J. Phys. Chem. 1994,98, 6120-6127
6120
Quantum Mechanical Theory of Dissociative Electron Transfer in Polar Solvents E. D. German' and A. M. Kuznetsov The A. N . Frumkin Institute of Electrochemistry, Russian Academy of Sciences, 31 Lminsky Prospect, I 1 7071 Moscow, Russia Received: October 19, 1993; In Final Form: March 14, 1994'
A quantum mechanical theory of nonadiabatic dissociative electron transfer processes in polar media is presented. Expressions for the transition probability in quantum and classical limits are derived for different model molecular potentials. It is shown that in the quantum limit the dependence of the transition probability on the dissociation energy of the chemical bond to be broken is involved in the temperature independent tunneling factor, whereas in the classical limit it comes from the activation free energy. Quantum effects lead to a weaker dependence on the reaction free energy as compared to that for the purely classical case. Simple equations for activation free energy as a function of the free energy of the reaction are obtained in the classical limit, which generalizes earlier results.
1. Introduction
U
Intermolecular electron transfer or charge separation in a molecule may be accompanied by a rupture of the reactants' chemicalbonds. This situation often occurs for reactants in polar solvents.'" Two main mechanisms of the reactions of the dissociative electron transfer (DET) may be distinguished: (1) two-step processes and (2) one-step (concerted) reactions. The process of the first type may be described by a reaction scheme
+ N- = [RXI- + N [RXI- R' + X-
RX
-
(1.1) (1.2)
where N- is the electron donor (in particular, an electrode may serve as N-). These reactions are characterized by the formation of a metastable intermediate [RXI- with the relaxation of virbational energy in this state. The first step (eq 1) may be considered as a usual outersphere electron transfer process if one uses a generalized definition of ref 4. The second step (eq 2) leads to a dissociation of the anion-radical (A/R). This mechanism may be described using a set of potential energy surfaces shown in Figure 1. The rate of the overall process of this type may be determined either by the first step or by the second one. Then the calculation of rate is largely reduced to the calculation of the rate of the slower step. Corresponding theory was developed both in harmonic approximation for molecular vibrations7-15and for more complicated potentials.16J7 The concerted reactions of the electron transfer occur simultaneously with the formation or rupture of chemical bonds either without the formation of any intermediate state or without the relaxation of energy in the intermediate state. Reactions of this type proceed by means of a dynamicmechanism as one elementary act: 18.19
N-+ RX -.* N
+ R' + X-
(1.3)
They belong to the class of the innersphere proce~ses.~The potential energy surfaces corresponding to concerted processes of electron transfer are shown schematically in Figure 2. In general the curve Ufmay have a shallow minimum at a rather long distance from the minimum of the curve corresponding to the initial state. Abstract published in Advance ACS Abstracts, May 1, 1994.
0022-3654/94/2098-6120$04.50/0
Figure 1. Scheme of the transition on the free energy surfaces for the sequentialstepwisemechanism. The relaxationof the energy takes place in the intermediate state (Im.
IU
Figure 2. Free energy surfaces for the concerted mechanism of the
reaction. This paper presents a theory of nonadiabatic reactions of the dissociativeelectron transfer proceeding by means of a concerted mechanism. We shall use earlier results obtained for the electrontransfer processes in the systems with nonparabolic molecular potentials both in the classical limit and with due account of quantum effects.9916 The application of the theory of DET in a series of reactions of halogenalkanes will be given in the next paper.20 The structure of the paper is as follows. In section 2 a model for quantum mechanical calculations of the rate constant is described. The calculation of the probability of the nonadiabatic electron transfer is performed in section 3. A classical limit for 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 24, 1994 6121
Dissociative Electron Transfer in Polar Solvents decaying A/R and harmonic intermolecular vibrations of the reactant is considered in section 4. Section 5 involves the results for Morse-like molecular potentials in the classical limit. A quantum mechanical model for Morse-like potentials is described in section 6 . Adiabatic reactions are discussed ip.section 7.Section 8 concludes the paper with a sumnary of the results.
equilibrium free energies of solvation of the reactants are involved implicitly in molecular potentials "ui(r)and "udr). The model described is rather simple; however, it incorporates all characteristic features that are of importance for these reactions. It takes into account both the reorganization of the solvent polarization and the change of the intramolecular structure of the reactants.
