2736
J . Phys. Chem. 1990, 94, 2136-2140
Role of Solvent Electronic Polarization in Electron-Transfer Processes Hyung J. Kim and James T. Hynes* Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309-021 5 (Receiced: Nouember 27, 1989)
The role of solvent electronic polarization in electron-transfer processes is investigated. The solvent electronic polarization is assumed to be instantaneously equilibrated to the quantum charge distribution of the transferring electron and the fluctuating solvent orientational polarization;this yields a nonlinear Schriidinger equation for the electron wave function. The transition between nonadiabatic and adiabatic regimes is found to be governed by both electronic coupling and electronic polarization. Activation and reorganization free energies are obtained; in some regimes, they differ considerably from some conventional predictions. The physical origin and consequences of these features are described.
1. Introduction Activation barriers for electron-transfer (ET) reactions can be significantly influenced by the s ~ l v e n t . l - ~While considerable attention has been focused on weakly electronically coupled reaction systems,'-4 related issues arise in more strongly coupled ET systems,s including photochemical electron transfers?' and in more general ET processes, including sN1and s N 2 reaction^.^,^ Here we discuss solvent contributionsI0 to ET activation barriers over a wide electronic coupling range, focusing on the role of solvent electronic polarization. One of the following-approximations related to the solvent electronic polarization P, has been a d ~ p t e d l ~ . " -to' ~simplify calculation of the reaction activation energy: the transferring electron is treated as a classical charge distribution;I*l3OJ the electron is described quantum m_echanically but P, is "adiabatically" _eliminated, so that P, in_teracts with a point electron3-li (or P, is neglected14). Here P, merely screens the electric field by the optical dielectric constant t, so that only the ( I ) Marcus, R. A. J. Chem. Phys. 1956,24,966,979; Faraday Discuss. Chem. Soc. 1960.29,21; J . Chem. Phys. 1963,38,1858;Annu. Reo. Phys. Chem. 1964,I S , 155. (2)Hush, N. S. Trans. Faraday Soc. 1961,57,557;frog. Inorg. Chem. 1967,8, 391. (3) Levich, V. G. Adv. Eleclrochem. Eng. 1966, 4, 249; In Physical Chemistry: an Advanced Treatise; Henderson, D., Yost, W., Eds.; Academic: New York, 1970;Vol. 9B. Levich, V. G.;Dogonadze. R. R. Dokl. Akad. Nauk. SSSR 1959,124, 9 (Engl. Transl.). (4) For recent reviews, see: (a) Newton, M. D.; Sutin, N. Annu. Reo. Phys. Chem. 1984,35.437.(b) Sutin, N. frog. Inorg. Chem. 1983,30,441. ( 5 ) Hush, N. S. In Mixed-Valence Compounds; Brown, D. B., Ed.; Reidel: Dordrecht, 1980. Creutz, C. f r o g . Inorg. Chem. 1983,30, I . (6)Meyer, T. J. f r o g . Inorg. Chem. 1983,30, 389. (7)(a) Beens, H.; Weller, A. Chem. Phys. Left. 1%9,3,666. (b) Beens, H.; Weller, A. In Organic Molecular Photophysics; Birks, J. B., Ed.; Wiley-Interscience: London, 1975;Vol. 2. (c) Mataga, N. In The Exciplex; Gordon, M.,Ware, W. R., Eds.; Academic: New York, 1975. (d) Mataga, N. In Molecular Interactions; Ratajczak, H., Orville-Thomas, W. J., Eds.; Wiley: New