* < 1 the radii of the localization of the electron near the donor and the acceptor in the transitional configuration are larger than those in corresponding equilibrium configurations.More complicated functions may also be used to approximate qi and qf. The explicit dependence of qi and pf on the nuclear coordinates is necessary for the calculations in the ICA and BCA. The model considered assumes, in particular, that the major contribution to the transition probability comes from symmetrical polarization fluctuations near the donor and the acceptor of the type &PA = (c/4z)t D?(r,Poi), SPB = ( c / ~ z ) v DeB(r,Pof> (13) where t and 7 are independent variables, and D2(r,Poi) and D:(r,Pof) are the electro- static inductions due to the electron at the equilibrium values of the polarization. This model enables us to take into account the dependence of pi and qf on the polarization vibrations and the change in shape of the free-energy surfaces Ui and U,, and to go beyond the 'Condon a p p r o x i m a t i ~ n .l ~ * ~ ~ ~ ~ ~ Calculations beyond the Condon approximation have shown that the effects of deviation from the CA are important only for long-range tran~fer.'~.'~,'~ At short transfer distances the ICA is satisfactory. More exact results may be obtained by considering polarization fluctuations of a more complicated shape. PATH-INTEGRAL METHOD The above analysis uses the assumption that the major contribution to the transi- tion probability arises from polarization fluctuations of a certain type [see eqn (1 3)].54 A QUANTUM THEORY OF CHARGE TRANSFER A more general and more consistent method has been This starts from the Fermi golden rule, and may be expressed in terms of Feynman path integrals of the initial and final states for the interacting electron and nucleus subsystems.The calculation gives the following expression for the transition probability14*15 W = const x p?(x*,q*)Zi’ pr dxdx’ q;”(x,O)qi(x’,O)qf(x,O)pf*(x’,O) c-ica where pi is the off-diagonal part of the interaction leading to the transition, pi(x,z) and pf(x,z) are the electron wavefunctions of the donor and the acceptor, respectively, zi = 1 - 8, zf = 6, and Zi is the statistical sum of the initial state. Rif(m,z,z’,zi,zf) = ([sinhphco(zi - ~)~inh~hwz’~inh~hmz~sinh~hwz~cos(~hw/2) + sinhphmzsinhphm(zf - z’)sinh~hwzisinh~htwzfcosh(~hw/2) + sinh~hwzsinh~h~r’sinh~hmzisinh~hwzfcosh(~w/2)] /2sin h(Phw/2)sin h/lhwz, sin hphwz,) (20) where qi0(z) and qio(z) are the equilibrium values of the coordinates of the nuclear subsystem in the initial and final states corresponding to the electron wavefunctions pi(x,z) and p,(x,z). wK is the frequency spectrum of the nuclear subsystem, and Hi and I€! are the Hamiltonians of the electron in the initial and final states.In the simplest case they take the form Hi = -(h2/2m)A + VeA Hof = -(h2/2m)A + V,,A. M. KUZNETSOV 55 where VeA and VeB are the interaction energies of the electron with the donor and the acceptor. In a more rigorous consideration Hi involves in addition the diagonal part of the interaction of the electron with the acceptor and Hi involves the diagonal part of the interaction of the electron with the donor.Eqn (14)-(22) are valid for the case when the nuclear subsystem can be described by the harmonic approximation and for the case of the electron interaction with the fluctuations of the polarization or of the density of the medium in the long-wave approximation. In the case when the vibrations of the medium polarization play the major role in the transition, the quantities involved in eqn (16)-(22) may be expressed in terms of the complex dielectric function of the medium, ~ ( k , m ) , using the summation where Dak(7) is the Fourier component of the electrostatic induction Da(r7Z) due to an electron in the state qa(x,z). In the classical limit similar expressions for the transition probability are obtained for the case of a linear interaction of the electron with any fluctuating field, provided the fluctuations of the field are Gaussian. In this limit more general expressions may be obtained for the free-energy surfaces of arbitrary shape.The electron wavefunctions qi(x,z) and qf(x,z) are determined by the condition of the maximum value of the transition pr0babi1ity.l~ In particular in the improved Condon approximation the equations for the calculation of qi and qf have the form & i S f [ ~ i ~ i l + &iSp[~i,qf,zi,rf] = 0 & S [ ~ f , ~ f l + ~ & j ' p [ ~ i ~ ~ f , ~ i , ~ f l = 0 (25) (26) where Bpa denotes the variation with respect to qa. Note that the equations for pi and pf are coupled and describe a self-consistent state of the electrons in the donor and in the acceptor.This reflects the fact that in the transitional configuration the electron interacts with the polarization fluctuation near its own location centre and with that near the other centre. In the classical limit the expression for the transition probability takes the form14*15 T f * - 5 k l . r {[ T&Di(r,z)]' + [ 1 dzDf(r,z)12 + 2 lo:z ,/~z'Di(r,z)Of(r,zl)) 871 0 0 1 c 1 - /odr(CiiHi]f+) + E-/ d3r{\ 0 d~D~(r,r)}~]}; a = i, f where Ti* = 1 - 8*, T ~ * = O* [8* is the symmetry factor determined as a saddle point in the integral over 0 in eqn (14)], pi is the electron wavefunction of the donor in the initial equilibrium configuration, and Di the corresponding electrostatic induction. In particular for the symmetrical transition 8* = zi* = zf* = 1/2.We may use a direct variational method rather than have to solve eqn (23) and (24) for the wave- functions in the transitional configuration and similar equations for the wavefunctions56 A QUANTUM THEORY OF CHARGE TRANSFER in the equilibrium configurations. Using the exponential functions of the type in eqn (12) as probe functions for pi and pi and varying the expressions for the transition probability and for the statistical sum of the initial states over a and a. we may find these parameters and thus calculate the free energy of activation. The calculation in this case, making the assumption that the mutual influence of centres A and B on the zeroth-order wavefunctions qi and pf may be neglected, gives for the free energy of activation14J5 where F," = (5/32)ce2a0 is the free energy of activation calculated at R + co neglecting the modulation of the electron wavefunctions by the phonon field; a.is the inverse localization radius of the electron in the equilibrium configuration. As one can see from eqn (28) Fa may differ significantly from F,", especially for the transfer of weakly bound electrons between neutral or negatively charged centres. CONCLUSIONS The above analysis shows that the interaction of the electron with the vibrational subsystem leads to various effects which should be taken into account when calculating the probability of electron transfer. Some of these effects (viz. the effect of diagonal dynamic disorder, respresenting the dependence of the electron energies on the nuclear coordinates) have been considered earlier. Two new methods of calculation of the transition probability presented in this paper enable us to incorporate into the theory other effects which are due to the modulation of the electronic wavefunctions caused by the change in the nuclear configuration. These effects are of especial importance for processes involving weakly bound, trapped and solvated electrons and for long- range electron transfer. The results obtained provide deeper understanding of the mechanism of elementary charge-transfer processes in polar media. ' R. R, Dogonadze and A. M. Kuznetsov, in Itogi Nauki i Techniki, ser. Kinetika i Katalis (VINITI, Moscow, 1978), vol. 5 . R. R. Dogonadze and A. M. Kuznetsov, Prog. Surf: Sci., 1975,6, 1. J. 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