Time-dependent wavepacket propagation

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 vibronic coupling model Hamiltonian is well suited for a combination with the MCTDH method as it has the required product form (see Sect. 4.2.3). Usually, molecular systems affected by strong vibronic couplings have complicated spectra with very dense bands. Therefore, a detailed analysis of the spectrum in terms of individual vibronic states is in general impossible and one is more interested in the overall electronic band profiles. In this case, the use of a time-dependent fl-amework can be advantageous since the absorption profile can be obtained from the time-dependent wavepacket propagated over relatively short times, as exposed below. 

A method which has been used in several studies [155,157] involves a time-dependent wavepacket propagation in the two-dimensional space Z,r using a grid... 

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-dependent theoretical treatment of the electronic emission spectrum is very similar to that of the absorption spectrum because the two potential surfaces involved are the same. The principal difference is that the initial wavepacket starts on the upper (excited state) electronic surface and propagates on the ground electronic state surface. The overlap of the initial wavepacket with the time-dependent wavepacket is given by Eq. (4). The emission spectrum is given by (49)... 

The key to performing a wavepacket calculation is the propagation of the wavepacket forward in time so as to solve the time-dependent Schrodinger equation. In 1983, Kosloff proposed the Chebyshev expansion technique [5, 6, 7, 8] for evaluating the action of the time evolution operator on a wavepacket. This led to a huge advance in time-dependent wavepacket dynamics [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Several studies have compared different propagation methods [30, 31, 32] and these show that the Chebyshev expansion method is the most accurate. 

Other classical-path treatments have been formulated for atom-molecule [43,95-101] and molecule-molecule [102-107] collisions however, they were mostly based on internal-state expansions that become computationally impractical as the collision energy increases. Semiclassical calculations have also been implemented by means of time-dependent wavepackets [39], whose propagation can become expensive for motions with widely differing time scales. The following TCF-semiclassical approach encompasses very high densities of internal states as well as fast and slow motions. 

The wavepacket is propagated until a time where it is all scattered and is away from the interaction region. This time is short (typically 10-100 fs) for a direct reaction. Flowever, for some types of systems, e.g. for reactions with wells, the system can be trapped in resonances which are quasi-bound states (see section B3.4.7). There are eflScient ways to handle time-dependent scattering even with resonances, by propagating for a short time and then extracting the resonances and adding their contribution [69]. 

Our main concern in this section is with the actual propagation forward in time of the wavepacket. The standard ways of solving the time-dependent Schrodinger equation are the Chebyshev expansion method proposed and popularised by Kossloff [16,18,20,37 0] and the split-operator method of Feit and Fleck [19,163,164]. I will not discuss these methods here as they have been amply reviewed in the references just quoted. Comparative studies [17-19] show conclusively that the Chebyshev expansion method is the most accurate and stable but the split-operator method allows for explicit time dependence in the Hamiltonian operator and is often faster when ultimate accuracy is not required. All methods for solving the time propagation of the wavepacket require the repeated operation of the Hamiltonian operator on the wavepacket. It is this aspect of the propagation that I will discuss in this section. 

This equation is exact and constitutes an iterative equation equivalent to the time-dependent Schrodinger equation [185,186]. The iterative process itself does not involve the imaginary number i therefore, if h(f) and )( — x) were the real parts of the wavepacket, then (f + x) would also be real and would be the real part of the exact wavepacket at time (f + x). Thus, if )( ) is complex, we can use Eq. (4.68) to propagate the real part of 4>(f) forward in time without reference to the imaginary part. 

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. 

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