Generalized Langevin theory for many‐body problems in chemical dynamics: The method of partial clamping and formulation of the solute equations of motion in generalized coordinates
1.S. A. Adelman, Adv. Chem. Phys. 53, 61 (1983). Also see references to earlier work cited in this article.
2.For recent reviews with many references to earlier work on condensed phase spectroscopic processes see: (a) For liquid state spectroscopic processes D. W. Oxtoby, Adv. Chem. Phys. 40, 1 (1979);
2.(b) For solid state spectroscopic processes, see L. E. Brus and V. E. Bondybey, in Radiationless Transitions, edited by S. H. Lin (Academic, New York, 1980).
3.For a recent review concerning liquid state energy relaxation processes, with many references to earlier work, see D. W. Oxtoby, Adv. Chem. Phys. 51, 487 (1981).
4.For some recent experimental studies of liquid solution chemical reaction dynamics see, for example: (a) For photolysis reactions, T. J. Chuang, G. W. Hoffman, and K. B. Eisenthal, Chem. Phys. Lett. 25, 201 (1974);
4.C. A. Langhoff, K. Gnadig, and K. B. Eisenthal, Chem. Phys. 46, 117 (1980);
4.C. A. Langhoff, B. Moore, and M. De Meuse, J. Am. Chem. Soc. 104, 3576 (1982);
4.D. F. Kelley and P. M. Rentzepis, Chem. Phys. Lett. 85, 85 (1982);
4.P. Bado, P. H. Berens, and K. R. Wilson, Proc. Soc. Photo‐Opt. Instrum. Eng. 322, 230 (1982);
4.(b) For isomerization reactions D. L. Hasha, T. Eguchi, and J. Jonas, J. Chem. Phys. 75, 1571 (1981)
4.and D. L. Hasha, T. Eguchi, and J. Jonas, J. Am. Chem. Soc. 104, 2290 (1982);
4.S. P. Velsko and G. R. Fleming, Chem. Phys. 65, 59 (1982)
4.and S. P. Velsko and G. R. Fleming, J. Chem. Phys. 76, 3553 (1982).
5.(a) R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957);
5.(b) R. Kubo, Rep. Prog. Theor. Phys. 29, 235 (1966).
6.H. Mori, Prog. Theor. Phys. 33, 423 (1965);
6.H. Mori, 34, 399 (1965)., Prog. Theor. Phys.
7.R. Zwanzig, J. Chem. Phys. 33, 1338 (1960).
8.For numerical aspects of stochastic classical trajectory methodology see J. C. Tully, G. H. Gilmer, and M. Shugard, J. Chem. Phys. 71, 163 (1979). Also see Appendix A of Ref. 10(a).
9.(a) C. L. Brooks III, M. W. Balk, and S. A. Adelman, J. Chem. Phys. 79, 784 (1983);
9.(b) M. W. Balk, C. L. Brooks III, and S. A. Adelman, J. Chem. Phys. 79, 804 (1983).
10.For applications of the MTGLE theory to other important problems in condensed phase chemical kinetics see: (a) M. Olson and S. A. Adelman, Phys. Rev. (to be published);
10.(b) H. L. Nguyen and S. A. Adelman, J. Chem. Phys. (to be published);
10.(c) R. Stote and S. A. Adelman, ibid. (to be published).
11.For application of the MTGLE theory (a) to calculation of reagent configuration dependent correlation functions see C. L. Brooks III and S. A. Adelman, J. Chem. Phys. 76, 1007 (1982);
11.(b) to modeling of reagent configuration‐dependent correlation functions see J. Chem. Phys. 77, 484 (1982)., J. Chem. Phys.
12.The fundamental role played by the solvent frequency spectrum in liquid state chemical reaction dynamics has also been recognized by Wilson and co‐workers. See, for example, Fig. 9 of P. Bado, P. H. Berens, J. P. Bergsma, M. H. Coladonto, C. G. Dupery, P. M. Edelsten, J. D. Kahn, and K. R. Wilson, in Proceedings of the International Conference on Photochemistry and Photobiology, edited by A. Zewail (Harwood Academic, New York, 1983).
13.These limiting cases include the case of solutes for which the solute‐solvent mass ratio tends to infinity and the case of solutes linearly coupled to perfectly harmonic solvents.
14.The clamping approximation was suggested by the rigorous mathematical operation of chain atom clamping which is MTLE analog of the Mori (Ref. 6) projection process (see Ref. 1).
15.See, for example, the numerical tests of the clamping approximation presented in Ref. 12.
16.An example of this breakdown was given in Ref. 10(b) where the small excursion or linear response model was shown to only crudely describe the very fast initial cage breakout step of the photolysis process. A solid state example is discussed in the paper of Brus and Bondybey [Ref. 2(b)], They mention that the rotation of diatomic hydrides in low temperature rare gas matrices is nearly free.
17.For example, for molecular iodine in simple solvents only four of the 36 matrix elements of the response function surfaces are distinct and nonnegligible.
18.For a detailed analysis of the dynamics of molecular vibrations see, for example, E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations, The Theory of Infrared and Raman Spectra (Dover, New York, 1965).
19.For a quantitative formulation of the concept of a reaction coordinate see R. A. Marcus, J. Chem. Phys. 45, 4493, 4500 (1966)
19.and R. A. Marcus, J. Chem. Phys. 45, 4500 (1968).
19.For related work, see R. A. Marcus, J. Chem. Phys. 49, 2617 (1968).
19.For a recent formulation see W. H. Miller, J. Chem. Phys. 72, 99 (1980).
20.Analytic solution is possible within the parabolic well and barrier approximations for the potential of mean force.
21.Barrier crossing within the Langevin equation framework is treated by H. A. Kramers. See, for example, S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
22.By including a solvent sheath explicitly, extended MTGLE models which should give results of comparable accuracy to full molecular dynamics calculations may be developed. Within the extended models, however, the simple solvation shell picture of the basic models is obscured.
23.See, for example, H. Goldstein, Classical Mechanics (Addison‐Wesley, Reading, Mass., 1980).
24.These formulas will be presented elsewhere in applications papers.
25.The matrix notation is defined as follows: and “(see PDF for diagram)”.
26.Equation (2.5) may be readily derived from the results presented in Ref. 23, Chap. 1.
27.All matrix square roots taken in this paper will be evaluated by the Choleski prescription as described in E. Bodewig, Matrix Calculus (North‐Holland, Amsterdam, 1956), Part I.
28.See the derivation of Eq. (6.36) given in Ref. 1.
29.A rigorous derivation of the form of may be made by following the method of derivation of the Liouville equation from Hamilton’s equations for the case of unconstrained dynamics given in Sec. IV B of Ref. 1.
30.An analysis shows that Eq. (5.16) holds exactly or approximately if molecular coordinates, of mass and orientational coordinates; set of normal modes, normal modes plus center of mass and orientational coordinates; coordinate, and internal coordinates and center of mass and orientational coordinates.
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