York, 1981;Vol. 2. (e) Barbara, P. F.; Kang, T. J.; Jarzeba, W.; Fonseca, T. In Perspectives in Photosynthesis; Jortner, J., Pullman, B., Eds.; Kluwer: Dordrecht, 1990. (f) Rettig, W. Angew. Chem., Inr. Ed. Engl. 1986,25, 971. (9) Lippert, E.; Rettig, W.; BonaEiE-Koutecky, V.; Heisel, F.; Mieht, J. A. Ado. Chem. Phys. 1987,68, I. (h) Meyer, T. J. In MixedValence Compounds; Brown, D. B., Ed.; Reidel: Dordrecht, 1980. (8)For discussion of some electronic aspects of SN1 and SN2 reactions, see e.g.: Pross, A. Adu. Phys. Org. Chem. 1985,21, 99. Shaik, S.S.frog. Phys. Org. Chem. 1985, 15. 197. (9)(a) Zichi, D. A.; Hynes, J. T. J. Chem. Phys. 1988,88,2513.(b) Lee, S.; Hynes, J. T. J . Chem. Phys. 1988,88,6863.(c) Gertner, 8. J.; Bergsma, J. P.; Wilson, K. R.; Lee, S.; Hynes, J. T. J. Chem. Phys. 1987,86,1377. (d) Kim, H. J.; Hynes, J. T. To be submitted for publication. (e) Keirstead, W.; Wilson, K. R.; Hynes, J. T. To be submitted for publication. (IO) Of course there are other contributions to these barriers as well; cf. refs 1-6. ( I I ) Efrima, S.; Bixon, M. J . Chem. Phys. 1976,64,3639. (12)Cannon, R. D. Chem. Phys. Left. 1977,49,299. (13) (a) Tembe, B. L.; Friedman, H. L.; Newton, M. D. J . Chem. Phys. 1982,76,1490. (b) Newton, M. D.; Friedman, H. L. J. Chem. Phys. 1988, 88,4460;Erratum 1988,89, 3400. (14)Calef, D. F.; Wolynes, P. G. J . Phys. Chem. 1983,87,3387;J . Chem. Phys. 1983. 78, 470. These authors use free energy functional and Schrijdinger equation techniques but neglect pe; this results in a linear SchrGdinger equation.
0022-3654/90/2094-2736$02.50/0
solvent orientational polarization contributes to the ET activation free energy. In a correct general formuktion, the electron should be treated quantum mechanically and P, included. Since the transferring electron and the bound solvent molecule electrons' binding energies are comparable, +e "adiabatic" approximation above seems inappr~priate.'~P, should be instead treated self-consistently, Le., equilibrated to the quantum charge distribution determined by the electron wave f ~ n c t i o n , ' ~ - 'Here ~ we report results for ET activation barriers when P, is included self-consistently in a quantum description for the electron. While our discussion is couched in solvent dielectric continuum terms, the aspects we stress should translate to a molecular description.
2. Formulation We consider for simplicity a symmetric reaction system consisting of an electron, a donor (D) and an acceptor (A) immersed in a dielectric continuum. D and A, taken as identical and electrically neutral, are placed in cavities. The system free energy with an electron wavefunction \k in the presence of arbitrary electronic polarization P, and orientational polarization Pois"*'
GIQ,Fe,Fo]=
where @ is the vacuum Hamiltonian, eo thq static dielectric constant, V the volume outside cavities, and &[Q] the vacuum electric field arising from \k
We assume Feto be equilibrated instantaneously to Z[\k] and p,: I5,17
At this stage, the wave function 9,which depenjs on the solvent, is unknown. To determine it, we substitute into eq 1 and minimize G relative to \k variations. We then obtain a time-in-
e
(15) Jortner, J. Mol. Phys. 1962,5, 257.
