Scattering calculations dynamics

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

Based on this physical view of the reaction dynamics, a very broad class of models can be constructed that yield qualitatively similar oscillations of the reaction probabilities. As shown in Fig. 40(b), a model based on Eckart barriers and constant non-adiabatic coupling to mimic H + D2, yields out-of-phase oscillations in Pr(0,0 — 0,j E) analogous to those observed in the full quantum scattering calculation. Note, however, that if the recoupling in the exit-channel is omitted (as shown in Fig. 40(b) with dashed lines) then oscillations disappear and Pr exhibits simple steps at the QBS energies. As the occurrence of the oscillation is quite insensitive to the details of the model, the interference of pathways through the network of QBS seems to provide a robust mechanism for the oscillating reaction probabilities. 

Discrete Fourier transform (DFT), non-adiabatic coupling, Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 153-155 Discrete variable representation (DVR) direct molecular dynamics, nuclear motion Schrodinger equation, 364-373 non-adiabatic coupling, quantum dressed classical mechanics, 177-183 formulation, 181-183... 

Electron dynamic scattering must be considered for the interpretation of experimental diffraction intensities because of the strong electron interaction with matter for a crystal of more than 10 nm thick. For a perfect crystal with a relatively small unit cell, the Bloch wave method is the preferred way to calculate dynamic electron diffraction intensities and exit-wave functions because of its flexibility and accuracy. The multi-slice method or other similar methods are best in case of diffraction from crystals containing defects. A recent description of the multislice method can be found in [8]. 

The theory of molecular scattering has now been developed to the point that scattering calculations can be made with an accuracy sufficient for comparison with current experiments. Thus any discrepancy between theory and experiment should be traced to an inadequate knowledge of the interaction potentials, or to experimental errors, rather than to approximations in the collision dynamics. This tighter coupling of theory and experiment should permit a much more fruitful utilization of the results of molecular beam scattering. 

The development of theory for reliable calculations of chemical dynamics has two components the construction of accurate, ab initio, multidimensional potential energy surfaces (PESs) and the performance of reactive scattering calculations, either by time-independent (TI) or time-dependent (TD) methods, on these surfaces. Accurate TI quantum methods for describing atom-diatom reactions, in particular for the benchmark H + H2 reaction, have been achieved since 1975.[1,2,3] Many exact and approximate theories have been tested with the H + H2 reaction.[4,5]... 

The first block, INTERACTION, is devoted to the calculation of electronic energies determining the potential energy surface (PES) on which the nuclear morion takes place. The second l)lock, DYNAMICS, is devoted to the integration of the scattering equations to determine the outcome of the molecular process. The third block, OBSER WBLES. is devoted to the reconstruction of the ol)serr al)le properties of the beam from the calculated dynamical quantities. All these blocks reejuiro not only different skills and expertise but also specialized computer software and hardware. 

For all but the very smallest systems, (such as HeH and even there it is very expensive), it is not possible in practice to calculate the full potential surface, with a grid fine enough that it can be directly used for solving the (nuclear) dynamical problem in Van der Waals molecules (or for scattering calculations). Moreover, such a numerical potential would not be convenient for most purposes. Therefore, one usually represents the potential by some analytical form, for instance, a truncated spherical expansion (1) or another type of model potential (cf. sect. 2). The parameters in this model potential can be obtained by fitting the ab initio results for a limited set of intermolecular distances and molecular orientations. Since we have encountered some difficulties in this fitting procedure which we expect to be typical, we shall describe our experience with the ( 2114)2 and (N2)2 cases in some detail. At the same time, we use the opportunity to make a few comments about the convergence of the spherical expansion used for (Njjj and about the validity of the atom-atom model potential applied to both ( 2114)2 and (Njjx. 

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