Hierarchy of models

The hierarchy of models is complemented by a variety of methods and tecluiiques. Mesoscopic models tliat incorporate some fluid-like packing (e.g., spring-bead models for polymer solutions) are investigated by Monte Carlo... 

Over the last decade, the improvements in our understanding of atmospheric processes have pushed the development of a hierarchy of models, which were designed to... 

In a recent survey [19] it was noted that a realistic model for catalytic oxidation reactions must include equations describing the evolution of at least two concentrations of surface substances and account for the slow variation in the properties of the catalyst surface (e.g. oxidation-reduction). For the synchronization of the dynamic behaviour for various surface domains, it is necessary to take into consideration changes in the concentrations of gas-phase substances and the temperature of the catalyst surface. It is evident that, in the hierarchy of modelling levels, such models must be constructed and tested immediately after kinetic models. On the one hand, the appearance of such models is associated with the experimental data on self-oscillations in reactors with noticeable concentration variations of the initial substances and products (e.g. ref. 74) on the other hand, there was a gap between the comprehensively examined non-isothermal models with simple kinetics and those for the complex heterogeneous catalytic reactions... 

In an attempt to understand the factors that determine the feedbacks from the global nature-society system of the cycles of carbon and other chemicals, we construct a hierarchy of model units to parameterize all the known physical and biogeochemical processes that are responsible for the transport of various substances. We substantiate these units by means of partial models which estimate the balance between relationships at the boundaries of different media. The correlations between biogeochemical cycles and the many activities of human society are the basic objectives of this book. 

The relevance of resonance-theoretic wave-functions naturally suggests at least one more derivative VB model, obtained by restriction of the covalent-space (Heisenberg) VB model to the subspace of (neighbor-paired) Kekule structures. In fact a whole hierarchy of models begins to emerge, as discussed a little more in the next section. 

A hierarchy of models can often be derived from a more detailed model under certain assumptions. This approach was discussed above in the case of deterministic, continuum models (see Fig. 3a). Such hierarchical models can be valuable in multiscale modeling. Let us just mention two cases. First, one could use different models from a hierarchy of models for different situations or length scales. This approach plays a key role in hybrid multiscale simulation discussed extensively below. Second, one could easily apply systems tasks to a simpler model to obtain an approximate solution that is then refined by employing a more sophisticated, accurate, and expensive model from the hierarchy. 

In this respect, chemistry does not differ from other sciences. Contemporary chemical research is organized around a hierarchy of models that aid its practitioners in their everyday quest for the understanding of natural phenomena. The building blocks of the language of chemistry, including the representations of molecules in terms of structural formulae [1], occupy the very bottom of this hierarchy. Various phenomenological models, such as reaction types and mechanisms, thermodynamics and chemical kinetics, etc. [2], come next. Quantum chemistry, which at present is the supreme theory of electronic structures of atoms and molecules, and thus of the entire realm of chemical phenomena, resides at the very top. 

A variety of VB models arise, with possibly the most natural hierarchy [22] indicated in fig. 1. The hierarchy of models occur in the column on the left and corresponding methods of solution are indicated on the right (though in some cases the methods have been little explored to date and are then identified with a "question mark") - the abbreviation Cl refers to "configuration interaction". The Pauling-Wheland... 

MODEL.LA. is a language with infinite extensibility of its vocabulary, enabled by a fixed set of she modeling hierarchies and a fixed set of 13 semantic relationships. In Section IV.C, we discussed the hierarchies of modeling subclasses emanating from the six basic modeling elements. What is far more important for the modeling power of MODEL.LA. is its... 

Rich hierarchies of modeling elements, which can be used to represent any conceivable quantity of interest in chemical engineering science. 

A. The Chemical Engineering Science Hierarchies of Modeling Elements... 

Fig. 19. Partial view of the variable hierarchy of modeling subclasses.

Fig. 21. Partial view of the hierarchy of modeling classes representing various types of equations m chemical engineering science.

With the above comments in mind, we first briefly discuss a hierarchy of models for VP (Section 2), then dwell on the quantum (Section 3), quasiclassical (Section 4) and classical (Section 5) theories of VP. After that, we compare different approaches (Section 6) and summarize our results on the quantum-classical correspondence for isolated resonances, which is the case for vibrational predissociation. 

This Hamiltonian ean be simplified without losing any interesting features of VP by using speeial eonditions pertinent to the VP event. The simplification is carried out in several steps thus ereating a hierarchy of models. 

Figure 2. Hierarchy of models in terms of their resolution in time and space.

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