Fourier transform time-dependent wavepacket

Raman Spectroscopy The time-dependent picture of Raman spectroscopy is similar to that of electronic spectroscopy (6). Again the initial wavepacket propagates on the upper excited electronic state potential surface. However, the quantity of interest is the overlap of the time-dependent wavepacket with the final Raman state 4>f, i.e. < f (t)>. Here iff corresponds to the vibrational wavefunction with one quantum of excitation. The Raman scattering amplitude in the frequency domain is the half Fourier transform of the overlap in the frequency domain,... 

The time-independent total dissociation wavefunction tot Ef) is the Fourier transform of the time-dependent wavepacket y (t). 

The time-independent and the time-dependent approaches are completely equivalent. Equation (4.11) documents this correspondence in the clearest way the time-dependent wavepacket f(t), which contains the stationary states for all energies, and the time-independent wavefunction tot(Ef), which embraces the entire history of the fragmentation process, are related to each other by a Fourier transformation between the time and the energy domains,... 

Resonance Raman (RR) spectra yield valuable additional information on the excited-state dynamics and allow for a further test of the theoretical model. RR excitation profiles and spectra can be computed via a straightforward generalization of the Lanczos algorithm to allow for several initial states (namely, all those states where RR transitions are sought for). This amounts to a block-Lanczos or band-Lanczos procedure, both of which are well established in the literature. Alternatively, RR amplitudes may be obtained through Fourier transformation of appropriate cross-correlation functions which can be directly extracted from the time-dependent wavepacket 4 (t). di4... 

The matrix element for resonance Raman scattering thus is proportional to a half-Fourier transform of flie overlap of the final vibrational wavefunction (Xh(g)) with the time-dependent wavepacket X(t) created by exciting the molecule with white light in the ground state. See [6] and [8] for more complete proofs of this relationship, and [10] for a review of some of its extensions and applications. The resonance Raman excitation spectrum is proportional to as explained above. 

The time-dependent wavepacket method for state to state reactive scattering is based on the following remarkable expression for the reactive scattering matrix element 5cu,au( ) involving the Fourier transform of a correlation function Ccv. aaff) between an initial reactant wavepacket Xw and a... 

The time-dependent quantity in the integrand of Eq. (4.24), (,(r, R, f = 0) ,(r, R, f)), is called the autocorrelation function. It is the integral over all space of the product of the initial wavepacket with the wavepacket at time t. Rama Krishna and Coalson [86] have shown that the Fourier transform over time in Eq. (4.24) can be replaced by twice the half-Fourier transform where the time integral mns from f = 0 to f = oo. Using this result we obtain the final expression ... 

Time-independent and time-dependent theories are not really separate disciplines. This should be clear from the work of Kouri [188,189] and Althorpe [136,158], who use time-independent wavepacket techniques. These are easily derived from the more natural time-dependent versions by Fourier transforming the propagator over time. This is equivalent to transforming from the time domain to the energy domain at the beginning rather than the end of the calculation. 

On the other hand, additional spectroscopic information can be obtained by making use of this technique The Fourier transform of the frequency-filtered transient (inset in Fig. 8) shows that the time-dependent modulations occur with the vibrational frequencies of the A E and the 2 IIg state. In the averaged Na2+ transient there was only a vanishingly small contribution from the 2 IIg state, because in the absence of interference at the inner turning point ionization out of the 2 IIg state is independent of intemuclear distance, and this wavepacket motion was more difficult to detect. In addition, by filtering the Na2+ signal obtained for a slowly varying pump-probe delay with different multiples of the laser frequency, excitation processes of different order may be resolved. This application is, however, outside the scope of this contribution and will be published elsewhere. 

However, the total dissociation wavefunction is useful in order to visualize the overall dissociation path in the upper electronic state as illustrated in Figure 2.3(a) for the two-dimensional model system. The variation of the center of the wavefunction with r intriguingly illustrates the substantial vibrational excitation of the product in this case. As we will demonstrate in Chapter 5, I tot closely resembles a swarm of classical trajectories launched in the vicinity of the ground-state equilibrium. Furthermore, we will prove in Chapter 4 that the total dissociation function is the Fourier transform of the evolving wavepacket in the time-dependent formulation of photodissociation. The evolving wavepacket, the swarm of classical trajectories, and the total dissociation wavefunction all lead to the same general picture of the dissociation process. 

In the time-independent approach one has to calculate all partial cross sections before the total cross section can be evaluated. The partial photodissociation cross sections contain all the desired information and the total cross section can be considered as a less interesting by-product. In the time-dependent approach, on the other hand, one usually first calculates the absorption spectrum by means of the Fourier transformation of the autocorrelation function. The final state distributions for any energy are, in principle, contained in the wavepacket and can be extracted if desired. The time-independent theory favors the state-resolved partial cross sections whereas the time-dependent theory emphasizes the spectrum, i.e., the total absorption cross section. If the spectrum is the main observable, the time-dependent technique is certainly the method of choice. 

In the present work, we monitor the laser-driven dynamics designed by the present formulation by numerically solving the time-dependent Schrodinger (5.2). It is solved by the split operator method [52] with the fast Fourier transform technique [53]. In order to prevent artificial reflections of the wavepacket at the edges, a negative imaginary absorption potential is placed at the ends of the grid [54]. The envelope of the pulses employed is taken as... 

This last equation shows that the absorption spectrum can be computed from the Fourier transform of the autocorrelation function C(t) = ( l (O)l l (f)), which measures the time-dependent overlap of the evolving wavepacket with the initial wavepacket. Assuming an hemitian Hamiltonian operator, one may write... 

One fruitful application this approach is to relate the absorption spectrum to the spatial overlap of the wavepacket initially created by the excitation (X(0)) with the moving wavepacket at later times (X(t)). HeUer and coworkers [132-137] showed that the spectrum can be obtained by a Fourier transform of the time-dependent overlap integral (X(0)LT(0) ... 

One assumes here that the molecular Hamiltonian H is the same for all electronic states j, still the notation Hj with index j is useful to identify the electronic shell in which the wavepacket evolves. One can then also apply directly the general theory to nuclear degrees of freedom with the nuclear Hamiltonian depending on the electronic state j. A corresponding time-dependent representation for the RXS cross section (3.93) can be obtained by a Fourier transform of the spectral function,... 

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