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Green function theory of charge transfer processes in solution
1.(a) R. A. Marcus, J. Chem. Phys. 24, 966 (1956);
1.(b) R. A. Marcus, 24, 979 (1956); , J. Chem. Phys.
1.(c) R. A. Marcus, 38, 1858 (1963). , J. Chem. Phys.
1.(d). An alternative but equivalent approach ws taken by N. S. Hush, Trans. Faraday Soc. 57, 557 (1961).
2.B. Tembe, H. L. Friedman, and M. D. Newton, J. Chem. Phys. 76, 1490 (1982), Appendix C.
2.See also the erratum in B. Tembe, H. L. Friedman, and M. D. Newton, J. Chem. Phys. 86, 4297 (1987).
3.P. G. Wolynes, J. Chem. Phys. 86, 5133 (1987).
4.H. L. Friedman, M. D. Newton, and G. Stell (to be submitted).
5.S. H. Northrup and J. T. Hynes, J. Chem. Phys. 73, 2715 (1980).
6.E. Efrima and M. Bixon, J. Chem. Phys. 70, 3531 (1979).
7.L. D. Zusman, Chem. Phys. 49, 295 (1980).
8.(a) G. van der Zwan and J. T. Hynes, J. Chem. Phys. 76, 2993 (1982);
8.(b) G. van der Zwan and J. T. Hynes, 78, 4174 (1983); , J. Chem. Phys.
8.(c) J. T. Hynes, J. Phys. Chem. 89, 4181 (1985);
8.(d) J. T. Hynes, 90, 3701 (1986)., J. Phys. Chem.
9.H. L. Friedman and M. D. Newton, Faraday Discuss. Chem. Soc. 74, 73 (1982), Eq. (12) of this reference should be multiplied by 1/2.
10.H. Sumi and R. A. Marcus, J. Chem. Phys. 84, 4894 (1986).
11.D. F. Calef and P. G. Wolynes, J. Phys. Chem. 87, 3387 (1983).
12.I. Rips and J. Jortner, J. Chem. Phys. 87, 2090 (1987).
13.J. D. Jackson, Classical Electrodynamics, 2nd. ed. (Wiley, New York, 1975), Eq. (5.32) ff.
14.P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw‐Hill, New York, 1953).
15.G. Rickayzen, Green Functions and Condensed Matter (Academic, New York, 1980). There is a difference between the usual Green function and the dielectric Green function defined here which typically depends upon r and R individually.
16.R. R. Dogonadze and A. M. Kuznetsov, J. Electroanal. Chem. Interface. Electrochem. 65, 545 (1975).
17.A. A. Ovchinnikov and M. Y. Ovchinnikova, Soviet JETP 29, 688 (1969).
18.M. Y. Ovchinnikova, Theor. Exp. Chem. 17, 507 (1982).
19.P. Madden and D. E. Kivelson, Adv. Chem. Phys. LVI, 467 (1984).
20.(a) R. P. Feynman, R. D. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison‐Weseley, Reading, MA, 1964), Vol. II, Sects. 10.3 and 10.4.
20.(b) To suppress surface terms we imagine to be a smoothly varying function of location such that in the interfacial regions, including the boundaries of cavities, ε changes rapidly but continuously.
20.(c) Whether a given ket represents a charge density field or a potential field is indicated by the notation.
21.B. Blaive and J. Metzger, Phys. Rev. A 30, 1600 (1984).
22.Typically (vacuum in cavity) or represents the dielectric response due to the electronic polarizability of material in the cavity. In either case it is assumed to be independent of frequency.
23.J. G. Kirkwood and F. H. Westheimer, J. Chem. Phys. 6, 506 (1938). Eq. (6).
24.H. L. Friedman, Mol. Phys. 29, 1533 (1975), where [In the general case (i.e., any r and R) for a spherical cavity of radius a in an infinite dielectric medium, the monopole term may be written as, Here max selects the argument of the largest magnitude.]
24.M. D. Newton, J. Phys. Chem. 79, 2795 (1975).
25.Y. T. Mazurenko, Opt. Spectrosc. 36, 283 (1974).
26.(a) A typical missing term has the form Its contribution to is smaller than the leading terms by a factor of or less. However, an interesting aspect is that the functional of whose limit is given in the preceding equation relaxes with more than one time constant even if itself has the Debye form. (See Sec. V.)
26.(b) S. Levine and H. E. Wrigley, Discuss. Faraday Soc. 24, 43 (1957).
26.(c) P. S. Ramanathan and H. L. Friedman, J. Chem. Phys. 54, 1086 (1971).
27.A. M. Kuznetsov and J. Ulstrup, J. Elektrochim. 21, 632 (1985).
28.For the case of charge distributions that conform to Eq. (2.7) we obtain the conventional two‐sphere result (Ref. 1), Eq. (3.9). As an alternative to the reorganization energy defined following Eq. (3.7) one can in the symmetric exchange case, associate this energy with the energy change due to a sudden transfer of charge to A from B in a system that is initially at equilibrium with and
29.A side from the dielectric contribution to the difference in energy of the ground state systems P,P and S,S may have an additional term due to differences in intrinsic ionization potentials. It is assumed that any such term has been included in calculations of
30.In subsequent use the symbol RC refers to a functional of and but these density fields will not be displayed in the notation. When is time dependent we write and as in Eq. (4.7).
