Wave packet propagation

In recent years, the wave packet approach has proved to be a powerful tool to describe and analyze the real-time dynamics of molecular motion [2, 3]. Many of the theoretical investigations in this field [7, 8, 299, 313] depend on the work of Heller, who formulated a basic description in 1981 [284]. Parallel to this, magnificent experimental studies in the field of laser femtochemistry, pioneered by Zewail, were successfully performed [20]. 

The choice of the ground state of the ion as the final state j)f) may have several conceptual and practical advantages. First, the detection of charged particles is extremely sensitive. Second, the detection of the ions provides mass information, and third, the ionization is always an allowed process. Any molecular state can be ionized, whereas the electronic spectroscopy relies on the existence of optically allowed transitions. Besides this, no Rabi oscillation between bound states, which can interfere with wave packet measurements, can occur in ionization. Further information can be obtained by analyzing the photoelectron (e.g. as to its kinetic energy), analogous to dispersed fluorescence methods. 

To observe the pure principles of wave packet propagation up to now, one has had to concentrate the investigations mainly on small, i.e. di- and tri-atomic, molecular systems. Therefore, this chapter is divided into two parts. First, in Sect. 3.1, the ultrafast dynamics of different alkali dimers is investigated in greater detail. Besides some basics, several features, for example energy dependence, isotopic effects, controlled molecular dynamics, and revivals of wave packets, are discussed. The use of the spectrogram technique nicely visualizes the different phenomena. 

In the second part of this chapter (Sect. 3.2), different wave packet propagation phenomena in excited alkali trimers are discussed. The time-resolved pseudorotation of the sodium trimer is presented in Sect. 3.2.2. Last but not least, applying laser pulses of the same wavelength but of different pulse width enables a mode-selective preparation of the trimer, hence controlling its dynamics (Sect. 3.2.4). Wave packet propagation on a repulsive PES (Sect. 3.2.5), studied on the potassium trimer, leads to the phenomena of ultrafast photodissociation, which then is the topic of the subsequent chapter. 

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Beck M H and Meyer H D 1998 Extracting accurate bound-state spectra from approximate wave packet propagation using the filter-diagonalization method J. Chem. Phys. 109 3730... 

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. 

Note that there are a number of quantum number indices in Eq. (38) and the computation of observables, depending on the degree of averaging associated with the observable, can require many separate wave packet propagations. 

It can be seen from the algorithm model stated above that in the Krotov method the electric field obtained in the feth iteration is used immediately to propagate /(f), which has a direct contribution to the new electric field in the next time step. In one iteration, the Krotov method involves three wave packet propagations, that is, the forward propagations of and x (t) in Steps 8.3... 

The term scar was introduced by Heller in his seminal paper (Heller, 1984), to describe the localization of quantum probability density of certain individual eigenfunctions of classical chaotic systems along unstable periodic orbits (PO), and he constructed a theory of scars based on wave packet propagation (Heller, 1991). Another important contribution to this theory is due to Bogomolny (Bogomolny, 1988), who derived an explicit expression for the smoothed probability density over small ranges of space and energy... 

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-... 

State Spectra from Approximate Wave Packet Propagation Using the Filter-Diagonalization Method. 

Koppel [180] has performed exact time-dependent quantum wave-packet propagations for this model, the results of which are depicted in Fig. 2A. He showed that the initially excited C state decays irreversibly into the X state within 250 fs. The decay is nonexponential and exhibits a pronounced beating of the C and B state populations. This model will allow us to test mixed quantum-classical approaches for multistate systems with several conical intersections. 

The ionisation process of the NaK dimer is investigated by applying evolution strategies to optimize the spectral phase of fs pulses interacting with the molecules. The obtained optimal pulse structure with three intensity maxima is presented. As an explanation of the ionization process a simple model of wave packet propagation on given potential energy surfaces is proposed. 

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. 

A statistical relationship between the above description and the standard one can be obtained. In a molecular sample at time t, the nuclei are statistically s-tributed. Each molecule shows its own particular energy gap between the electronic states involved in the process. On the average, things may look like as an electro-nuclear adiabatic process which could be modelled as a wave packet propagating on an adiabatic potential energy surface. This is the point where standard BO simulations of reaction processes [40] and the present view can be tied together. Individual systems are sensing electronic processes while the molecular... 

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. 

J. Breidbach, L.S. Cederbaum, Migration of holes Formalism, mechanisms, and illustrative applications, J. Chem. Phys. 118 (2003) 3983 A.I. Kuleff, J. Breidbach, L.S. Cederbaum, Multielectron wave-packet propagation General theory and application, J. Chem. Phys. 123 (2005) 044111. 

In the calculations reported below, the MCTDH method has been employed for all calculations involving more than a single 2/i electronic state, i.e., involving PJT interactions. As a drawback, vibronic line spectra are not directly obtained from this (as with any wave-packet propagation) method. The spectral envelope is, however, easily obtained as a Fourier transform according to Ref. [26] ... 

Kucar, J. and Meyer, H.-D. (1989). Exact wave packet propagation using time-dependent basis sets, J. Chem. Phys. 90, 5566-5577. 

Lee, S.-Y. and Heller, E.J. (1982). Exact time-dependent wave packet propagation Application to the photodissociation of methyl iodide, J. Chem. Phys. 76, 3035-3044. 

The information about the decay of the resonance states can be extracted from a wave packet propagation. The initial state y(i2, to) = (-R x(0)) is chosen such that... 

An advantage of the local control method described above is that it can be applied to wave packet propagation starting from an initial, nonstationary state, in contrast to ordinary wave packet control, which begins with the initial condition of a stationary state. An example where starting from such an initial condition is useful is the control of a localized state of a double-well potential. In this case, by propagating the final-state wave packet backward to the initial state, pulses that are optimized for forward... 

Specifically, as shown in Figure 4.4, Na2 is pumped from the ground electronic state to the 21IIf state in a two-photon process by an initial pulse. The wave packet propagates on this potential curve until an additional pulse carries it to the ionized state. The Franclc-Condon factors favor production of the Na2+ product if excitation is when the packet is at the inner turning point of the 2 llg curve, whereas the excited Na2 decays to the Na+ + Na + e product if excitation is at the outer turning point. Tire experimental results on the ratio of the Na+ and Na2+ signals, as a... 

S. Nanbu, M.S. Johnson, Analysis of the ultraviolet absorption cross sections of six isotopically substituted nitrous oxide species using 3D wave packet propagation, /. Phys. Chem. A 108 (41) (2004) 8905-8913. 

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