J . Phys. Chem. 1992,96, 3337-3345 pendence of the rate of formation of radicals, the hypothesis is advanced that a biphotonic step, possibly involving the absorption of a photon by the lowest triplet, is operative in producing a state X, more reactive than the triplet. In spite of the kinetic evidence produced, no definite spectroscopic and kinetic identifications of
3337
species X are reported. Acknowledgment. I thank L. Minghetti, M. Minghetti, and R. Cortesi for technical assistance and G. Gubellini for the drawings.
Role of High-Frequency and Low-Frequency Polarization of the Medium in the Kinetics of Electron Transfer and Thermodynamics of Solvation Alexander M. Kuznetsov The A. N . Frumkin Institute of Electrochemistry, Academy of Sciences of the USSR,Leninskii prospect 31, Moscow 1 1 7071, USSR (Received: August 26, 1991)
The behavior of high-frequencyand low-frequency polarizationsof the medium in the processes of electron transfer is considered in the framework of a rigorous analysis. It is shown that for the adiabatic reactions the solvation of the transition state by classical (and hence inertial with respect to transferable electron) polarization is considerably weaker than the solvation of the initial state due to delocalization of the electron density over both reactants in the former case. The solvations of the transition state and of the initial one by the high-frequency (Le., inertialess) polarization are close to each other. Possible influence of the high-frequency polarization on the activation barrier and the role of a retardation of the polarization with respect to the electron are discussed.
I. Introduction It is understood that the interaction of the charge with the polarization of the medium plays a great role in the processes of charge transfer in the condensed phase. The study of the dynamics of polarization in the course of these processes is a central point of the quantum mechanical theory of electron-transfer reactions, processes of the transfer of protons and atom groups.I-* It was assumed in first works on the theory of the electrontransfer reactions that the total polarization of the medium Pt may be split into two parts: fast (inertialess) P, and slow (inertial) P.1,495Similar to polaron theory, it was assumed that the fast polarization is characterized by much greater frequencies of fluctuations wfas compared to the characteristic frequency of the motion w, of the transferable electron. Therefore, it follows adiabatically the motion of the latter. This physical idea was used to exclude the inertialess polarization from a dynamic description of the transition process. On the contrary, the slow polarization cannot follow the electron motion and creates a Franck-Condon barrier for its transition from a donor to an acceptor. For electron-transfer processes in polar liquids, e.g., in water, fast (inertialess) polarization P,in fact is related to the electronic polarization of the medium. The inertial polarization P is formed mainly by orientational polarization of the dipole molecules, and also by vibrational and librational polarizations, and partially by electronic polarization. Thus the electronic polarization gives a contribution to both inertialess (fast) and inertial (slow) polarizations of the medium. It was understood in later works that the inertialess polarization plays an essential role in electron transfer.+” It screens partially ( I ) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, 15, 155. (2) Dogonadze, R . R.; Kuznetsov, A . M . In Comprehensive Treatise of Elecrrochemistry; Conway. B. E., Bockris, J. 0. M., Yeager, E., Eds.; Plenum Press: New York, 1983; Vol. 7, p 1 . (3) Kuznetsov, A. M.; Ulstrup, J.; Vorotyntsev, M. A . In Chemical Physics of Solvation, Part C; Dogonadze, R. R., Kalman, E., Kornyshev, A . A , , Ulstrup, J., Eds.; Elsevier: Amsterdam, 1988; p 163. (4) Dogonadze, R. R.; Kuznetsov, A . M.; Levich. V. G. Electrochim. Acta 1968. .., 13. ., 1025. ..-. ~
(5) Hush, N . S. Trans. Faraday Soc. 1961, 57, 557. (6) Calef, D. F.; Wolynes, P.G.1.Phys. Chem. 1983, 87, 3387. (7) Zusman, L. D. Chem. Phys. 1980, 49, 295. (8) Grote, R. F.; Hynes, J . T. J . Chem. Phys. 1980, 73, 2715.
