Adiabatically reduced coupled equations for intramolecular dynamics calculations
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17.The term dissipative here is meant to describe the exponential decay of the initially prepared nonstationary state into the intramolecular “bath” states. This behavior alone is not truly dissipative (i.e., irreversible) since reoccurrences of the initial state probability would eventually occur if the molecule did not undergo other relaxation processes such as collisions, radiative decay, etc.
18.An iterative numerical approach for determining the time dependence of zeroth‐order state amplitudes that is not based on the diagonalization of large matrices has been developed in A. Nauts and R. E. Wyatt, Phys. Rev. Lett. 51, 2238 (1983);
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21.See, for example, D. M. Larsen and N. Bloembergen, Opt. Commun. 17, 254 (1976);
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22.(a) In general, these states satisfy the criterion where and are, respectively, the coupling matrix element and the zeroth‐order energy difference between the state in the resonant/strongly coupled manifold of states and the state in the manifold of states off‐resonant and/or weakly coupled to the states, (b) The rapidly oscillating derivatives and the resulting negligible magnitudes of the are related to the fact that those off resonant states are detuned in energy [cf. Ref. 22(a)] from the nonstationary state energy 〈H〉 More specifically, a substantial buildup of probability for any length of time in the off‐resonant states would require the violation of energy conservation of the nonstationary state. In the language of quantum field theory, the off‐resonant states are energy‐nonconserving “virtual” states, and the probability in those states is modulated, in some sense, by the time‐energy uncertainty principle (i.e., the system can violate energy conservation, but it must do so on a very short time scale).
23.The matrix is the matrix representation of the total Hamiltonian operator H in the off‐resonant manifold of the basis states and is not to be confused with the zeroth‐order Hamiltonian matrix
24.An approximation equivalent to Eq. (2.7) is used by S. N. Dixit and P. Lambropoulos [Phys. Rev. A 27, 861 (1983)]. to help solve the dynamical equations for the density matrix describing multiphoton absorption and ionization.
24.Similar approximations have been applied to the dynamical equations for these processes written in the Heisenberg representation[e.g., J. L. F. de Meijere and J. H. Eberly, Phys. Rev. A 17, 1416 (1978);
24.P. W. Milonni and J. H. Eberly, J. Chem. Phys. 68, 1602 (1978), and references cited therein].
24.A related approximation [cf. discussion in P. W. Milonni and W. A. Smith, Phys. Rev. A 11, 814 (1975)]
24.that relies on a simplification of the poles of the resolvent operator for these systems has been used by, e.g., B. L. Beers and L. A. Armstrong, Jr., Phys. Rev. A 12, 2447 (1975);
24.S. N. Dixit and P. Lambropoulos, Phys. Rev. A 21, 168 (1980), and references cited therein., Phys. Rev. A
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26.H. Feshbach, Ann. Phys. (NY) 5, 357 (1958);
26.H. Feshbach, 19, 287 (1962)., Ann. Phys. (N.Y.)
27.For a review, see J. Killingbeck, Rep. Prog. Phys. 40, 963 (1977).
28.The constant matrix has been deleted and would simply introduce a constant shift in the eigenvalues of In the integration of the coupled equations [Eq. (2.9)], however, this factor is of particular value since the diagonal terms are small and the amplitudes thereby have less oscillatory character. The coupled equations are thus easier to integrate numerically.
29.See also the related treatment in P. R. Certain and J. O. Hirschfelder, J. Chem. Phys. 52, 5977 (1970);
29.P. R. Certain, D. R. Dion, and J. O. Hirschfelder, J. Chem. Phys. 52, 5987 (1970); , J. Chem. Phys.
29.J. O. Hirschfelder, Chem. Phys. Lett. 54, 1 (1977).
30.J. H. Choi, Prog. Theor. Phys. 53, 1641 (1975), and references cited therein.
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31.see also, J. D. Cresser and B. J. Dalton, J. Phys. A 13, 795 (1980).
32.The Hamiltonian matrix partitioning discussed in Sec. III A may also be cast in a projection operator form using similar techniques (e.g., Refs. 25 and 26).
33.M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964), Chap. 8, and references cited therein.
34.To include the radiative decay dynamics in the equations developed in Sec. II, a phenomenological radiative damping matrix γ is included in a new Hamiltonian such that More details of this procedure are given in Ref. 1.
35.L. Mower, Phys. Rev. A 22, 882 (1980);
35.see also, S. Swain, J. Phys. A 8, 1277 (1975);
35.S. Swain, 9, 1811 (1976)., J. Phys. A
36.H. Schultheis, R. Schultheis, and A. B. Volkov, Ann. Phys. 141, 179 (1982), and references cited therein.
37.See, for example, J. E. Wollrab, Rotational Spectra and Molecular Structure (Academic, New York, 1967), Appendix 7; see also.
37.B. Kirtman, J. Chem. Phys. 49, 3890 (1968).
38.A straightforward illustration of this point may be seen in a system related to that presented in Sec. IVA. If one sets and equal to zero, the coupled local mode system of Ref. 15 has only “sequential” coupling between zeroth‐order levels. The form of the solutions for the |0,3〉 and |3,0〉 states probabilities are the same as in Eq. (4.1), but the result based on the reduced coupled equations [Eq. (2.9)] now gives a frequency Ώ equal to A Van Vlecktreatment (e.g., Ref. 37) gives Ω equal to In the limit of (i.e., strong coupling in the manifold), the frequency obtained from Eq. (2.9) equals whereas the frequency based on the Van Vleck treatment remains the same. Numerical calculations for this model have verified that the latter frequency differs significantly from the exact result, whereas the result based on Eq. (2.9) showed good agreement for the parameters examined. Presumably, the Van Vleck perturbation expansion for this strong coupling case does not converge or converges slowly.
39.The authors of Refs. 15 and 16 assume an approximate time‐dependent solution for the local mode probability of the form (in their notation) where δ is the splitting between the symmetric and asymmetric local mode eigenstates. Their exact and approximate values for δ (our Ω) are given in the text.
40.S. M. Lederman, G. A. Voth, V. Lopez, and R. A. Marcus (to be submitted).
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