diE%!U LA K

LIQUIDS Journal of Molecular Liquids, 60 (1994) 147-160

ON THE VALIDITY OF MARCUS RELATIONSHIP FOR DISSOCIATIVE ELECTRON TRANSFER

REACTIONS IN SOLUTION

Victor P&es, Jose M. Lluch* and Juan Bert&n. Departament de Quimica, Universitat Autbnoma de Barcelona. 08193 Bellaterra, Barcelona, Catalonia, Spain.

Summary

In this paper Monte Carlo simulations of a dissociative electron trans- fer reaction in solution have been performed in order to test to what extent the deviation of Marcus’ relationship for this kind of reactions can be attributed to the features of the solute internal potential energy. It is shown that Marcus’ equation does not apply to this kind of dissocia- tive electron transfer processes if the ab initio solute internal potential energies are used. Conversely, if a Morse curve and its repulsive part are employed to describe the solute internal potential energies, Marcus’ rela- tionship is recovered. Then, the problem is reduced to what extent the ab initio solute internal potential energies can be represented by Morse curves.

Introduction

The key role of the solvent fluctuations in electron transfer reactions in solution was recognized in the pioneering work of Marcus (refs. l-S), who showed the importance of using free energy rather potential energy in order to understand the kinetics of electron transfers. One of the main results of the Marcus’ theory is the well known Marcus’ relationship (refs. 1, 2, 5 and 7)

AF# = (A& + a)2

4cr!

that relates the activation free energy AFif, with the reaction free en-

0167-7322/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDI 0167-7322 (94) 00743-G

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ergy AF,, and the reorganization free energy CY (that is, the free energy released when the system evolves from equilibrium configurations cor- responding to reactants to those corresponding to products, while an electronic wave function, that can be directly related to products in a valence-bond structure sense, is maintained for describing the solute).

Marcus’ theory, based on a macroscopic solvent continuum model, was originally devised for outer-sphere electron transfer reactions, in which no chemical bonds are broken or formed. Within this theory, it is very surprising at first glance that the quadratic expression given in equation (1) holds despite the fact that the motions of individual solvent dipoles are very anharmonic. However, it is clear that in order for AFf to be related to the reorganization free energy by equation (l), what is really necessary is the existence of an unique one-dimensional reaction coordinate against which the free energy curves corresponding to reactants and products present parabolic shape with equal curvature (refs. 8 and 9).

From the microscopical point of view, in recent years several computer simulations (refs. 9-15) h ave confirmed the quadratic behavior of the free energy curves for outer-sphere electron transfer reactions, and therefore the validity of Marcus’ relationship for this kind of reactions in solution. As a matter of fact it has been shown that the harmonicity of the free energy curves comes from the gaussian character (refs. 16 and 17) of the probability distribution of solvent configurations versus a suitable reaction coordinate, that will be described below.

Recently, Saveant (ref. 18) has developed an empirical model that leads to Marcus’ equation for a special case of inner-sphere electron trans- fer reactions. Concretely, he has studied dissociative electron transfer reactions, that are reactions in which the transfer of the electron and the breaking of a bond are concerted processes. His model is based on Morse curves for the solute internal potential energy and a dielectric continuum approximation for the solvent fluctuational reorganization, both degrees of freedom being treated as independent.

On the other hand, we have carried out a Monte Carlo simulation (ref. 19) of a dissociative electron transfer reaction in a polar solvent, as a first attempt to test the applicability of Marcus’ equation to this class of reactions. In particular, we chose as a very simple model the electrochemical reduction of hydrogen fluoride to give a hydrogen atom and a fluoride anion in a dipolar solvent. The main conclusion of that work was that the Marcus’ relationship seems to fail for this kind of inner-sphere processes.

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It is clear that this previous Monte Carlo simulation for a dissociative electron transfer reaction (ref. 19), disagrees with the above mentioned computer simulations for outer-sphere electron transfer reactions. As a matter of fact, the practical difference between both kind of microscopical simulations comes from the solute internal potential energy that is used to describe the breaking of the corresponding bond. This term is present in dissociative electron transfer simulation, but absent in the outer-sphere electron transfer simulations. Then, the purpose of this paper consists of testing to what extent the deviation from Marcus’ relationship for dissociative electron transfer reactions can be attributed to the’features of the solute internal potential energy.

