Resonance stationary wavefunction

Fig. 7.4. Schematic illustration of the stationary wavefunction P(E) for energies off and on resonance. In contrast to Figure 7.2, here the horizontal line marks a particular energy.

The time-dependent wavepacket accumulates in the inner region of the PES while it oscillates back and forth in the shallow potential well as illustrated in Figure 7.8. This vibrational motion leads to an increase of the stationary wavefunction in the inner region, however, only if the energy E is in resonance with the energy of a quasi-bound level. If, on the other hand, the energy is off resonance, destructive interference of contributions belonging to different times causes cancelation of the wavefunction. 

Thus, the stationary wavefunction becomes on resonance essentially a bound-state wavefunction with an amplitude which is proportional to the corresponding lifetime. The larger the survival time in the well region the larger is the magnitude of the stationary wavefunction. 

Resonances in half and in full collisions have exactly the same origin, namely the temporary excitation of quasi-bound states at short or intermediate distances irrespective of how the complex was created. In full collisions one is essentially interested in the asymptotic behavior of the stationary wavefunction L(.E) in the limit R —> 00, i.e., the scattering matrix S with elements Sif as defined in (2.59). The S-matrix contains all the information necessary to construct scattering cross sections for a transition from state i to state /. In the case of a narrow and isolated resonance with energy Er and width hT the Breit- Wigner expression... 

The excited complex breaks apart very rapidly and only a minor fraction performs, on the average, one single internal vibration. Therefore, the total stationary wavefunction does not exhibit a clear change of its nodal structure when the energy is tuned from one peak to another (Weide and Schinke 1989). In the light of Section 7.4.1 we can argue that the direct part of the total wavefunction, S dir-, dominates and therefore obscures the more interesting indirect part, Sind- The superposition of the direct and the indirect parts makes it difficult to analyze diffuse structures in the time-independent approach. In contrast, the time-dependent theory allows, by means of the autocorrelation function, the separation of the direct and resonant contributions and it is therefore much better suited to examine diffuse structures. 

Periodic orbits also explain the long-lived resonances in the photodissociation of CH.30N0(S i), for example, which we amply discussed in Chapter 7. But the existence of periodic orbits in such cases really does not come as a surprise because the potential barrier, independent of its height, stabilizes the periodic motion. If the adiabatic approximation is reasonably trustworthy the periodic orbits do not reveal any additional or new information. Finally, it is important to realize that, in general, the periodic orbits do not provide an assignment in the usual sense, i.e., labeling each peak in the spectrum by a set of quantum numbers. Because of the short lifetime of the excited complex, the stationary wavefunctions do not exhibit a distinct nodal structure as they do in truly indirect processes (see Figure 7.11 for examples). 

Resonance phenomena have been shown to play a significant role in many electron collision and photoionization problems. The long lived character of these quasi-stationary states enables them to influence other dynamic processes such as vibrational excitation, dissociative attachment and dissociative recombination. We have shown it is possible to develop ab initio techniques to calculate the resonant wavefunctions, cross sections and dipole matrix elements required to characterize these processes. Our approach, which is firmly rooted in the R-matrix concept, reduces the scattering problem to a matrix problem. By suitable inversion or diagonalization we extract the required resonance parameters. 

In a time-dependent picture, resonances can be viewed as localized wavepackets composed of a superposition of continuum wavefunctions, which qualitatively resemble bound states for a period of time. The unimolecular reactant in a resonance state moves within the potential energy well for a considerable period of time, leaving it only when a fairly long time interval i has elapsed x may be called the lifetime of the almost stationary resonance state. 

At the center of the edifice of any theory of resonances in the continuous spectrum of a many-particle Hamiltonian, regardless of what type of process produces their excitation, is the concept of transient wavefunction localization in the N-particle configuration space, which takes place as the reaction proceeds from the stationary states (reactants) at f = - oo to the stationary states (products) at f = oo. 

In Fano s [29] formal theory of resonance states, the energy-dependent wavefunctions are stationary, the energies are real, and the formalism is Hermitian. The observable quantities, such as the photoabsorption cross-section in the presence of a resonance, are energy-dependent and the theory provides them in terms of computable matrix elements involving prediagonalized bound and scattering N-electron basis sets. The serious MEP of how to compute and utilize in a practical way these sets for arbitrary N-electron systems is left open. 

Additional insight into the vibronic dynamics can be achieved by performing time-dependent calculations. The latter allow for a more direct visualization of the coupled electronic and nuclear motions. Moreover, given only the spectrum, Eq. (31), or a small number of resonance Raman amplitudes, the information obtained from the time-dependent wavefunction differs also in principle from that of stationary spectra. 

---