( I 6) Marcus, R. A. Faraday Symp. Chem. Soc. 1975,I O , 60. The selfconsistent treatment for the electron is in fact explicitly outlined in this reference but is not pursued. (Note that, despite the rather different appearance of the Marcus free energy functional, it is in fact equivalent to our eqs I and 3.) (17) Kim, H. J.; Hynes, J. T. To be submitted for publication. ( I 8) For the free energy functional method in general, see: Pekar, S.I. Unrersuchungen iiber die Elekrronentheorie der Krisfalle; Akademie-Verlag: Berlin, 19541 (19)Felderhof, B. U.J. Chem. Phys. 1977,67,493. (20) Lee, S.;Hynes, J. T. J . Chem. Phys. 1988,88,6853. (21)In eqsl-3; dielectric image effecG are ign_ored, so, that the electric displacement D is equal to the vacuum field 6 . * O P, and Po are assumed to be longitudinal.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990
Letters iependent Schriidinger equation for 9 in the presence of arbitrary
+
po16J7
where the energy eigenvalue is E.22 Due_to the equilibrated electronic polarization, the Hamiltonian H[\k,Po)is a functional of 9;eq 4 is thus a nonlinear Schrodinger equation. We solve this with the ansatz Q = cl$l ~ ~ $where 9, fii and $2 are symmetrically orthogonalized states constructed from nonorthogonal diabatic states 4Iand 4 ~ which ~ , represent charge localized states on D and A, r e ~ p e c t i v e l y . ~The ~ orthonormal states have a vacuum coupling p = -($21@I$1). The Schrtidinger equation then reduces to a quartic equation for cI and c2,whose real roots determine the solvent-dependent solute electronic structure, and the nonequilibrium solvation stationary states (NSS) and thus their energy eigenvalues E . The NSS free energies G are then obtained fromI7
2737
(a)
GeRq
+
I
solvent coordinate
G
G= G* (5)
G:
G and E differ by the solvent self-energies associated with Po and
e.24 The NSS and their free energies are, in general, none-
quilibrium in character because Po is arbitrary.25 Very special and important NSS, corresponding to extrema on the free energy surface,_are thgequilbrium solvation stationary states (ESS),for which Po and P, are in equilibrium with 9.The ESS wave functions 9- ~ a t i s f y ' ~ , ~ ~
= Ew-(F) (22) E is a Lagrange multiplier for the normalization condition (*Irk) =
I. (23) From the nonorthogonal diabatic states +I and &with overlap integral S = (C#Jl1&), we construct the orthonormal wave functions $I and &, = a 2q+ + a 2 , i C # J ~ ; 2 w i t ~ l t ~ ~ n s f o r mcoefficients ation lal,aJ: a1,2= [ & ( I + 2)1/2]-1. For large D-A separation and small S, + ( I - S) ] [ and $2 approach the diabatic states and &, respectively. As the D-A separation decreases and the overlap increases, $, and q2 progressively deform from the diabatic states. See Lowdin, P. J . Chem. Phys. 1950, 18, 365. (24) Since pe is in equilibrium, it consists of 8 v ]and Focontributions (eq 3). The second t e p in eq 5 is the_sum of the Po self-energy and a part of the self-energy for P, arising from Po in eq 3. The third term of eq 5 is the remaining self-energy for P, due to the solute electric field. (25) Nonlinear SchrGdinger equations for equilibrium solvation have been studied by several g r o u p ~ . " ~ *(However, ~~ ET requires a nonequilibrium solvation formulation.) There are, however, important exceptions: ( 1 ) energies of vertical Franck-Condon transitions out of equilibrium solvation states are sometimes calculated or discussed, e&, ref 26b. However, such transitions lead to special (rather than general) nonequilibrium solvation states. (2) In some treatments (e&, refs 26c and 26g) reaction fields experienced by the solute are formulated in terms of fixed solvent particle positions, with subsequent averages variously prescribed. (26) (a) Yomosa, S. J . Phys. Soc. Jpn. 1973,35, 1738; 1978.44.602. (b) Newton, M. D. J . Chem. Phys. 1973.58.5833; J . Phys. Chem. 1975.79.2795. (c) Thole, B. T.; Van Duijnen, P. T. Theor. Chim. Acta (Berlin) 1980, 55, 307. (d) Rivail, J.-L.;Rinaldi, D. Chem. Phys. 1976, 18, 233. (e) Rivail J.-L. In Chemical Reacfiuity in Liquids; Moreau, M., Turq, P., Eds.; Plenum: New York, 1988. (f) Miertus, S.;Tomasi, J. Chem. Phys. 1982,65,239. (g) Tapia, 0. In Molecular fnreracrions; Ratajczak, H., Orville-Thomas, W. J., Eds.; Wiley: New York, 1982; Vol. 3. (h) Karlstrom, G. J . Phys. Chem. 1988, 92, 1315. (i) Mikkelsen, K. V.; Dalgaard. E.;Swanstrem, P. J . Phys. Chem. 1987, 91, 3081.
