Wavepacket calculations, time-dependent

Hankel, M., Balint-Kurti, G.G. and Gray, S.K. (2003) Sine wavepackets A new form of wavepacket for time-dependent quantum mechanical reactive scattering calculations Int. J. Quant. Chem. 92, 205-211. 

To uniquely associate the unusual behavior of the collision observables with the existence of a reactive resonance, it is necessary to theoretically characterize the quantum state that gives rise to the Lorentzian profile in the partial cross-sections. Using the method of spectral quantization (SQ), it is possible to extract a Seigert state wavefunction from time-dependent quantum wavepackets using the Fourier relation Eq. (21). The state obtained in this way for J = 0 is shown in Fig. 7 this state is localized in the collinear F — H — D arrangement with 3-quanta of excitation in the asymmetric stretch mode, and 0-quanta of excitation in the bend and symmetric stretch modes. If the state pictured in Fig. 7 is used as an initial (prepared) state in a wavepacket calculation, one observes pure... 

An overview of the time-dependent wavepacket propagation approach for four-atom reactions together with the construction of ab initio potential energy surfaces sufficiently accurate for quantum dynamics calculations has been presented. Today, we are able to perform the full-dimensional (six degrees-of-freedom) quantum dynamics calculations for four-atom reactions. With the most accurate YZCL2 surface for the benchmark four-atom reaction H2 + OH <-> H+H2O and its isotopic analogs, we were able to show the following ... 

The initial wavepacket, described in Section III.B is intrinsically complex (in the mathematical sense). Furthermore, the solution of the time-dependent Schrodinger equation [Eq. (4.23)] also involves an intrinsically complex time evolution operator, exp(—/Ht/ ). It therefore seems reasonable to assume that aU the numerical operations involved with generating and analyzing the time-dependent wavefunction will involve complex arithmetic. It therefore comes as a surprise to realize that this is in fact not the case and that nearly all aspects of the calculation can be performed using entirely real wavefunctions and real arithmetic. The theory of the real wavepacket method described in this section has been developed by S. K. Gray and the author [133]. 

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. 

These photofragmentation T matrix elements contain all the information about the photofragmentation dynamics. We will now discuss how they may be extracted from the time-dependent wavepacket calculations. 

Similar transient signals were obtained from time-dependent quantum mechanical calculations performed by Meier and Engel, which well reproduce the observed behavior [49]. They show that for different laser field strengths the electronic states involved in the multiphoton ionization (MPI) are differently populated in Rabi-type processes. In Fig. 13 the population in the neutral electronic states is calculated during interaction of the molecule with 60-fs pulses at 618 nm. For lower intensities the A state is preferentially populated by the pump pulse, and the A state wavepacket dominates the transient Na2+ signal. However, for the higher intensities used in the... 

When two or more normal coordinates are coupled, the wavepacket dynamics depends on all of the coupled coordinates simultaneously. Thus, the (t) s for each coordinate computed individually and Eq. (3) cannot be used. Instead, the multidimensional wavepacket must be calculated by using the time dependent Schrodinger equation. In this section we show how to calculate the wavepacket for two coupled coordinates. The computational method discussed here removes all of the restrictive assumptions used in deriving Eq. (4). Any potential, including numerical potentials, can be used. 

In practice one does not proceed as we did in the above derivation. Instead of calculating first all stationary wavefunctions and then constructing the wavepacket according to (4.3), one solves the time-dependent Schrodinger equation (4.1) with the initial condition (4.4) directly. Numerical propagation schemes will be discussed in the next section. Since 4 /(0) is real the autocorrelation function fulfills the symmetry relation... 

One of the virtues of the time-dependent approach is the ease with which it can be implemented. In this section we shall briefly outline several methods that are nowadays routinely employed for the propagation of wavepackets. Rather than being exhaustive we attempt to give a more general understanding of such calculations. 

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. 

Figure 12.9 depicts a comparison between classical trajectory results and exact close-coupling calculations for He--Cl2 and Ne- -Cl2, respectively. In both cases, the classical procedure reproduces the overall behavior of the final state distributions satisfactorily. Subtle details such as the weak undulations particularly for He are not reproduced, however. As shown by Gray and Wozny (1991), who treated the dissociation of van der Waals molecules in the time-dependent framework, the bimodality for He CI2 is the result of a quantum mechanical interference between two branches of the evolving wavepacket and therefore cannot be obtained in purely classical calculations. 

Fig. 13.3. (cont.) and (13.7). The wavefunctions are calculated using the empirical fit PES of Sorbie and Murrell (1975). The arrows indicate qualitatively the route of the evolving time-dependent wavepacket in the excited state. The dashed contours in (a) and (b) represent the A-state PES. Right-hand side The corresponding absorption spectra as functions of the energy in the excited electronic state. E = 0 corresponds to three ground-state atoms, H + O + H. Within this normalization the lowest eigenvalue of H20(A) is -9.500 eV. 

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