Wavepacket propagation numerical calculation

As long as the photodissociation reaction is fairly direct, the time-dependent formulation is fruitful and provides insight into both the process itself and the relationship of the final-state distributions to the absorption spectrum features. Moreover, solution of the time-dependent Schrodinger equation is feasible for these short-time evolutions, and total and partial cross sections may be calculated numerically.5 Finally, in those cases where the wavepacket remains well localized during the entire photodissociation process, a semi-classical gaussian wavepacket propagation will yield accurate results for the various physical quantities of interest.6... 

As described in Sec.3.1, standard wavepacket propagation schemes can be employed for the evaluation of fiux correlation functions. These methods employ multi-dimensional grids or basis sets to represent the wavefunction. Thus, the numerical effort of these schemes increases exponentially with the number of degrees of freedom. Given the computational resources presently available, only systems with up to four atoms can be treated accurately. The extension of numerically exact calculations towards larger systems therefore requires other schemes for the solution of the time-dependent or time-independent Schrodinger equation. 

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... 

To examine the quality of the PSANB paths and transition probabilities, we have carried out the corresponding quantum wavepacket calculations. The numerical method is essentially the same as already described above. The right panel of Fig. (6.6) shows the time-propagation of the square modulus of the nuclear wavepackets, that is, a simple superposition of Ixi [R,t) and x2 (, i) for k = 32.1. At each time t, we have plotted points the height of xi R, and x2 of predetermined values... 

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