Wave packet dynamics/propagation

In the wave packet dynamics approach to unimolecular predissociation, the quantum dynamics is studied by directly propagating quantum wave packets. After numerically propagating the quantum wave packets, all the detailed information about the reaction dynamics can, in principle, be extracted from the numerical results. 

With this short overview of molecular wave packet dynamics in mind, the wave packet propagation in small prototype molecules will now be examined more concretely. Different theoretical methods exist to describe ultrafast multiphoton ionization processes in diatomics and have been discussed in detail in previous work [3, 33-35, 291-294]. The theoretical approach used here is adapted specially to the presented 3d problems and was improved in Manz s group, mainly by de Vivie-Riedle [295] assisted by Reischl [81]. A few special features of their theoretical ansatz applied to the pumpfeprobe experiments carried out on the model molecules K2 and Naa are now briefly summarized. 

This discussion makes clear that the spectrogram technique enables us to characterize the varied wave packet dynamics for the two different pumpfeprobe cycles. The discussion nicely demonstrates the value of spectrograms for the analysis of wave packet phenomena as total and fractional revivals. Moreover, the spectrograms allow the identification of ionization pathways in pump probe experiments. Hence, spectrograms are excellent tools for an improved analysis of wave packet propagation phenomena, compared to real-time and Fourier spectra only. 

Real-Time Dynamics of Excited NaK Molecules. Berg et al. measured the wave packet dynamics of NaK excited to its A state up to At = 40 ps [353]. They used a heat-pipe oven to produce a rather hot ensemble of NaK molecules in the electronic ground state. With an 85 ps laser pulse at 790 nm a wave packet was prepared on the A state PES. The wave packet s propagation was followed by a delayed probe pulse, which induced an excitation of the E Z" " state. By detecting the fluorescence of the E state, the real-time evolution of the wave packet could be recorded within the first 40 ps. 

In Chap. 3, wave packet propagation could be observed for nearly all of the alkali dimer and trimer systems considered, over a rather long time compared to the wave packet oscillation period. The wave packet dynamics - a fingerprint of the excited molecule - definitely characterize the excited bound electronic state of these molecules. However, with the results on K3 (excited with A 800 nm), another phenomenon, which often governs ultrafast molecular and cluster dynamics, comes into the discussion photodissociation induced by the absorption of single photons. This photoinduced dissociation permits detailed study of molecular dynamics such as breaking of bonds, internal energy transfer, and radiationless transitions. The availability of laser sources with pulses of a few tens of femtoseconds today opens a direct, i.e. real-time, view on this phenomenon. 

It is shown here that such photodetachment experiments are a powerful tool to observe and charaterize properties of the transition state of molecules and clusters. Neumark and coworkers nicely demonstrated this in cw experiments [414-417]. The approach using ultrashort laser pulse excitation enables the preparation of the neutral in a superposition of eigenstates. Hence, wave packet dynamics can take place. The corresponding wave packet propagation is probed by the ionization process, which results in the production of the cation and a further photoelectron. Here it is shown that the real-time detection of the cation signal yields new information about the ultrafast nuclear dynamics of the neutral molecules or clusters. 

The theoretical model for photodetachment is similar to that used to describe photodissociation outlined in the last section. As illustrated in Fig. 3.7, the initial wave packet on the neutral PES was chosen as the ground vibrational state of cis-HOCO, which has a lower energy than its tram counterpart. The anion vibrational eigenfunction was determined on a newly developed anion PES at the same CCSD(T)-F12/AVTZ level [130], as used to construct the neutral PES [100, 101]. The neutral wave packet was propagated to yield probabilities to both the HO-I-CO and H-I-CO2 asymptotes with a flux method [108] and the cosine Fourier transform of the Chebyshev autocorrelation function yielded the energy spectrum [44]. The discretization of the Hamiltonian and wavepacket, and the propagation were essentially the same as in our recent reaction dynamics study [107]. 

Ohrn and co-workers have developed a direct dynamics approach which incorporates both the electrons and nuclei dynamics (END).""" The complete electron-nuclear coupling terms are retained in the calculation and, as a result, the dynamics is not constrained to a single Born-Oppenheimer potential energy surface i.e., electronic non-adiabaticity is explicitly included. A complication in this approach is the computational demand in propagating an electronic wavefunction which is an accurate representation of the ground electronic state as well as multiple excited electronic states. This approach will become more widely used as computation becomes more powerful. In its initial development,""" Deumens et al. used END and treated the dynamics of the nuclei purely classical as in the above classical direct dynamics. More recently, a semiclassical description of the nuclear motion has been implemented by incorporating Heller s""" "" Gaussian wave packet dynamics."" ... 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

It is possible to propagate just the real part of a wave packet if the representation of H is real-valued, as it often is in chemical reaction dynamics. 

Another popular and convenient way to study the quantum dynamics of a vibrational system is wave packet propagation (Sepulveda and Grossmann, 1996). According to the ideas of Ehrenfest the center of these non-stationary functions follows during a certain time classical paths, thus representing a natural way of establishing the quantum-classical correspondence. In our case the dynamics of wave packets can be calculated quite easily by projection of the initial function into the basis set formed by the stationary eigen-... 

In this contribution we extend our previous studies on the HB spectroscopy and dynamics of salicylaldimine (SA, see Fig. 1) focussing on the influence of isotopic H/D substitution on the IVR dynamics. In the following Section 2 the CRS Hamiltonian is briefly introduced. In Section 3 numerical results of a seven-dimensional (7D) wave packet propagation are discussed. 

In summary, using a combination of a 7D reaction surface and a numerically exact wave packet propagation, we have shown that the IVR dynamics in SA after ultrafast IR laser excitation changes qualitatively upon deuteration of the hydrogen bond. 

R.H. Bisseling, R. Kosloff, J. Manz, Dynamics of hyperspherical and local mode resonance decay studied by time dependent wave packet propagation, J. Chem. Phys. 83 (1985) 993. 

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