The Relation Between the Barriers for Thermal and Optical Electron Transfer Reactions in Solution

References

• The term “mixed-valence system” includes both intramolecular electron transfer within a bridged or linked system and electron transfer within the precursor complex formed in a bimolecular reaction. Although transitions in mixed-valence systems have been selected for discussion for the sake of concreteness, the considerations are of much more general application
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• Creutz , C. 1983 . Prog. Inorg. Chem. , 30 : 1
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• Hush , N. S. 1967 . Prog. Inorg. Chem. , 8 : 391
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• Marcus , R. A. 1956 . J. Chem. Phys. , 24 : 966
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• Marcus , R. A. 1960 . Disc. Faraday Soc. , 29 : 21
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• Marcus , R. A. and Sutin , N. 1985 . Biochim. Biophys. Acta , 811 : 265
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• Marcus , R. A. 1956 . J. Chem. Phys. , 24 : 979 The harmonic (quadratic) behavior of the free-energy profile along the reaction coordinate (- m) bears a simple relationship to a harmonic dependence of the free energy on the charge distribution: the charge distribution used to generate the changes in dielectric polarization depends linearly on this quantity m, as shown in Refs. 4 and 5. For a discussion of the quadratic free-energy dependence on a change of charge distribution see, for example J. Chem. Phys. 39, 1734(1963). The dielectric continuum discussion refers to a constant temperature and pressure system, while the statistical-mechanical treatment is for a system at constant temperature and volume. However, a related statistical-mechanical treatment can be given for a constant temperature and pressure system
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• The equality of ΔE and ΔH for a vertical transition arises because there is no time for a volume change during the vertical transition (which occurs at constant nuclear coordinates in the Franck-Condon approximation): any volume change occurs in the relaxation process after the vertical transition. For a process in a condensed phase ΔH and ΔHW differ negligibly because the PΔV term is very small for a reaction in a condensed phase under normal conditions. Along the reaction coordinate there are nevertheless volume changes so that, at a given temperature and pressure, the volume of the system at -m = 1 will differ from that at m = 0 when ΔVo'≠ = 0
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• The assumption that λ is temperature independent has been made in order to simplify the resulting expressions. Expressions for thermally activated electron transfer derived without this assumption are presented in Ref. 10
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• This result can be obtained using the results in Appendix If. In particular, in terms of the quantities derived there, one uses Eq. (A9), first with m equal to its value at the intersection (m∗ = - (ΔGo′ + λ)/2λ) and then with in equal to its value at the minimum of the H' vs. in curve (m = TΔSo/2λ) and subtracting. The difference yields the first term on the right-hand side of Eq. (2). One can also obtain this result by noting that Hr and Hp are quadratic functions of in and for this reason the value of H at the intersection equals, as in the G plots, the square of the vertical difference, ΔHo′ + λ, between the products' and reactants' H curves at the minimum of the reactants H curve, divided by 4λ
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• This result follows immediately using Eq. (A9) with in equal to its value at the minimum of the Hr vs. m curve, namely m= TΔSo′/2λ, as in the preceding footnote
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• As we noted in Ref. 6, Sec. IIID, this equality of Eop, and λ + ΔGo′ was also shown in a previous paper on solvent effects on spectra, R. A. Marcus, J. Chem. Phys. 43, 1261 (1965): this equality is evident from Eq. (14) of that paper, where Fop e-g - Fe-g is the polar contribution to λ and Fe - Fg is the polar contribution to ΔFo (ΔGo′ in the present notation)
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• Marcus , R. A. 1984 . J. Chem. Phys. , 81 : 4944 In obtaining Eq. (14) from Eq. (13) in this reference it should have been stated there that cross-terms from different branches of the Wfi in Eq. (13) oscillate rapidly and can be (and were in Eq. (14)) neglected
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• In this discussion we have tacitly used, for simplicity, a distribution function for a constant volume system rather than for an isobaric system and, in so doing, we have neglected for these condensed-phase systems the (indeed negligible) difference between enthalpy and energy. The relationship between Helmholtz and Gibbs free energies for this problem is described in the article cited in Ref. 13
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• Liu , D. K. , Brunschwig , B. S. , Creutz , C. and Sutin , N. in press . J. Am. Chem. Soc. ,
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