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Long time tails in canonical ensemble unimolecular decay
1.See, W. L. Hase, in Dynamics of Molecular Collisions, Part B, edited by W. H. Miller (Plenum, New York, 1976), for an introduction to statistical theories of unimolecular reactions.
2.(a) R. S. Dumont and P. Brumer, J. Phys. Chem. 90, 3509 (1986).
2.(b) R. S. Dumont and P. Brumer, Chem. Phys. Lett. 188, 565 (1992).
3.The importance of the separation of reaction time scale from that of internal relaxation is recognized in (a) D. Chandler, J. Chem. Phys. 68, 2959 (1978), see also Ref. 2(b);
3.(b) R. S. Dumont and S. Jain, J. Chem. Phys. 97, 1227 (1992)., J. Chem. Phys.
4.N. B. Slater, Theory of Unimolecular Reactions (Cornell University, Ithaca, 1959);
4.E. Thiele, J. Chem. Phys. 36, 1466 (1961).
4.See also H. W. Schranz, L. M. Raff and D. L. Thompson, Chem. Phys. Lett. 171, 68 (1990), and references therein.
5.N. De Leon and B. J. Berne, J. Chem. Phys. 75, 3495 (1981).
6.A potential energy saddle point corresponds to unstable “variational equations” (or tangent dynamics). P. Brumer and J. W. Duff, J. Chem. Phys. 65, 3566 (1976).
7.See, for example, E. Pollak, in Theory of Chemical Reaction Dynamics, Vol. III, edited by M. Baer (Chemical Rubber, Boca Raton, 1985).
7.Also, E. Pollak and P. Pechukas, J. Chem. Phys. 69, 1218 (1978);
7.M. J. Davis, J. Chem. Phys. 86, 3978 (1987)., J. Chem. Phys.
8.(a) R. S. Dumont and P. Brumer, J. Chem. Phys. 90, 96 (1989).
8.(b) F. Vivaldi, G. Casati, and I. Guarneri, Phys. Rev. Lett. 51, 727 (1983) and Ref. 2(b).
9.Decomposition process population decay, or “survival probability” is related to the gap distribution, via
10.Any constrained (to the transition state) local minimum determines a threshold for a reaction path to a process. In this section, we assume only one such threshold. However, cases of multiple thresholds present no real difficulty. They are considered at the end of Sec. IV.
11.R. S. Dumont and P. Brumer, J. Chem. Phys. 87, 6437 (1987).
12.R. S. Dumont, J. Comp. Chem. 12, 391 (1991).
13.J. E. Straub, D. A. Hsu, and B. J. Berne, J. Phys. Chem. 89, 5188 (1985).
14.R. S. Dumont, J. Chem. Phys. 96, 2203 (1992).
15.D. M. Wardlaw and R. A. Marcus, J. Chem. Phys. 83, 3462 (1985).
16.D. R. Cox, Renewal Theory (Methuen, London, 1962).
16.Unimolecular kinetics is given a dynamical basis in terms of renewal theory in R. S. Dumont, J. Chem. Phys. 91, 4679 (1989).
17.This is essentially the content of Watson’s lemma which applies to Laplace transforms, but is easily adapted to Fourier transforms. See, for example, B. Davies, Integral Transforms and their Applications, 2nd ed. (Springer, New York, 1985), p. 10.
18.E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations (Dover, New York, 1955), p. 22.
19.It has recently been pointed out that that local rotation directions determine zero eigenvectors of the force constant matrix only at critical points of the potential. See I. Kolossvary and C. McMartin, J. Math. Chem. 9, 359 (1992).
20.M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (McGraw-Hill, New York, 1968), p. 95.
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