Propagation of wave packets

Propagation of Wave Packet with a Modified Split-operator Scheme... 

The cosine iterative equation [8] is the ancestor of the RWP method. Consider propagating a wave packet at time t forward in time to I + r. 

Equation (4) is a three-term recursion for propagating a wave packet, and, assuming one starts out with some 4>(0) and (r) consistent with Eq. (1), then the iterations of Eq. (4) will generate the correct wave packet. The difficulty, of course, is that the action of the cosine operator in Eq. (4) is of the same difficulty as evaluating the action of the exponential operator in Eq. (1), requiring many evaluations of H on the current wave packet. Gray [8], for example, employed a short iterative Lanczos method [9] to evaluate the cosine operator. However, there is a numerical simplification if the representation of H is real. In this case, if we decompose the wave packet into real and imaginary parts. 

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-... 

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. 

Z.G. Sun, S.Y. Lee, H. Guo, D.H. Zhang, Comparison of second-order split operator and Chebyshev propagator in wave packet based state-to-state reactive scattering calculations, J. Chem. Phys. 130 (2009) 174102. 

A very effective way to apply MCTDSCF methods to multidimensional systems is the use of propagated Gaussion wave packets for the majority of modes [18,44,45,46,15]. 

To calculate numerically the quantum dynamics of the various cations in time-dependent domain, we shall use the multiconfiguration time-dependent Hartree method (MCTDH) [79-82, 113, 114]. This method for propagating multidimensional wave packets is one of the most powerful techniques currently available. For an overview of the capabilities and applications of the MCTDH method we refer to a recent book [114]. Additional insight into the vibronic dynamics can be achieved by performing time-independent calculations. To this end Lanczos algorithm [115,116] is a very suitable algorithm for our purposes because of the structural sparsity of the Hamiltonian secular matrix and the matrix-vector multiplication routine is very efficient to implement [6]. 

Determination of the time-dependent nuclear wave function now becomes a straightforward exercise in the locally quadratic theory. We need to propagate Gaussian wave packets of the form... 

In order to illustrate the complexity of excited states reactivity in transition metal complexes two selected examples are reported in the next section dedicated to the ab initio (CASSCF/MR-CI or MS-CASPT2) study of the photodissociation of M(R)(CO)3(H-DAB) (M=Mn, R=H M=Re, R=H, Ethyl) complexes. Despite the apparent complexity and richness of the electronic spectroscopy, invaluable information regarding the photodissociation dynamics can be obtained on the basis of wave packet propagations on selected 1-Dim or 2-Dim cuts in the PES, restricting the dimensionality to the bonds broken upon visible irradiation (Metal-CO or Metal-R). The importance of the intersystem crossing processes in the photoreactivity of this class of molecules will be illustrated by the theoretical study of the rhenium compound. 

An illustration of wave packet propagation on two-coupled electronic states in diabatic representation. Nonadiabatic transition occurs at the intersection region of the two potential energy surfaces, resulting in the observed two portions of the wave packet (initiated on state 1) on the two electronic states, with the larger portion remaining on the initiated state. 

State-to-state Analysis of the Propagated Real Wave Packet... 

In propagation, the wave packet is expanded in terms of a body-fixed translational-vibrational-rotational basis functions as follows ... 

Final analysis is carried out using the time-independent part of the propagated quantum wave packet, obtaining through the half-Fourier transformation ... 

Inspection of Figure 1.11a displaying the time-dependent wave function with starting point [I -> Ily] confirms the above scenario. After starting its propagation, the wave packet separates into two parts in a rather short time. One part... 

Concerning the observation of wave packet propagation phenomena, the Ka molecule is another promising candidate. Theoretical calculations [82] predicted an electronic state comparable to the Naa B state at about 800 nm. Different, highly sensitive methods, such as MPI and depletion spectroscopy, were applied but failed [83]. However, as presented in Sect. 3.2.5, with femtosecond real-time spectroscopy it is possible to observe both the vibrational and the dissociation dynamics of this system. This, once again, is an example of the complementarity of cw and femtosecond spectroscopy. 

The basic technique used to propagate the wave packet in the spatial domain is the fast Fourier transform method [287, 288, 299, 300]. The time-dependent Schrodinger equation is solved numerically, employing the second-order differencing approach [299, 301]. In this approach the wave function Sit t = t St is constructed recursively from the wave functions at t and t" = t — St. The operator including the potential energy is applied in phase space and that of the kinetic energy in momentum space. Therefore, for each... 

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. 

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. 

Fig. 3.51. Snapshots of wave packet propagation in Nas B for a 120 fs excitation at P = 16 021cm" demonstrating the dephasing of the 310 fs breathing mode (taken from [382])...

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