(a) Initial wave packet with and placed at the point on the potential function for . (b) The half spectrum calculated after SWP propagation of this initial wave packet through time with classical trajectories and .
Barrier region of the potential function. Initial wave packet with and is placed at Wave functions of under-the-barrier and over-the-barrier metastable states were calculated by propagating this initial wave packet and then using Eq. (19). Solid line-quantum propagation; dashed line-SWP propagation with and . Shaded areas show the classically forbidden regions; the barrier top is at
(a) The autocorrelation function and (b) the half spectrum obtained from propagation of the initial wave packet shown in Fig. 2. Solid line-quantum propagation; dashed line-SWP propagation with and . Dotted line-SWP propagation with very small and very large ; see text for details. Vertical lines in frame (b) show energy of the barrier top and energies of two metastable states ( and ) extracted from the autocorrelation function using the Prony analysis at time .
Diagram of the energy transfer mechanism in a three-level system. Two metastable states and with energies and width (, ) and (, ), respectively, are stabilized by third body collisions to give stable in the bound state at energy . Stabilization rate coefficients are , , and . The rates of formation and decay of the metastable states are determined by , , , and .
Energies of several upper states in the model system of Eq. (4) for the collision. The dashed line shows energy of the barrier top as a function of . At there are no metastable states; as increases, more and more metastable states appear. The majority of states are under-the-barrier states, but some over-the-barrier states are also present. States are labeled by the vibrational quantum number . For each value of the transition is shown (by arrow) from the not narrow metastable states to the upper bound state, or to the upper narrow metastable state, when present. See text for details.
Widths of not narrow resonances in our model. The SWP propagation and Prony analysis were used. Metastable states are labeled by the corresponding values of . The dotted line corresponds to . Upper and lower dashed lines correspond to and , respectively. The majority of the metastable states fall in the region and should be assigned as broad. The contour map shows the value of which determines the contribution of a metastable state to the recombination rate. Empty circles show the metastable states with a large uncertainty in the SWP result for . See text for details.
Contributions of different values to the total recombination rate of Eq. (44). Filled bars-SWP results; empty bars-WKB results. Transitions from maxima to minima occur every time the upper metastable state disappears due to centrifugal distortion of the potential (compare to Fig. 5).
Energies and lifetimes of the metastable states in potential (4) calculated via Prony analysis of the autocorrelation function found from the semiclassical (SWP) and quantum (QM) propagation of wave packets. Values designated as WKB has been calculated with traditional semiclassical time-independent WKB method.
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