the electric field of charged species and weakens their interaction, influencing thereby the value of a perturbation operator leading to the electron transfer. In nonadiabatic reactions this effect is involved in the value of the electron transmission coefficient. In the adiabatic reactions it is involved in the value of lowering of the potential barrier due to the resonance splitting of the potential energy surfaces. In past years the problem of the participation of the electronic polarization in the electron-transfer reactions was under discussion in relation to two aspects: (a) In some cases one may expect that the characteristic frequencies of a part of the electronic polarization of the medium and of the motion of the transferable electron may be comparable (Wf
-
,4).12
(b) It is supposed that since the electronic polarization is fast, it must always be in equilibrium with the distribution of the charge in the reacting system’2J3and hence it may influence the value of the activational barrier for the adiabatic reactions due to the change of the solvation energy while going from the equilibrium configuration to the transitional one.I2 The present paper is aimed at analyzing in more detail the role played by high-frequency (electronic) and low-frequency polarizations in the kinetics of electron transfer and thermodynamics of the reactant solvation. A way of rigorous decomposition of the polarization into inertial and inertialess components is presented. 11. Formulation of the Problem
A process of electron transfer is considered from a donor A to an acceptor B located at fixed distance R from each other in a polar medium. The total Hamiltonian of the system has the form
cA
where T, is the kinetic energy of the transferable electron, and are the interactions of the electron with the donor and the acceptor, Vcptis the interaction of the electron with total
cB
(9) Dogonadze, R. R.; Kuznetsov, A . M. Zh. Vses. Khim. Ova 1974, 19, 242. (10) Kuznetsov, A. M . N o w . J . Chim. 1981, 5 , 427. ( I I ) Kuznetsov, A . M . Elektrokhimiya 1982, 18, 594, 598, 736. (12) Kim, H . J.; Hynes, J . T. J . Phys. Chem. 1990, 94, 2736. (13) Newton, M. D.; Friedman, H. L. J . Chem. Phys. 1988, 88, 4460.
0022-3654/92/2096-3337%03.00/00 1992 American Chemical Society
3338 The Journal of Physical Chemistry, Vol. 96, No. 8, 1992
Figure 1. Frequency spectrum of the medium polarization (schemati-
cally). polarization of the medium, and Hp, is the Hamiltonian of the medium polarization with due account of its interaction with A and B. Hp, involves also the Hamiltonian of other degrees of freedom (if any) interacting with the polarization. The medium is assumed to be a local linear dielectric characterized by a complex dielectric function c(w). It is assumed that a transparency band separating classical (w > kBT/h) parts of the polarization exists in the frequency spectrum of the dielectric losses (Figure I). The fluctuations of the electronic polarization of the medium are surely entirely quantum (wf >> kBT/h). The transferable electron is also a quantum particle. However, it will not be supposed so far that we ken,the system remains on the lower adiabatic free energy surface Uad(Pc) in the process of transition from the initial equilibrium configuration P, = PG,corresponding to the localization of the electron in the donor, to the final one Pc = Pof, corresponding to the localization of the electron in the acceptor. In this case the transition probability has the form14J5 where weff is the effective frequency of the motion along the classical degrees of freedom which is largely determined by the form of the spectrum of the dielectric lossesI6 and in some cases by the relaxation time T for the medium polarization (wefr 1 / ~ ) , ” . ’ ~and Fa is a configurational free energy of activation determined by the height of the barrier on the lower adiabatic free energy surface uad(pc), Such reactions are called adiabatic ones.
-
(14) Dogonadze, R. R. In Reactions of Molecules ar Elecrrodes; Hush, N. S . , Ed.;Interscience: London, 1971; p 135. ( I 5 ) Dogonadze, R. R.; Kuznetsov, A. M. In Irogi Nauki Tekh. Ser.: Fiz. Khim. Kiner.; VINITI: Moscow, 1973; Vol. 2, p 3. (16) Kuznetsov, A. M. Elekrrokhimiya 1971, 7 , 1067. (17) Dakhnovskii, Yu.I.; Ovchinnikov, A. A. Chem. Phys. 1983, 80, 17. (18) Hynes, J. T. J . Phys. Chem. 1986, 90, 3701.
