3697
J. Phys. Chem. 1986, 90, 3697-3700
Potential Surfaces for Gas-Phase Electron-Transfer Reactions David E. Richardson Department of Chemistry, University of Florida, Gainesville, Florida 3261 1 (Received: January 13, 1986; I n Final Form: April 16, 1986)
Potential surfaces for some recently studied inefficient gas-phase electron-transfer ion-molecule reactions are estimated. The surfaces are generally double wells, with the potential minima resulting from stabilizing ion-molecule interactions. The central barrier is equated to the Marcus inner-sphere reorganization energy commonly applied to condensed-phase electron-transfer reactions. The experimental rates of electron self-exchange reactions for metallocenes (Cp2Mt/O) and SF6-/0 are considered in view of the estimated inner barriers, and the thermodynamics of cross-reactions of SFC/O with organic acceptors are used to estimate enthalpic and free energy barriers. The comparison of theory and experiment shows a strong relationship between the magnitude of the estimated central barriers and the efficiencies of the electron-transfer processes.
Interest in gas-phase electron-transfer processes has increased recently with reports of inefficient charge-transfer reactions involving organometallic,l,2 ~ r g a n i c ,and ~ , ~main-group system^.^ The purpose here is to provide a unifying theoretical framework for the interpretation of these slow reactions through an adaptation of the Marcus which has been used extensively to interpret solution kinetics for many years. Traditionally, Marcus theory has been applied to solution redox reactions involving metal centers, but the general approach is valid for all types of electron-transfer systems. The principal goal is to derive the energetics of the appropriate potential surface for gas-phase electron-exchange reactions involving ion-molecule collisions. The general kinetic scheme is given in (1). The rate of formation of the precursor ion-neutral
A+
+ B &k
k-i
A+ - B
k k 2 A - B+ - 3 ’ + ~B+ k-3 k-2
(1)
complex, A+ - B, is given by k , and can be estimated with some confidence by using the Langevin, ADO, or transition-state approaches.* In the high-pressure (>1 Torr) mass spectrometer experiments used by some workers, the precursor complex may suffer multiple collisions before dissociating and become thermalized. In the ion cyclotron resonance approach, the low pressures (lO-*-lO” Torr) require that the precursor complex retain the total energy of the reactant particles as essentially no third-body collisions occur. The formation of the precursor complex is a chemical activation process, and in statistical theories the assumption is made that the total energy will be redistributed into available degrees of freedom in the c ~ m p l e x . ~ The precursor complex may dissociate into reactants without or the electronic state may change to A charge transfer (kI), B+, the successor complex, upon electron transfer from B to A+ ( k 2 ) . The overall kinetic scheme in (1) parallels the approach used extensively by other workers to interpret gas-phase displacement and proton-transfer reactions.’+l2 The usual dou(1) Eyler, J. R.; Richardson, D. E. J . Am. Chem. SOC.1985, 107, 6130. (2) Christ, C.; Sharpe, P.; Eyler, J. R.; Richardson, D. E., unpublished
results. (3) Mautner, M.; Nelsen, S.F.; Willi, M. R.; Frigo, T. B. J . Am. Chem. SOC.1984, 106, 7384. (4) Mautner, M.; Rumack, D.; Nelsen, S . Abstracts of the 10th Interntaional Mass Spectrometry Conference, Swansea, U.K., 1985; No. 338. (5) Grimsrud, E. P.; Chowdhury, S.; Kebarle, P. J. Chem. Phys. 1985.83, 1059. (6) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, IS, 155. (7) Sutin, N. Prog. Inorg. Chem. 1983, 30, 441.
(E) (a) Su,T.; Bowers, M. T. In Gas Phase Ion Chemistry; Bowers, M. T., Ed.; Academic: New York, 1979; p 84. (b) Su, T.; Chesnavich, W. J. J. Chem. Phys. 1982, 76, 5183. (9) Forst, W. Theory of Unimolecular Reactions; Academic: New York, 1973. (10) (a) Farnath, W. E.; Brauman, J. I. J. Am. Chem. Soc. 1976, 98,7891 (b) Olmstead, W. N.; Brauman, J. I. J . Am. Chem. SOC.1977, 99, 4219.
0022-3654/86/2090-3697$01.50/0
ble-well surfaces are illustrated in Figure 1, where the attractive ion-neutral interaction initially lowers the potential energy (increasing kinetic energy) and the central border is related to the activation energy of the electron-transfer processes k2 and k-2 in (1). It has been recognized for other types of reactions that the size of the central barrier is crucial to the overall rate.1s12 When no barrier is present, the overall rate is known to approach the collision rate. Even if the central barrier is below the separated reactants total potential energy (as in Figure l ) , the overall rate can be reduced because of increased tightness in the transition state for k2 and the relatively fewer microstates compared to the loose orbiting complex. It is suggested here that the central barrier for electron-transfer reactions can be estimated by using Marcus theory. Let us consider the reaction coordinate of Figure 1 more closely. When a nonpolar neutral is used as an example, the attractive potential for the ion-neutral complex has the well-known d-4 dependence on the reactant separation ( d ) and E,(d) = -(q2a/ 2d4), where q is the charge on the ion and a is the molecular polarizability. At small values of d , repulsive forces become important, and the potential energy for the ion-molecule interaction can be represented by
+
E,,(d) = - ( q 2 a / 2 d 4 ) be-d/c
(2)
where the values of b and c are chosen to yield the potential minimum a t do, the equilibrium distance between the ion and molecule in the precursor complex. For electron transfer in the gas phase, only the “inner” reorganization component of the total Marcus free energy of activation is considered. The derivation of potential surfaces for electron transfer assuming one significant normal mode has been given in detail el~ewhere.’~’~ The surface is shown in Figure 2, where the harmonic approximation has been used. When the notation of ref 13 is used, the lower and upper surfaces are defined by
[
E”(q,€) = hv q 2 / 2 +
((
Xq
+-
c~)”~]
(3b)
where e, the electronic interaction parameter, can be given approximately by 44 = €0 exP[-P(d - do)] (4) Typical values for /3 are in the range 0.5-1.3 A-’,I4 do is the (1 1) Caldwell, G.; Magnera, T. F.; Kebarle, P. J. Am. Chem. SOC.1984, 106, 959. (12) Jasinski, J. M.; Brauman, J. I. J. Am. Chem. SOC.1980, 102, 2906. (13) (a) Piepho, S.B.; Krausz, E. R.; Schatz, P. N. J . Am. Chem. SOC. 1978,100, 2996. (b) Wong, K. Y.; Schatz, P. N.; Piepho, S . B. J. Am. Chem. SOC.1979, 101, 2793.
0 1986 American Chemical Society
3698 The Journal of Physical Chemistry, Vol. 90, No. 16, 1986
Richardson
TABLE I: Calculated Inner Reorganization Energies for Self-Exchange Reactions SFn-lo Cp,MntIo CP,CO+~O 0.25c 0.1 I d 0.145' Ar, A.l 4.889 1.95h 2.8' f(av)/ mdyn k' 8.7 2.4' 22.0 A&*, kcal mol-' 0.006 0.36