Methodology

Electron transfer reactions can be understood in terms of a suitable diabatic two-state basis. It consists of the electronic wave functions \E, and \E,, which can be directly identified in a valence-bond structure sense with the precursor (reactants) and the successor (products) complexes, respectively (refs. 20 and 21). With the Q, and %I?* functions specified,

the integrals Hpp =< \E, 1 I? 1 Q, > and H,, =< X&S 1 l? 1 \E, > versus the nuclear coordinates of the system give the corresponding diabatic poten- tial energy hypersurfaces. Assuming a classical frame (refs. 4, 22 and 23), the actual electron transfer must occur at the intersection region S* of both diabatic potential energy hypersurfaces. This region S* is reached by a suitable thermal fluctuation in the nuclear configurations of the pre- cursor complex. The electronic coupling integral between both diabatic states is assumed to be large enough for the reactants to be converted into products with unit probability in the intersection region, but small enough to be neglected in calculating the amount of energy required to reach S*.

As mentioned in the introduction, the aim of this paper is to test if the validity of Marcus’ relationship for dissociative electron transfer reactions depends on the features of the solute internal potential energy. Then for the sake of comparison with the preceding simulation we have also chosen here the electrochemical reduction of hydrogen fluoride to give a hydrogen atom and a fluoride anion in a dipolar solvent, as a very simple model of this kind of processes.

In order to calculate the energies of the diabatic hypersurfaces Hpp and H,, three kinds of pairwise additive potential functions have been

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considered, corresponding to the solute internal potential energy (that is the one associated with the gas phase reaction), and the solute-solvent and solvent-solvent interactions.

In order to describe the solute-solvent and the solvent-solvent interac- tions, solvent has been simulated by a point dipole, with a dipole moment of 2.15 D, which is the dipolar moment of a water molecule at the 6-31G** ab initio level. The pairwise potential functions used in the calculations of the present work are the same as the ones used in our simulation done in reference 19.

On the other hand, regarding the solute internal potential energy, the precursor complex has been modeled by a hydrogen fluoride plus an elec- tron inside an electrode. Its energy can be calculated as the sum of the hydrogen fluoride internal energy and a constant value that represents the Fermi level energy of the electrode. Note that the solution exchanges electrons with the metal at the Fermi level because these are the most energetic electrons for reduction. The hydrogen fluoride internal poten- tial energy in function of H-F distance, has been previously obtained by means a truncated configuration interaction (CI) expansion that includes singly and doubly excited determinants (SDCI) (refs 19 and 24). Then, the potential corresponding to the precursor complex can be considered diabatic in the sense that the additional electron is kept apart, that is inside the electrode. In addition, the successor complex, once the electron has been transferred from the electrode, is assumed to be the hydrogen fluoride anion in the electronic state that leads to the diabatic dissociation in F- and H.

We have done two separate simulations using two different gas phase reaction potentials. For the first simulation we have used the analytical potential functions previously developed (refs. 19 and 24) for both the energy profiles corresponding to the precursor complex (the hydrogen fluoride plus the electron inside the electrode) and the successor complex (the hydrogen fluoride anion). This analytical potential functions were obtained by cubic spline fitting of a set of ab initio calculations done for several values of the H - F distance. For this simulation the Fermi level energy has been chosen as the value that makes the gas phase energy barrier equal to 57.63 kJ/mol. Hereafter in this paper, this simulation will be called lower energy barrier (LEB) simulation. It should be commented that in the aforementioned former simulation (ref. 19) exactly the same analytical functions were used, but with the Fermi level energy being chosen in such a way that the gas phase reaction energy is zero, this fact

151

imposing an energy barrier of 94.52 kJ/mol. Then, from now on that simulation will be designated as higher energy barriers (HEB) simulation (refs. 19 and 24).

For the second simulation performed in the present paper, hereafter called Morse simulation, the ab initio calculations corresponding to the precursor complex have been fitted to a Morse curve,

U = D [l - exp(-@y)]’ (2)

were y stands for the H-F distance minus the equilibrium bond distance

in the reactant. Then the gas phase potential energy of the successor com- plex (following the hypothesis introduced in Saveant’s model) is assumed to be the repulsive part of the reactant Morse curve, that is D exp( -2py). In this case the Fermi level energy is chosen as the value that makes the gas phase reaction energy equal to zero, analogously to HEB simulation.