2. Model Hamiltonian for a Homogeneous Reaction A homogeneous electron-transfer reaction will be considered in detail. The generalization for the electrochemical reaction may be done in a straightforward way. The reaction of the type
N-
+ RX
-. + N
R
+ X-
(2.1)
is considered. It is assumed that the reactants are located at some distance from each other. Unlike concerted reactions of SN2type, the dynamics of the motion along the N-R coordinate in the nonadiabatic reaction above plays a minor role and may be reduced to subsequent averaging transition probability over the N-R d i ~ t a n c e . ~ J ~ The Hamiltonian of the initial state corresqonds to individual species N- and RX, and the off-diagonal part Vof the interaction between them serves as a perturbation leading to the reaction. The Hamiltonian of the reaction products describes the species N and the decaying anion-radical RX-. Using the Born-Oppenheimer approximation for the separation of the motions of the electrons and heavy species, we end up with the adiabatic Hamiltonians of the initial and final states describing the dynamics of the nuclei:
Hi = h
+ vi
Hf = h+
u,
(2.2)
where h is the kinetic energy and vi and Uf are the potential energy surfaces of the initial and final states. Below we accept the simplest possible model in which
X
(1/2)tio,f[q - q0Xr)I2+ 9, (2.4) where the first term in the right-hand side of eq 2.3 describes the potential energy for the polarization of the solvent, which in the method of the effective Hamiltonian is represented as a set of harmonic oscillators with the dimensionless coordinates {, and vibrational frequences w,. The second term, "ui, is the potential energy of the chemical bond R-X. The third term describes the vibrations along the quantum intramolecular degrees of freedom; for example, it may be the proton vibrations in the molecules. A harmonic approximation is used here with a constant frequency of vibration wqi. However, it is taken into account that the equilibrium position of the quantum oscillator q@depends on the location roftheatomx. Thelast term 3irepresents theminimum value of the potential energy. The terms in eq 2.4 have a similar meaning. The first term describes the vibrations of the polarization of the solvent around the new equilibrium value {df,which depends on the position r of the anion X-. The second term, "uf, describes the repulsion between the group R and the anion X-. The third term describes the vibrations of the quantum oscillator in the group R around the final equilibrium position qor(r). It should be emphasized that only fluctuations of the solvent polarization are shown explicitly in eqs 2.3 and 2.4. The
3. Transition Probability
In this section we present a general expression for the transition probability per unit time assuming that the reaction is nonadiabatic, Le. that the reactants are not located too close to each other. The reasons for starting from the nonadiabatic case are as follows: (1) On the basis of preliminary estimations for some typical potentials studied in the paper,*' we expect that effects of subbarrier transitions may be important for some reactions under consideration. The calculations in the framework of the perturbation theory (which is valid for nonadiabatic reactions) allow us in a rather simple way to take into account these effects. (2) For adiabatic reactions which involve the tunnel effects, the transition probability may be calculated by means of introduction of some "zero-order states" with subsequent use of a perturbation theory. The tunnel effects here may be also estimated as the overlapping of the wave functions of zero-order states. (3) The results obtained for nonadiabatic reactions involve many qualitative features which are inherent to adiabatic ones. We shall return to the discussion of adiabatic reactions later in one of the subsequent sections. Now we shall modify the results obtained in refs 18 and 19 for reactions with parabolic molecular potentials. Starting from the Fermi golden rule (see Appendix A), we can transform the expression for the transition probability per unit time to obtain eq 3.1:
where p ( q ) is the density of the states, elriand ef are the energies of the reactants (ef represents a continuous manifold of energies in the decaying potential), and and xif are the nuclear wave functionsfor the relative motion of R and X. For one-dimensional motion of a particle with mass M in a potential "box" of length C the density of the states p(cf) is equal to
W,is the transition probability for the outersphere electron transfer reaction, which in the simplest case is
[E, + AF aP( -
+
'f
- (fPi - C o i ) ] 2
4EskT
where Es is the reorganization energy of the solvent polarization and A F is the reaction free energy (see Appendix A),
AF = A 3 - eoi
(3.3)
Here A 3 is the difference of the minimum values of Vi and Uf (Siand 3f, respectively). The partition function 2 is determined as follows:
6122 The Journal of Physical Chemistry, Vol. 98, No. 24, 1994
tu
I
German and Kuznetsov classical wave functions xri and
I I I
I
I
I
B
I
YM
Figure 3. Profiles of the free energy surfaces along the coordinates Cx describing the polarization of the medium. I;% and J;or are the initial and final equilibrium values, and AF, is the partial free energy of the transition.
