Wavepacket techniques

Taken together, the use of the real part of the wavepacket and the mapping of the time-dependent Schrodinger equation lead to a very significant reduction of the computational work needed to accompfish the calculation of reactive cross sections using wavepacket techniques. 

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. 

So far we have exclusively discussed time-resolved absorption spectroscopy with visible femtosecond pulses. It has become recently feasible to perfomi time-resolved spectroscopy with femtosecond IR pulses. Flochstrasser and co-workers [M, 150. 151. 152. 153. 154. 155. 156 and 157] have worked out methods to employ IR pulses to monitor chemical reactions following electronic excitation by visible pump pulses these methods were applied in work on the light-initiated charge-transfer reactions that occur in the photosynthetic reaction centre [156. 157] and on the excited-state isomerization of tlie retinal pigment in bacteriorhodopsin [155]. Walker and co-workers [158] have recently used femtosecond IR spectroscopy to study vibrational dynamics associated with intramolecular charge transfer these studies are complementary to those perfomied by Barbara and co-workers [159. 160], in which ground-state RISRS wavepackets were monitored using a dynamic-absorption technique with visible pulses. 

To add non-adiabatic effects to semiclassical methods, it is necessary to allow the trajectories to sample the different surfaces in a way that simulates the population transfer between electronic states. This sampling is most commonly done by using surface hopping techniques or Ehrenfest dynamics. Recent reviews of these methods are found in [30-32]. Gaussian wavepacket methods have also been extended to include non-adiabatic effects [33,34]. Of particular interest here is the spawning method of Martinez, Ben-Nun, and Levine [35,36], which has been used already in a number of direct dynamics studies. 

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literature on interference effects, associated with Aharonov-Bohm and Berry phases [8-14]. In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techniques to achieve these are also anchored in the phases of the wave function components. This trend is manifest in such diverse areas as coherent or squeezed states [15,16], electron transport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23]. Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar methods [25],... 

Techniques similar to the those described in this section and in Ref. 133, but used within a time-independent framework, have been developed by Kouri and coworkers [188,189] and by Mandelshtam and Taylor [62,63]. Kroes and Neuhauser [65-68] have used the methods developed in these papers to perform time-independent wavepacket calculations using only real arithmetic. The iterative equation that lies at the heart of the real wavepacket method, Eq. (4.68), is in fact simply the Chebyshev recursion relationship [187]. This was realized by Guo, who developed similar techniques based on Chebyshev iterations [50,51]. 

Exciting new developments, not discussed in the review are the extension of time-dependent wavepacket reactive scattering theory to full dimensional four-atom systems [137,199-201], the adaptation of the codes to use the power of parallel computers [202], and the development of new computational techniques for acting with the Hamiltonian operator on the wavepacket [138]. 

We have employed this phase-sensitive pump-probe technique to further investigate the multiphoton ionization of Na2 with 618-nm femtosecond pulses as discussed in the previous paragraph and have observed the interference of the A E and 2 Tlg wavepackets created by the first pulse and those created by the second pulse in the Na2+ signal. The amplitude of the high-frequency oscillations in the Na2+ signal was obtained as a function of pump-probe delay by filtering the transient with the laser frequency. It is shown in Fig. 8 (top). Below the averaged Na2+ transient of Fig. 4 is... 

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. 

In this chapter Prof. Rice has advocated two techniques that should be useful for evaluations of optimal fields for laser control of chemical reactions (i) reduced space of eigenstates for representations of nuclear wavepackets and (ii) the use of effective reaction coordinates. Both techniques have already been used for efficient evaluations of reaction probabilities in model reactions. See, for example, Ref. 1 for the prediction of population inversion and Ref. 2 for the demonstration of rather strong deviations of chemical reactions from the reaction path, specifically in the case of hydrogen transfer reactions. 

These selective transitions (1), (7), and (9) may be achieved by proper optimization of the parameters eo and w, as described elsewhere [13, 18, 21]. Extensions to IR femtosecond/picosecond laser-pulse-induced dissociation or predissociation have been derived in Ref. 16, using either the direct or the indirect solutions of the Schrodinger equation (2) the latter requires extensions of the expansion (5) from bound to continuum states [16,31]. (The consistent derivation in Ref. 16 is based on S. Fliigge in Ref. 31). The same techniques can also be used for IR femtosecond/picosecond laser-pulse-induced isomerization as well as for more complex systems that are two dimensional, three dimensional, and so on, at the expense of increasing numerical efforts due to the higher dimensionality grid representations of the wavepackets f/(t) or the corresponding expansions (5) (see, e.g., Refs. 18, 20, and 21). 

The time-dependence of the wavepacket evolving on any potential surface can be numerically determined by using the split operator technique of Feit and Fleck [10-15]. A good introductory overview of the method is given in Ref. [12]. We will discuss a potential in two coordinates because this example is relevant to the experimental spectra. The time-dependent Schrodinger equation in two coordinates Qx and Qy is... 

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. 

As demonstrated for CS2, the pump-and-probe technique with CMI is a promising tool for investigating how a nuclear wavepacket evolves in real time. Considering that few-cycle laser pulses are now becoming available, we are now entering into a new era when it will be possible to see molecules in intense laser fields in real time with highest temporal resolution [34,35]. 

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... 

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