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Quasidissipative intramolecular dynamics: An adiabatically reduced coupled equations approach
1.E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), p. 481.
2.C. Cohen‐Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. II (Wiley, New York, 1977), p. 1343.
3.R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw‐Hill, New York, 1965), p. 159.
4.V. F. Weisskopf and E. P. Wigner, Z. Phys. 63, 54 (1930).
5.M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964), Chap. 8, and references cited therein.
6.Minor variations of the definition of this quantity occur in the literature (see, e.g., Refs. 1–5, 7, and 13): Sometimes it is defined as the average of for only the receptor states at the energy of the initial state, while at other times it is defined as the average for all of the couplings to the receptor states.
7.For reviews, see K. F. Freed, Top. Appl. Phys. 15, 23 (1976);
7.P. Avouris, W. M. Gelbart, M. A. El‐Sayed, Chem. Rev 77, 793 (1977);
7.S. Mukamel and J. Jortner, in Excited State, edited by E. C. Lim (Academic, New York, 1977), Vol. 3, p. 57;
7.W. Rhodes, J. Phys. Chem. 87, 30 (1983).
8.For an application of the traditional Wigner‐Weisskopf method (Ref. 4) to the problem of radiationless transitions, see M. Btxon, J. Jortner, and Y. Dothan, Mol. Phys. 17, 109 (1969)
8.[see also, A. Nitzan, J. Jortner, and P. M. Rentzepis, Mol. Phys. 22, 585 (1971)]. This approach is most closely related to the present theory, although in the present theory explicit dynamical assumptions are employed in order to derive a generalized exponential decay law., Mol. Phys.
9.In an actual molecular system, there exists a discrete set of eigenstates, so truly dissipative exponential decay behavior can never occur. However, the density of states is so great in a typical large molecule that the recurrence time of the initial state probability is much longer than any relevant experimental time scale (and hence one uses the term “quasidissipative” behavior).
10.See, e.g., P. M. Felker and A. H. Zewail, Chem. Phys. Lett. 108, 303 (1984);
10.P. M. Felker and A. H. Zewail, J. Chem. Phys. 82, 2975 (1985);
10.P. M. Felker, W. R. Lambert, and A. H. Zewail, J. Chem. Phys. 82, 3003 (1985); , J. Chem. Phys.
10.N. Bloembergen and A. H. Zewail, J. Phys. Chem. 88, 5459 (1984).
11.See, e.g., K. W. Holtzclaw and C. S. Parmenter, J. Chem. Phys. 84, 1099 (1986), and references cited therein.
12.D. A. Dolson, K. W. Holtzclaw, D. B. Moss, and C. S. Parmenter, J. Chem. Phys. 84, 1119 (1986), and references cited therein.
13.See,e.g., D. F. Heller, Chem. Phys. Lett. 61, 583 (1979);
13.M. L. Sage and J. Jortner, Chem. Phys. Lett. 62, 451 (1979); , Chem. Phys. Lett.
13.D. F. Heller and S. Mukamel, J. Chem. Phys. 70, 463 (1979);
13.P. R. Stannard and W. M. Gelbart, J. Phys. Chem. 85, 3592 (1981).
13.For a review of some IVR theories for bond mode excitations, see M. L. Sage and J. Jortner, Adv. Chem. Phys. 47, 293 (1981).
13.Several random coupling models are applicable to certain IVR problems: See, e.g., E. J. Heller and S. A. Rice, J. Chem. Phys. 61, 936 (1974);
13.J. M. Delory and C. Trie, Chem. Phys. 3, 54 (1974);
13.K. G. Kay, J. Chem. Phys. 61, 5205 (1974);
13.W. M. Gelbart, D. F. Heller, and M. L. Elert, Chem. Phys. 7, 116 (1975);
13.M. Muthukumar and S. A. Rice, J. Chem. Phys. 69, 1619 (1978);
13.B. Carmeli and A. Nitzan, Chem. Phys. Lett. 58, 310 (1978);
13.B. Carmeli and A. Nitzan, J. Chem. Phys. 72, 1928, 2054 (1980).