Statistical simulation

We have simulated the dissociative electron transfer reaction between the precursor and the successor complexes in a dilute polar solution in- cluding 200 point dipoles at 2’ = 298K by means of the Monte Carlo method (refs. 25 and 26) within the Metropolis’ algorithm (ref. 27). In the simulations the F atom is held fixed and the H atom as well as the dipoles were moved. The size of the box has been chosen in such a way that 200 water molecules would give a density of about lgr/cm3. Pe- riodic boundary conditions under the minimum image convention have been employed. For all simulations a preliminary equilibration of the system followed by statistical analysis performed on additional configu- rations, has been carried out. These simulations were run using a program written by us.

For each generated configuration the value AE = Hbd - Hpp has been calculated. This parameter has been shown to behave as a good reac- tion coordinate for the diabatic free energy curves in electron transfer reactions (refs. 8, 13, 14, 16, 28 and 29). The configuration space was partitioned in different subsets S, each one being associated with a par- ticular value AEs of the reaction coordinate AE. For practical purposes, the criterion 1 AE - AEs ]< 0.5 kJ/moZ has been adopted in order to classify a given configuration as belonging to a subset S. We have iden- tified the reactants’ region (SR) with the most populated interval when the Hpp potential is used. Conversely, the products’ region (Sp) is ‘the

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most populated interval corresponding to the H,, potential. The in- tersection region S* corresponds to the interval centred at the value of AEs = 0 k Jlmol. Th ere f ore, this value of the reaction coordinate explic- itly characterizes the transition structures for the electron transfer. The activation free energy AF# corresponds to the transition from SR to S’, in such a way that the factor exp(-AFf/kT) expresses the probability that the reaction system will be in the transition state region S* relative to the probability of being in SR.

Due to the high value of AFf for our reaction, the complete sampling of the configuration space in order to obtain the diabatic free energy curves as a function of AEs would require an extremely long simulation. To circumvent this problem we have used a strategy previously developed by Warshel (refs. 9, 10 and 30). A mapping potential energy hypersur- face of the form H, = (1 - X,)H, + X,H,, is defined, Hpp and H,, being calculated from the potential functions obtained above. The parameter X, changes from 0 to 1 on movement from the precursor to the successor states. With the H,,., potential corresponding to X, = 0 the most pop- ulated subspace is SR. As X, increases, the system is forced to evolve towards the intersection region S*. With the H,,, potential corresponding to X, = 1 the most populated subspace is Sp. Now the diabatic free en- ergies corresponding to the precursor and the successor complexes along the reaction coordinate are obtained by using the expressions (refs. 9, lo,15 and 30) as a function of AEs:

AF,(AEs) = AF o-+~ - %ATs - kT ln[(@/&,,,) (QJqp))] (3)

AF,,(A&) = AFl-., + (1 - hJA& - kTln[(q~)/Q,)(Q,,/q!~))]

(4) The last term of equations (3) and (4) involves ratios of partition func-

tions. So, the factor qg)/Qm is the probability that, using the H,,, po- tential, the configurations generated belong to the subset S. The ratio

&?‘/Q, in the equation (3) represents the probability that, using the Hpp diabatic potential, the configurations generated belong to the re- actants’ subset SR. Analogously, the factor q$p)/QIa in the equation (4) represents the probability that, using the H,, diabatic potential, the configurations generated belong to the products’ subset Sp.

The values AFO,, and AFr,, have been calculated by using statisti- cal perturbation theory (refs. 9, lo,15 and 30-37). This approach follows

153

from equation (5) which expresses the free energy difference between sys- tems with mapping potentials Hk and Hj by :

ALF(Xj + &) = -!zTln(exp - (Hkk;Hj))Hj (5) The average is for sampling based on the potential Hi, so the Hk potential

corresponds to a perturbed system. Therefore:

m-l

AJ’0-m = AF(Xo = 0 + X,) = C AF(Xj + Xj+i) (6) j=O

In order to transform the potential smoothly and to avoid large pertur-

bations, Xj has been increased in small steps from 0 to 1. Simulations have been run in both directions, Aj + Xj+i and Xj+i -t Xi, except at the two end points. This is known as double-ended sampling. It is facilitated by using double-wide sampling (ref. 33). Specifically, the free energy differences for Xj + Xj+i and Xi --$ Xi-1 can be obtained simultaneously, since both require sampling based on the Xj system.