31.S. E. Efrima and M. Bixon, J. Chem. Phys. 64, 3639 (1979).
32.R. A. Marcus and N. Sutin, Biochim. Biophys. Acta 811, 265 (1985).
33.(a) In the case that independent of u, then Eq. (4.1) reduces to at any t, as expected. To see this, perform the integration over u [Ref. 33(b)] before the integration over ω. We also notice that, in the special case that both Eqs. (4.2) and (5.4) apply, we obtain where is given following Eq. (5.5).
33.(b) We notice that causality implies that the ω integral in Eq. (4.1) vanishes if It follows that the upper limit on the u integral may be moved to without effect and that the poles of all lie in the upper half complex ω plane.
34.M. Bixon, Discuss. Faraday Soc. 74, 103 (1982).
35.D. Y. C. Chan, D. J. Mitchell, and B. W. Ninham, J. Chem. Phys. 70, 2946 (1979).
36.J. S. Hoye and G. Stell, J. Chem. Phys. 77, 5173 (1982).
37.D. Chandler, K. S. Schweitzer, and P. G. Wolynes, Phys. Rev. Lett. 49, 100 (1982).
38.M. J. Thompson, K. S. Schweitzer, and D. Chandler, J. Chem. Phys. 76, 1128 (1982).
39.S. Engstrom, B. Jonsson, and R. W. Impey, J. Chem. Phys. 80, 5481 (1984). An interesting aspect is that the rms field gradient at the ion center has negligible contribution from water molecules outside of the first hydration shell.
39.[S. Engstrom, B. Jonsson, and B. Jonsson, J. Magn. Reson. 50, 1 (1982)].
40.This form for is chosen to represent experimental dielectric data. It may not be consistent with the sum rules that characterize the exact dielectric function of a real system.
41.(a) D. W. Davidson and R. H. Cole, J. Chem. Phys. 18, 1417 (1950);
41.D. W. Davidson and R. H. Cole, 19, 1484 (1951); , J. Chem. Phys.
41.(b) D. W. Davidson, Can. J. Chem. 42, 971 (1961);
41.(c) C. J. F. Boettcher and P. Bordewijk, Theory of Dielectric Polarization, 2nd. ed. (Elsevier Scientific, New York, 1978), Vol. II, Chap. IX.
42.The averaging employed here is appropriate for a homogeneous dielectric system. An alternative averaging procedure based on the assumption of dielectric inhomogeneity has been discussed by I. Rips and J. Jortner, Chem. Phys. Lett. 133, 411 (1987)
42.and applied to the analysis of experimental electron transfer data by H. Heitele, M. E. Michel‐Beyerle, and P. Finckh, Chem. Phys. Lett. 138, 237 (1987).
43.D. Huppert, H. Kanety, and E. M. Kosower, Discuss. Faraday Soc. 74, 161 (1982).
44.M. Maroncelli and G. R. Fleming, J. Chem. Phys. 86, 6221 (1987).
45.S. Okuyama and D. W. Oxtoby, J. Chem. Phys. 84, 5824 (1986).
46.F. Marchesoni and P. Griglolini, J. Chem. Phys. 78, 6287 (1983).
47.The fluctuating cavity field (Ref. 9) may play the role in the kinetics which here is taken by the fluctuating reaction coordinate.
48.J. M. Deutch, J. Chem. Phys. 73, 4700 (1980).
49.A. Szabo, K. Schulten, and Z. Schulten, J. Chem. Phys. 72, 4350 (1980).
50.K. Schulten, Z. Schulten, and A. Szabo, J. Chem. Phys. 74, 4426 (1981).
51.M. D. Newton and N. Sutin, Annu. Rev. Phys. Chem. 35, 437 (1984).
52.S. Northrup and J. T. Hynes, J. Chem. Phys. 73, 2700 (1980).
53.H. A. Kramers, Physica 7, 284 (1940).
54.This procedure follows our earlier work (Ref. 9) in which it was assumed that the electron transfer centers were in a cavity in the dielectric medium and the reaction coordinate was taken to be the field in the cavity due to the fluctuating polarization density fields in the solvent.
55.As defined here, κ is a velocity. It is not the same as the first order rate constant appearing in Ref. 9 which, in the present notation, is just
56.A. M. Kuznetsov, J. Elektrochim. 17, 84 (1981).
57.B. S. Brunschwig, S. Ehrenson, and N. Sutin, J. Phys. Chem. 90, 3657 (1986).
58.H. Frohlich, Theory of Dielectrics (Oxford, New York, 1949).
59.H. L. Friedman, J. Chem. Soc. Faraday Trans. 2 79, 1465 (1983).
60.(a) J. B. Hubbard, J. Chem. Phys. 68, 1649 (1978);
60.(b) E. F. Caldin, Discuss Faraday Soc. 74, 200 (1982).
61.M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. GPO, Washington, D.C., 1946).
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