Fourier transform propagation method

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectmm [108, 109, 150, 244]. In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator [104-111, 245-248], which circumvent the cumbersome root-search problem in boundary-value-based semiclassical methods, have been successfully applied to a variety of systems (see, for example, Refs. 110, 111, 161, and 249 and references therein). The mapping procedure introduced in Section VI results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore it... 

In the calculations reported below, the MCTDH method has been employed for all calculations involving more than a single 2/i electronic state, i.e., involving PJT interactions. As a drawback, vibronic line spectra are not directly obtained from this (as with any wave-packet propagation) method. The spectral envelope is, however, easily obtained as a Fourier transform according to Ref. [26] ... 

This expansion is valid to second order with respect to St. This is a convenient and practical method for computing the propagation of a wave packet. The computation consists of multiplying X t)) by three exponential operators. In the first step, the wave packet at time t in the coordinate representation is simply multiplied by the first exponential operator, because this operator is also expressed in coordinate space. In the second step, the wave packet is transformed into momentum space by a fast Fourier transform. The result is then multiplied by the middle exponential function containing the kinetic energy operator. In the third step, the wave packet is transformed back into coordinate space and multiplied by the remaining exponential operator, which again contains the potential. 

Standard methods are used to propagate each Om in time. For the z and Z coordinates we make use of the fast fourier transform [99], and for the p coordinate we use the discrete Bessel transform [100]. The molecular component of asymptotic region at each time step, and projected onto the ro-vibrational eigenstates of the product molecule, for a wide range of incident energies included in the incident wave packet [82]. The results for all ra-components are summed to produce the total ER reaction cross section, a, and the internal state distributions. 

It is therefore natural to accept the GCLT, which establishes, in fact, that all the propagators with the asymptotic form 1/ E, fall in the same basin, given by Eq. (103). The anti-Fourier transform of p(k, n) does not admit in general any analytical expression. However, the anti-Fourier transform method applied to Eq. (103) [46] shows that... 

Using whatever propagation method, one has to evaluate the action of the Hamiltonian operator on the wavefunction P(r). This is normally carried out by expanding P(f) in a suitable basis set and then evaluates the operator action on basis functions. One can use the FFT (fast Fourier transform) techniques (7,14), discrete variable representation (DVR) (15,16) techniques, or simply calculate matrix elements of the operator in a given basis set. 

Fig. 1. The X/A spectrum of the butatriene radical cation, (a) Experimental results from Ref. 30. (b) 2-mode model from Ref. 10. (c) 18-mode model from Ref. 12. The model spectra are the Fourier Transform of the autocorrelation function calculated using the MCTDH wavepacket propagation method. A damping function of r = 55 fs has been used.

It is noted that in order to perform a spectral analj is of P (t, At) with a standard fast Fourier transform routine, one needs to assure that the polarization has decayed completely. In practice, this requirement results either in invoking a phenomenological damping oc of the polarization (which artificially broadens the spectrum) or in a large propagation time of the polarization (which is computationally expensive). To avoid this problem, it has been suggested to employ a filter-diagonalization method " for the spectral analysis of P t, At). ... 

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... 

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