In all, a total of about 92,000,OOO configurations have been generated for the LEB simulation, being distributed in twenty three runs. Each run consisted of an equilibration phase of 2,010,OOO configurations followed by averaging over 2,010,OOO additional configurations. The Morse simulation involved an equivalent number of generated configurations and runs. To evaluate numerically AF( AEs) with equation (3) and equation (4), we have used in each case the mapping potential H,,, for which the most populated subset S is the one centred at AEs. Following this procedure, each calculation converges very fast.

Results and Discussion

We present first the results corresponding to the LEB simulation. The data required to calculate the diabatic free energy corresponding to the precursor complex (AF,) along the reaction coordinate by means of equation (3) are exhibited in table 1. The accumulated free energy values AFo+~ from equations (5) and (6) appears in the second column. The values AEs for the most populated subset S when each mapping potential H,,,, associated with A,.,,, is used, are done in the third column. The ratio of partition functions q2/Qm is shown in the last column. Note that the terms qg/Qm and qz/Q,, correspond to the values of the first

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L A&-m A&T d?/Qm

0.000 0.000 135.0 0.02623 0.100 12.893 129.0 0.02637 0.200 24.467 106.0 0.02229 0.300 34.191 92.0 0.02026 0.350 38.445 82.0 0.01875 0.400 42.210 73.0 0.01646 0.420 43.548 71.0 0.01607 0.450 45.339 56.0 0.01452 0.470 46.349 59.0 0.01472 0.500 47.575 46.0 0.01300 0.530 48.439 34.0 0.01188 0.550 48.757 13.0 0.01129 0.580 48.723 -4.0 0.01038 0.600 48.332 -18.0 0.00938 0.620 46.792 -138.0 0.00864 0.650 42.541 -146.0 0.00830 0.700 32.252 -280.0 0.00642 0.750 16.330 -361.0 0.00716 0.800 -4.131 -470.0 0.00713 0.850 -28.947 -568.0 0.00647 0.900 -58.882 -654.0 0.00466 0.950 -98.612 -837.0 0.02049 1.000 .141.057 -850.0 0.01913

Table 1: Accumulated precursor free energy (in kJ/mol), AEs (in kJ/mol) for the most populated subset S when each mapping potential II,,,, associated with X,, is used, and ratios of partition functions for LEB simulation.

and last rows , respectively, of this fourth column. Taking into account that

AF,-w,, = AFo+m - AFo+I (7)

equation (4) supplies the diabatic free energy corresponding to the suc-

cessor complex (AFSS) along the reaction coordinate. We have attempted to carry out a least-square quadratic fitting of AFpp

and AF,, values, the curves obtained along with the values to be adjusted (marked as squares) being displayed in figure 1. Two parabolic functions of the same curvature are achieved. The minimum corresponding to the successor complex coincides (at -850.3 kJ/mol) with the correct value

155

1000 0 -1000

Ab(kJ/mol)

Figure 1: Fitted diabatic free energy curves for the precursor and the successor complexes

with respect to the AEs values of the reaction coordinate AE for LEB simulation.

corresponding to products’ region Sp (AEs = -850 kJ/mol) as seen in table 1. However, the minimum corresponding to the precursor complex appears very far (at 1036.1 kJ/mol) from the correct value correspond- ing to the reactants’ region SE (AEs = 135 kJ/mol). This fact seems to indicate that an important deviation of the Marcus’ relationship can be predicted. In fact, the LEB Monte Carlo simulation leads to values of -140.28 kJ/mol and 275.28 kJ/mol for the reaction free energy AFO and the reorganization free energy cr, respectively. These numbers, once introduced in equation (l), provide an activation free energy AF# of 16.55 kJ/mol, which differs very much from the true value AF# = 55.40 kJ/mol, determined by the intersection point between both parabolic curves in figure 1. Indeed Marcus’ relationship does hold if AF, and Q are deduced taking as origin the two minima that arise from both fitted functions, but the values obtained in this way are not related at all with the true values corresponding to the real chemical process.