14.E. L. Sibert III, W. P. Reinhardt, and J. T. Hynes, Chem. Phys. Lett. 92, 455 (1982);
14.E. L. Sibert III, W. P. Reinhardt, and J. T. Hynes, J. Chem. Phys. 81, 1115 (1984).
15.J. S. Hutchinson, J. T. Hynes, and W. P. Reinhardt, Chem. Phys. Lett. 108, 353 (1984).
16.G. A. Voth and R. A. Marcus, J. Chem. Phys. 84, 2254 (1986);
16.G. A. Voth, Chem. Phys. Lett. 129, 315 (1986);
16.S. J. Klippenstein, G. A. Voth, and R. A. Marcus, J. Chem. Phys. 85, 5019 (1986).
17.P. W. Milonni and W. A. Smith, Phys. Rev. A 11, 814 (1975).
18.This is perhaps intuitively obvious: The initial state is expected to be coupled to a limited number of states by virtue of the relevant selection rules which apply for that particular potential energy function and zeroth‐order basis set. Those states which are coupled to the initial state will then be coupled to other states within the constraints of the selection rules, and so on.
19.E. S. McGinley and F. F. Crim, J. Chem. Phys. 85, 5741, 5748 (1986).
20.Rovibrational states from other electronic manifolds may also contribute to the various tiers of states in the intramolecular coupling scheme. In that case, the IVR process stimulates electronic energy redistribution. Experimental results indicating this interesting effect have been obtained for several large molecules [see, e.g., U. Schubert, E. Riedle, H. J. Neusser, and E. W. Schlag, J. Chem. Phys. 84, 6182 (1986);
20.A. Amirav, J. Jortner, S. Okajima, and E. C. Lim, Chem. Phys. Lett. 126, 487 (1986), and references cited therein].
21.It is assumed here that one only includes states in the calculations that are within a certain energy width around the energy of the initial state. Enough states should be included so that most, if not all, possible space couplings are included.
22.S. N. Dixit and P. Lambropoulos, Phys. Rev. A 27, 861 (1983).
23.Equation (2.20) has been suggested as a criterion for having an effective quasicontinuum of states in electronic radiationless transitions theory(see, e.g., Ref. 7). The widths in that theory arise from radiative decay processes. In the present analysis, this situation implies that the Q‐space states will undergo rapid intramolecular decay before any complicated dynamics can occur between them.
24.The expression for given in Eq. (2.27) is also the sort of equation predicted from approximate resolvent operator approaches when the Q‐space states have finite decay widths (see, e.g., Ref. 7 for the case when the Q‐space states have radiative decay widths).
25.By using a resolvent operator approach, Heller and Mukamel (Ref. 13) have derived a similar line shape expression for the specific case of bond mode overtone excitation in molecules with several such modes (e.g., benzene).
26.By visually inspecting Fig. 9 of Ref. 14 for the and 6 C‐H overtones, it was estimated that the first tier (Q‐space) states decayed with a larger rate in the numerical calculations than did the third tier states for which Sibert et al. has assigned the decay widths of Therefore, the appropriate widths of the first tier Q‐space states were estimated to be roughly
27.D.‐H. Lu, W. L. Hase, and R. J. Wolf, J. Chem. Phys. 85, 4422 (1986);
27.R. J. Wolf, O. S. Bhatia, and W. L. Hase, Chem. Phys. Lett. 132, 493 (1986).
28.For example, by using the Whitten‐Rabinovitch formula [see, e.g., P. J. Robinson and K. A. Holbrook, Unimolecular Reactions (Wiley, New York, 1972), p. 131].
29.The parameter is somewhat different from the parameter N employed by Dolson et al. (Ref. 12). The latter parameter, defined as is based on a model having an initial state coupled to a single tier of states with an average density ρ. The parameter N provides a measure of the number of states that participate in the IVR of the state.
30.For a review, see E. B. Stechel and E. J. Heller, Annu. Rev. Phys. Chem. 35, 563 (1984). It is noted here that one is usually concerned with the molecular eigenstate, rather than zeroth‐order, energy level distributions.
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