Vibrational methods reduced-dimensionality

For larger hydrogen-bonded systems, rigorous calculations are far more difficult to carry out, both from the point of view of obtaining full-dimensional potentials and the subsequent quantum vibrational calculations. Reduced dimensionality approaches are therefore often necessary and several chapters in this volume illustrate this approach. With increasing computational power, coupled with some new approaches, it is possible to treat modest sized H-bonded systems in full dimensionality. We have already briefly reviewed the approach we have developed for potentials for the vibrations we have primarily used the code Multimode (MM). The methods used in MM have been reviewed recently [24 and references therein, 25], and so we only give a very brief overview of the method here. 

This book describes the proceedings of a NATO Advanced Research Workshop held at CECAM, Orsay, France in June, 1983. The Workshop concentrated on a critical examination and discussion of the recent developments in the theory of chemical reaction dynamics, with particular emphasis on quantum theories. Several papers focus on exact theories for reactions. Exact calculations on three-dimensional reactions are very hard to perform, but the results are valuable in testing the accuracy of approximate theories which can be applied, with less expense, to a wider variety of reactions. Indeed, critical discussions of the merits and defects of approximate theories, such as sudden, distorted-wave, reduced dimensionality and transition-state methods, form a major part of the book. The theories developed for chemical reactions have found useful extensions into other areas of chemistry and physics. This is illustrated by papers describing topics such as photodissociation, electron-scattering, molecular vibrations and collision-induced dissociation. Furthermore, the important topic of how to treat potential energy surfaces in reaction dynamics calculations is also discussed. 

Another valuable use of accurate quantum dynamics calculations is testing the validity of classical simulations for predicting product-state distributions, and reduced-dimensionality studies of this issue are available for both Cl + H2 [67] and H + CI2 [104], In the present case extensive quasiclassical trajectory (( CT) calculations have been carried out for the full-dimensional Cl + D2 reaction by Aoiz and Bahares [105]. An example of how the QCT results compare to the accurate quantum ones is given in Fig. 4, which shows differential cross sections for Cl + D2(v=0J=1) —> DCl(v ) + D, where v and v are initial and final vibrational quantum number, respectively, j is initial rotational quantum number, and the results are summed over final rotational quantum number j. The comparison in Fig. 4 is for an initial relative translational energy of 10.1 kcal. The agreement is quite good. Notice, however, that the QCT method overestimates the amount of vibrationally excited product. 

Figure 20. The (So —> S2) absorption spectrum of pyrazine for reduced three- and four-dimensional models (left and middle panels) and for a complete 24-vibrational model (right panel). For the three- and four-dimensional models, the exact quantum mechanical results (full line) are obtained using the Fourier method [43,45]. For the 24-dimensional model (nearly converged), quantum mechanical results are obtained using version 8 of the MCTDH program [210]. For all three models, the calculations are done in the diabatic representation. In the multiple spawning calculations (dashed lines) the spawning threshold 0,o) is set to 0.05, the initial size of the basis set for the three-, four-, and 24-dimensional models is 20, 40, and 60, and the total number of basis functions is limited to 900 (i.e., regardless of the magnitude of the effective nonadiabatic coupling, we do not spawn new basis functions once the total number of basis functions reaches 900).

The TDAS is a methodology which reduces the dimension of the wavelet matrix due to applying the geometric mean to output from the wavelet transform. It is expected that the TDAS application will allow fault patterns to be identified better as shown in HALIM et al. (2008). However, the complexity involved in the vibration signals may impede this identification. This is one limitation of the TDAS method that needs to be pointed out. Additionally, this method requires a visual interpretation of the three-dimensional graph of the U(s,p), a hard task to be computationally automated. Besides, in some cases, vibration signals can be acquired from more than one source simultaneously. Therefore, the dimension of the vibration data can make use of the TDAS method difficult. 

Note we have reduced the coupling terms by the method of NF so that the action variables determined by the NF are constants of motion. This enabled us to discuss the effect of vibrational excitation by simply fixing the bath mode actions and looking at the change of the one-dimensional motion along the reaction mode. 

From the theoretical point of view, this relaxation process has been the subject of a large number of quantum dynamics investigations, based on reduced and full dimensional models. Farly works [13-17] reported three- and four-mode models and showed that a simple two-state four-dimensional model provides a qualitatively correct simulation of the UV absorption spectrum [17], These models were used to simulate various spectroscopic signals, including time-resolved transient absorption [18-20], and ionization [21] spectra, fluorescence [22] and resonance Raman spectra [23]. Worth et al. [24-27] performed accurate quantum dynamics simulations based on a model including the twenty-four vibrational modes of the molecule using the MCTDH method. These benchmark results have then been used to test various approximate methods for the simulation of non-adiabatic dynamics of molecular systems [28 0]. 

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