Kuznetsov
Figure 2. Profile of the potential energy of the transferable electron at
fixed value of classical polarization P, considered as an external field. (-) Profile of electron potential energy in the fields of the donor and the acceptor. (- - -) Uniform shift of the electron potential energy due to the interaction with the electronic polarization of the medium (6,(x) = constant). (-.-) Retardation of the polarization (6,(x) < 1) in the for c) the ground and barrier region. The eigenvalues E,d(Pc) and g!j((P first excited states as functions of P, serve as potential energy for the classical polarization, and their difference AE = q i ( P , ) - E&’,) represents the splitting of the free energy surfaces. If the resonance splitting AE is small, the probability for the system to remain on the lower free energy surface Uad(Pc) in the process of the variation of Pc from Pa to Pofis also small. In the neighborhood of the top of the barrier a jump to the FES qi occurs with a great probability. The expression for the transition probability in this case differs from that of eq 4 only by the presence of a small factor which is called the electronic transmission coefficient x, and which value is largely determined by the squared value of the resonance splitting V = ( A E / 2 ) 2 .Such reactions are called nonadiabatic, and the activation free energy here is determined by the point of minimum free energy on the crossing of the diabatic free energy surfaces Vi and Vf which are related with the adiabatic free energy surfaces according to eqs 5 and 6 . uad
= (1/2)(ui
GX,= (l/2)(Vi
+ uf- [(Vi - Vf)’ + ( h E ) 2 ] ’ / 2 ) + Vf + [(Vi- Vf)2 + ( A E ) 2 ] ’ / 2 )
(5) (6)
In what follows it is convenient to use the method of the effective Hamiltonian to describe the medium polarizati~n.~J~ According to this method the medium polarization may be described by a set of harmonic oscillators with normal coordinates q k and frequencies wk. The parameters of the effective oscillators involved in final expressions for the observables may be related to the complex dielectric function c(w) using the summation rules derived in refs 3 and 15. Below Uk, q k and ok,Qkwill denote the frequencies and normal coordinates of the effective oscillators corresponding to classical (wk > kBT/h) parts of the polarization, respectively. 111. Adiabatic Reactions As mentioned in section 11, the configurational free energy of the activation for the adiabatic reactions is determined by eq 7 Fa
= Vad(Pc*) - Vad(P0i)
(7)
where Pc*is the value of the classical polarization corresponding to the top of the potential barrier (saddle point) on the FES u a d (it is the transitional configuration). Thus the free energies u,d(pc*) and uad(pOi) must be found for the calculation of the activation free energy. (a) Free Energy in the Transitional Configuration. According to eq 3 we have =
(8) Ead(Pc*) The quantity V(P,) is determined only by the properties of classical polarization. The contribution of all the quantum part of the polarization (including the electronic polarization) is involved in &d(Pc). The quantity uad(Pc) may be interpreted as the configurational free energy of the system at a given value of the classical polarization of the medium Pc provided that all the Vad(Pc*)
V(Pc*)
-k
(19) Dogonadze, R. R.; Kuznetsov, A. M . Elekrrokhimiya 1971, 7 , 763.
The Journal of Physical Chemistry, Vol. 96, No. 8, 1992 3339
Polarization of Medium in Electron-Transfer Kinetics quantum subsystem occupies the unexcited (ground) state. Let us introduce another quantity F(Pc)according to eq 9 where exp[-@~(Pc)I= Tr PIP,
(9)
where the boundary conditions for the trajectories x(r) and Qk(r) have the same form as those in eq 15. The path integral for a harmonic oscillator Qk in the external field uk(x(t))Qk is known.*O As the result we have
the trace is calculated at fixed Pc and p is the density matrix defined by eq 10.