~2 are defined as follows:22
where a and C are the normalization constants: a is on the order of the amplitude of zero vibrations in the initial well, and C corresponds to normalization of the wave function X: in the potential "box". Therefore, we rewrite the overlapping integral (taking into account the above expressions for xri and xe,9in the form (x;(r)lxfp)) = -Jdr1
Iu
exp[-~;S'(2M(U~ 1 - e,'))'I2dr r,
where uQ = $J"'A''"(2M(Ui P) I
- E , , ~ ) ) ' / ~ dr + - A j - E , , ~ ) ) ' / ~ dr] (3.7)
J*(aj,d 'dA"P)(2M(Uf
Figure 4. Profiles of the free energy surfaces along the coordinate r of the chemical bond C-X. D is the dissociation energy, ra is the equilibrium length of the chemical bond, :e is the energy of the initial vibrational state, cy is the energy of the final state, and Aj = cy - ( c i - eoi).
Z = xexp[-(c,,'-
q,')/kT)
and 6r is the interval of the r values giving the major contribution to the overlapping integral (3.6). It should be emphasized that the potentials "ui(r) and "Urjr) are defined here in such a way that
ai(-)= D
(3.4)
P O
Equations 3.1 and 3.2 may be used for numerical calculations; however, since the calculation of the matrix element ( xri(x{J is rather difficult, we transform eq 3.1 into a more convenient form for the application of a quasi-classical approximation. The physical meaning of eqs 3.1 and 3.2 consists of the following. The transition between two energy levels 6 2 and tf corresponds to the transition along the solvent degrees of freedom with an effective free energy of the transition equal to AF, = A F + Aj, where A j = t f - (E; - eoi) (see Figure 3). Since the total free energy of the reaction is equal to AF, the free energy of the transition along the molecular degrees of freedom is-Aj. As the transitions along quantum degrees of freedom occur between their unexcited states, the free energy -AJ is accepted by the degree of freedom describing the chemical bond R-X (see Figure 4). Hence, at a given p the integration over q is equivalent to the integration over Aj:
where D is the dissociation enegy for anharmonic potential (see Figure 4). 4. Classical Limit for Decaying Anion-Radicals: Harmonic Approximation for the R-X Bond
If the "decaying" potential "Urjr) is smooth, the relative motion of the species R and X-may be considered as classical. In this case, a simple approximate limiting expression for the transition probability may be obtained (see Appendix B) if a model of harmonic vibrations is used for the description of the molecule RX; that is, if "Ui(r) = (l/2)hQi(Qi- Q,J2 = (MQ,2/2)(r- r ~ ) 2 (4.1) where Qi is the dimensionless distance r = rm (Qi = (MQ/ h)%) and Qa denotes the equilibrium value of Q. Then we have
2 4 q2exp( h Qi/2k 7')(Mai/ h ) ' I 2
W =
exp(-[E,
+ AF + Aj]2/4E,kT]p(Aj)
(3.5)
where us(r*) is the tunneling factor for the quantum oscillators calculated at the "transitional configuration" r* and the overlapping integral (x;(r)lxzf(r)) is calculated for the position of the potential C U N ~ S shown in Figure 4 with the energy E:. This form is convenient for the calculation of the overlapping integral using the quasi-classical approximation. The quasi-
[lHlooGnl-1'2 x (ZhkT){2mh[ hai(1 - t9*)/kT])'/2 exp [-H( 9 *,r*(19 *))I (4.2)
where H(t9,r) = [t9M+ t9(1 - tl))E,(r)]/kT+ G(t9,r)
Mai hSql-8) G(0,r) = -(r - r,)2th h 2kT
+ S"UAr)/kT
(4.3) (4.4)
The Journal of Physical Chemistry, Vol. 98, No. 24, 1994 6123
Dissociative Electron Transfer in Polar Solvents and Ha, and G,, denote the second derivatives over 9 and r, respectively. The quantities r*(9)and 9* are determined by the following equations:
aG(9,r)/ar = 0
(4.5)
a H ( t s , r * ( s ) ) / a a= 0
(4.6)
The partition function Z is equal to
2 = 1/[1- exp(-hdi/2kT)]
go'.
Equations 4.2-4.6 take into account possible effects of quantum stretching vibrations of the R-X chemical bond in molecule RX. Two limiting cases may be considered: (1) Classical Intramolecular Vibrations (hQ( 1 - 9) > k2').
In this case we obtain
However, one may expect that the repulsion of the anion Xfrom the group R in general is different from that for the molecule RX being still exponential. Therefore, we accept below a more general form for Wr(r):
(4.10) and the transition probability also may be calculated as the probability of the Youterspherenelectron transfer at a fixed distance r with subsequent averaging over r. However, unlike the former case, the averaging is performed with the quantum distribution function
4,+(r) = exp[- T ( r - roi)2]
(4.1 1)
W d r ) = B exp(-20r)
+ eoi
(5.6)
In view of the above condition, toi