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Ln AFo-., AEs d?/Qm

0.000 0.000 603.0 0.00573 0.050 27.482 506.0 0.00799 0.100 51.448 464.0 0.00783 0.150 72.261 399.0 0.00798 0.200 90.093 315.0 0.00837 0.250 104.847 277.0 0.00816 0.300 116.915 200.0 0.00770 0.350 125.939 131.0 0.00814 0.400 132.026 86.0 0.00799 0.450 134.939 29.0 0.00820 0.475 135.085 -28.0 0.00690 0.500 133.614 -102.0 0.00812 0.550 127.519 -176.0 0.00720 0.600 118.339 -213.0 0.00767 0.650 105.902 -279.0 0.00787 0.700 90.261 -345.0 0.00746 0.750 71.336 -404.0 0.00773 0.800 47.382 -541.0 0.00678 0.850 17.880 -620.0 0.00632 0.900 -16.924 -800.0 0.01496 0.950 -56.544 -791.0 0.02147 1.000 -96.392 -802.0 0.02011

Table 2: Accumulated precursor free energy (in kJ/mol), AEs (in kJ/mol) for the most populated subset S when each mapping potential I&,,, associated with X,, is used, and ratios of partition functions for Morse simulation.

Regarding the Morse simulation, table 2 and figure 2 are completely analogous to table 1 and figure 1, respectively. For the present case two parabolic functions with the same curvature are obtained, but now their minima are close (at 650.7 kJ/mol and -825.2 kJ/mol for AF, and AF,,, respectively) to the values directly obtained by means of the simulation (603 kJ/mol and -802 kJ/mol for th e reactants’ region SR and the prod- ucts’ region Sp, respectively). Note that the relative errors are only of 7% and 3 %, respectively. From these results the ful6llment of Marcus’ relationship would be expected. As a matter of fact, the Morse Monte Carlo simulation provides values of -99.5 kJ/mol and 702.5 kJ/mol for the reaction free energy AFo and the reorganization free energy cy, respec- tively, which through equation (1) lead to an activation free energy AF#

1.57

1000 0 -1000

A&(kJ/mol)

Figure 2: Fitted diabatic free energy curves for the precursor and the successor complexes

with respect to the AEs values of the reaction coordinate AE for Morse simulation.

of 129.4 kJ/mol. This result compared with the true value AF# = 135.2 kJ/mol, determined by the intersection point between both parabolic curves in figure 2, involves only a 4 % of relative error.

From the results of this paper and the previous paper (ref. 19) we can conclude that Marcus’ equation does not apply to this kind of dissociative electron transfer reactions if the ab initio solute internal potential energies are used.

It is interesting to note that a similar deviation from the Marcus free energy relation has been recently found in a theoretical study of SNl ionic dissociation in solution (ref. 38), which has been interpreted in terms of electron transfer processes between two valence-bond states. The devi- ation from the Marcus’ behavior for both dissociative electron transfer reaction and SNl ionization can be probably attributed to the same ori- gin, e.g., the anharmonicity associated with the solute internal potential energy curves.

158

8OOJ

600-

=

E 3 400- Y

WV

200-

O-

Figure 3: Representation of the gas phase ab initio points (squares) along with the Morse

fitting according with the procedure described in Methodology section.

Conversely, if a Morse curve and its repulsive part are employed to de- scribe the solute internal potential energies for the dissociative electron transfer reaction studied in the present paper, as described above, Mar- cus’ relationship is recovered. This is probably due to the fact that if the coordinate (ref. 39) Q = exp(-py) is defined, both equation (2) and its repulsive part are quadratic functions of Q with the same curvature, in such a way that Marcus’ relationship holds even in gas phase. It is clear that if Marcus’ equation runs for outer-sphere electron transfer reactions, the addition of quadratic contributions of the solute does not modify its validity.

Then the problem is reduced to what extent the ab initio solute in- ternal potential energies can be represented by the Morse curves. In figure 3 we have shown a representation of the ab initio points (squares) along with the Morse fitting according with the procedure described in the Methodology section. It can be seen that the Morse solute internal poten- tial energy curve corresponding to the precursor complex describes quite

159

well the ab initio values. However, the repulsive part of the Morse curve differs noticeably from the ab initio successor complex values. This dis- agreement explains why Marcus relationship does not hold if the ab initio solute internal potential energies are used. Anyway, it is well known that a good description of ab initio potential energy curves for anionic species, specially for short distances, is difficult (refs. 19 and 40-42). Additional theoretical work in this direction is now in progress in our laboratory.

Acknowledgements

The authors gratefully thank professor J.T. Hynes for inviting us to participate in this Teresa Fonseca memorial issue. Those of us who had the privilege of discussing scientific matters with her, know the high qual- ity of her work. J.B. and J.M.Ll. had the joy of having been friends of Teresa. Her enthusiasm, humor, and essential humanity are sorely missed.

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