G:
solvent coordinate
*
Figure I . Free energy curves for ET: (a) nonadiabatic case; (b) adiabatic case; the splitting between the two curves a t the transition state is 2&. Various free energies defined in the text are indicated.
which is the equilibrium analogue of eq 4;the related equilibrium free energy is
When the effective equilibrium coupling 3(, -($21H-[$I]I$I) is not too large,28there exist four solutions to eq 6. Two have localized electronic structure, slightly more diffuse than the diabatic states and 42,due to the electronic coupling. These two ESS are the local minima on the free energy surface versus solvent coordinate29(Figure 1) and are the reactant and product states for ET r e a ~ t i o n s . l ~ *The ~ ' other two states are the symmetric and antisymmetric delocalized ESS = [ 1/2'/2] [$If $21. Due to and Po, the $I-$2 coueling is renormalized from its vacuum value (3 to @, 1 -($21H[$l,Po]I$l),which varies with Po. For activated ET, the renormalized coupling p', evaluated at the transition state (TS) is a natural coupling measure. However, the difference between /3*rand 3(, is small and is not emphasized here.32
e
(27) Equations 6 and 7 folloy from eqs 4 and 5 when Po is in _equilibrium with the charge distribution: = (l/4r)((l/em)- ( I / c o ) ) e , & [ W ] . (28) ET barrier existence requires 0, < I - (l/e,,))M, (see eq 8);I' we assume this condition is satisfied. When & > 1/2(1 - ( l / e o ) ) M s we , find only two, delocalized, ESS; activated ET vanishes and we have only ground- and excited-state wells. This situation is similar, in a general way, to that catalogued as Class 111 by Robin and Day (Robin, M. B.; Day, P. Adu. Inorg. Chem. Radiochem. 1967, IO, 247). (29) Our solvent coordinate differs somewhat from the conventional one.13b.14.20.30 This difference is due to the electric field generated by the quantum overlap charge distribution a $l(fi$2(fi. (30) Zusman, L. D. Chem. Phys. 1980, 49, 295. (31) (a) Basilevsky, M. V.; Chudinov, G.E. Mol. Phys. 1988, 65, 1121. (b) Knapp, E. W.; Fischer, S. F. J . Chem. Phys. 1989, 90, 354.
e
2738 The Journal of Physical Chemistry, Vol. 94, No. 7, I990 With these renormalized couplings, we can determine the TS wave functions and their free energies G* as the free ensrgy minimum states with the constraint that the associated Po is invariant under 1 2.
-
3. Reference Definitions and Classifications It is useful to define the vacuum difference field electrostatic energy
with RD and R A the radii of D and A, and R D A the DA centerto-center distance. Another important quantity is the reorganization free energy Gr,33the free energy difference between the product state and the Franck-Condon excited state reached in a vertical transition from the reac_tantstate (Figure I ) . We can obtain \kFc by replacing in eq 4 Po equilibrated to the reactant state, and solving the resulting Schrodinger equation. For reference, in the standard Marcus theory for weak overlap ET reactions, G, is given by’
Letters distinctly separatedi7(due to nonlinearity) at a critical value b* = 1 of the dimensionless coupling36
At this point the electronic coupling overcomes the electronic polarization and allows a delocalized TS with a smooth barriersz8 (Instead of the above criterion, one could consider a dynamical definition based on a Landau-Zener4 transition between reactant and product curves, addressed in section 6 . ) 4. Nonadiabatic ET Reactions 1 - ( I /cm))Ms,Le., b’ < I , we find that When /3’, (-@J < the TS has a charge localization structure similar to the reactant and product states.37 With our classification above, we thus have a nonadiabatic ET process.38 The activation free energy AG* = G* - G;P is found to be36 (Figure la) r
(9) Equation 9 was derivedl for a Franck-Condon transition between charge-localized states, with explicit neglect of electronic coupling. Marcus has arguedl that such weak overlap conditions (such that the electronic coupling is SksTand ignorable) describe many ET reactions, and in this case, has given the ET activation free energy as
Just as in the weak coupling Marcus theory, eq IO, only the orientational polarization, signaled by the Pekar factor (( 1/em) - (I/eo)), makes a large direct contribution to AG*; the TS maintains the localized charge distribution. However, the coupling contribution to AG* is O(@ 2, (rather than O(@ ) in AG*c (eq I I)); contributes to AGY indirectly by way the coupling. We postpone further discussion of AG* until section 6 . For highly polar solvents (e- N 2 , c,