P = exp[-@Wc) - @(H- HPJI
(10)
It follows from eq 10 that exp[-@WC)1 = exp[-BWc)1 Tr (exp[-@W - HpJ1l
(1 1)
The last factor in eq 11 determines the free energy of the quantum subsystem FQ(Pc) exp[-@FQ(Pc)] = Tr [exp(-@HQ)l
(12)
Thus F(Pc) = V(Pc)+ FQ(P,). In the limit of big @-valueswith respect to the electronic variables, FQ(Pc)is equal to the energy of the ground state since from eq 12 we have -(I/@) In ITr [exp(-@HQ)j~ (13) and therefore uad(Pc) F(Pc) = ~ ( P C+) FQ(Pc) (14) Thus the problem is to calculate the quantity FQ(Pc)defined by eq 12 in the limit of big @-values. It is convenient to use the path-integral representationZo ~XP[-BFQ(P~)I = S d x S d Q k S W r ) n Dk Q k ( r )
Uk(X(U)) Vk(X(U?) du du'
1
(19)
where the sum over k is performed over all the oscillators with the frequencies Qk >> kBT/h. Equation 17 shows that the interaction with the electron leads to an additional shift of the equilibrium coordinates of the effective oscillators Qk and to an additional lowering of the energy due to "solvation" of the electron by the quantum part of the polarization PQ: 1 HpQ= $jhQd[Q, - AQko(x(0)12 - a2/aQkZ1 -
X
x(0) = x(@h) = x QdO) = &(Oh) =
Qk
where x is the coordinate of the transferable electron and Qk are the coordinates of the effective oscillators corresponding to quantum part of the polarization. The path-integral analysis for this system allows us to perform a considerable part of the calculations in an exact rigorous form and to go beyond ordinary approximations like a Born-Oppenheimer approximation which was used a priori in earlier papers. It enables us also to see a detailed behavior of the system and to obtain former results as certain limiting cases. Let us split HQ into two parts:
HQ= He + HpQ
(16)
where He = Te + VeA +
VeB
+
VePc
@hQk/2)/sh(@hQk/2)1hQkZAQdX(t+S/2)) X AQ&(r-s/2))])
(22)
where g(r) = 2r at 0 Ir I@ h / 2and g(r) = 2(r - Oh) at @ h / 2 SrS@h.
Thus eq 22 determines the free energy of the electron moving in an effective potential which for a given trajectory of the electron x(r) has the form Verr(x(0) =
The form of the Hamiltonian HpQdefined by eq 17 assumes that the coordinates Qk are counted from their equilibrium values Qko,which are determined by the interaction VMQ+ VepQof the polarization PQ with the donor A and acceptor B, and the term -EunQko describing the interaction of the electron with the equilibrium values of the polarization PQ near the donor A and acceptor B (in the absence of the electron) is included in the potential VeA+ VeBQ The last term in eq 17 describes the interaction of the fluctuations of the quantum part of the polarization with the electron where uL(x(t)) are some functions depending on the electron coordinate at a given moment of time f . Inserting eqs 16 and 17 in eq 15, we obtain exp[-@Fq(Pc)l =
Jv
Jdx
For most trajectories x(t) giving the major contribution to the integral in eq 22, the potential V,, has the form of a double-well potential (Figure 2). At large @ the free energy FQ(Pc*)is approximately equal to FQ(Pc*)z Ead(Pc*)z ti(Pc*)- y M ( P C * )
exP[-(l/h)Soa*H~Q(Q&(r),x(r)) dr] (18)
(24)
where ti(Pc*)is the energy in the left or right potential well at the transitional configuration (ti(Pc*)= cf(Pc*)),AE(Pc*) is the resonance splitting of the energy levels, and y is related to the symmetry factor a:I5 y z [a(I
JDxW exp[-(l/h)Soa*He(x(f),Pc)dt] x
dQk JnDQk(r) k
(23)
- a)]'/2
(25)
(i) Bom-Oppenheimer Limit. The limiting case is considered below where (20) Feynman. R. Stotisficol Physics; Benjamin Inc.: New York, 1972.
3340 The Journal of Physical Chemistry, Vol. 96, No. 8, 1992 we
