Wavepacket recurrences

H. Rabitz The information in the recurrence time alone is minimal. However, the temporal structure of the recurrence signal contains detailed information on the surface explored by the wandering scout wavepacket during its excursion. Further experiments may be necessary to follow (i.e., track) the wavepacket through its excursion over the potential surface. Such pump-probe experiments go beyond conventional spectroscopy. 

The periodic motion of the wavepacket in the potential well naturally shows up in the autocorrelation function S(t) as depicted schematically in Figure 7.3. Each return of (<) to its origin leads to a maximum in the autocorrelation function, a so-called recurrence. Since the part that is temporarily trapped in the inner region gradually diminishes the overall amplitude of S(t) decays in time. Eventually, the entire wavepacket leaks... 

In the time-dependent picture, resonances show up as repeated recurrences of the evolving wavepacket. Resonances and recurrences reveal, in different ways, the same dynamical effect, namely the temporary excitation of internal motion within the complex. In the context of classical mechanics, the existence of quantum mechanical resonances is synonymous with trapped trajectories performing complicated Lissajou-type motion before they finally dissociate. The larger the lifetime, the more frequently the wavepacket recurs to its starting position, and the narrower are the resonances. 

The transition from direct to indirect photodissociation proceeds continuously (see Figure 7.21) and therefore there are examples which simultaneously show characteristics of direct as well as indirect processes the main part of the wavepacket (or the majority of trajectories, if we think in terms of classical mechanics) dissociates rapidly while only a minor portion returns to its origin. The autocorrelation function exhibits the main peak at t = 0 and, in addition, one or two recurrences with comparatively small amplitudes. The corresponding absorption spectrum consists of a broad background with superimposed undulations, so-called diffuse structures. The broad background indicates direct dissociation whereas the structures reflect some kind of short-time trapping. 

The analysis presented so far explains the energy dependence of the spectrum in terms of the recurrences of the wavepacket. It does not, however, elucidate what kind of molecular motion actually causes the recurrences. In this particular example, the evolving wavepacket by itself does not clearly reveal the origin of the recurrences. The major part of the wavepacket follows the shortest route to dissociation and buries the much smaller portion, which returns at least once to the starting position. 

According to Section 4.1.1 the wavepacket is a superposition of stationary wavefunctions corresponding to a relatively wide range of energies. This and the superposition of three apparently different types of internal vibrations additionally obscures details of the underlying molecular motion that causes the recurrences. A particularly clear picture emerges, however, if we analyze the fragmentation dynamics in terms of classical trajectories. 

The period of the anti-symmetric stretch periodic trajectory does not correspond, however, to any of the three recurrences we see in Figure 8.4. This is not at all surprising in order to come back to the FC region, which in this case is considerably displaced from the anti-symmetric stretch orbit, the trajectory must necessarily couple to the symmetric stretch mode. If we were to launch the wavepacket at the outer slope of the saddle point, the anti-symmetric stretch periodic orbit would support recurrences by itself without coupling to the symmetric stretch mode. An example is the dissociation of IHI discussed in Section 7.6.2. 

Because of the superposition of three distinct types of periodic motion with different periods the wavepacket by itself does not reveal a clear picture in the present case, i.e., the classical skeleton is hardly visible through the quantum mechanical flesh . The perfect agreement between the recurrence times of the quantum mechanical wavepacket and the periods of the classical periodic orbits, however, provides convincing evidence that the structures in the absorption spectrum are ultimately the consequence of the three generic unstable periodic orbits. This correlation is... 

The quantum mechanical wavepacket closely follows the main classical route. It slides down the steep slope, traverses the well region, and travels toward infinity. A small portion of the wavepacket, however, stays behind and gives rise to a small-amplitude recurrence after about 40-50 fs. Fourier transformation of the autocorrelation function yields a broad background, which represents the direct part of the dissociation, and the superimposed undulations, which are ultimately caused by the temporarily trapped trajectories (Weide, Kiihl, and Schinke 1989). A purely classical description describes the background very well (see Figure 5.4), but naturally fails to reproduce the undulations, which have an inherently quantum mechanical origin. 

Fig. 8.11. (cont.) the unstable periodic orbit, represented by the solid line, influences the dissociation dynamics all direct trajectories, which fragment immediately without any recurrence, are discarded. The times range from 0 fs in (a) to 50.8 fs in (h). The arrows schematically indicate the evolution of the classical wavepacket and the heavy dot marks the equilibrium of the R-state potential energy surface. Adapted from Weide, Kiihl, and Schinke (1989). 

This is inherently impossible in the time-independent approach because the wavefunction contains the entire history of the wavepacket. The real understanding, however, is provided by classical mechanics. Plotting individual trajectories easily shows the type of internal motion leading to the recurrences which subsequently cause the diffuse structures in the energy domain. The next obvious step, finding the underlying periodic orbits, is rather straightforward. 

The wavepacket in the lBi state, 3>b, performs large-amplitude symmetric stretch motion leading to recurrences in the autocorrelation function the recurrences in turn cause vibrational structures in the absorption spectrum. 

Figure 13. Result of the two-dimensional calculation for Ca-HBr. The calculation and the spectrum of the Pb(A A")transition are compared on the left. The two-dimensional surface built on the reaction coordinate Rh-x and the Ca-HX bending angle is displayed in the top-right panel. The propagation was performed over 1.25 ps. Each contour corresponds to 500 cm. The contour of the Gaussian wavepacket issued from the ground state by vertical excitation is also shown in this surface. The bottom panel on the right shows the resulting modulus of the autocorrelation function with clear recurrences. Adapted from Ref. [243].

Figure 5 shows a plot of the magnitude of the overlap for / = 0, K0/ (f)> sl. versus time. The magnitude of the overlap decreases as the wavepacket spreads out. There is no recurrence. The steeper the inverted potential (i.e., the higher coj), the faster the wavepacket spreads out and the faster the overlap decreases. Because the inverted harmonic potential surface can model only a small area around the Frank-Condon region, this model can only be applied to short time dynamics. 

The overlap for the absorption part (A) versus time is shown in Figure 15a. The magnitude of the overlap is 1 at time zero. It decreases with time as the wavepackets move away from their original positions. At later time as the wavepackets come back to their original positions, it increases giving a recurrence in the overlap. 

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