Hierarchy, of computational methods

In Chapter 2.1 a new section on computer-aided techniques has been introduced. This gives an overview of the hierarchy of computational methods available to heterocyclic chemists and a guide to some of the terminology used. This is followed by a glossary of general terms used throughout the structure chapters and an indication of sections where examples can be found. 

The second large class of computational methods that is most useful for predicting reactivity of zeolites is based on the quantum mechanical description of a chemical system. Quantum mechanics represents the highest level in the hierarchy of computational methods. By solving the electronic structure problem they provide us the energy and wave function of a system, from which all properties of all atoms in it can be derived. In practice, however, the exact solution of the electronic structure problem cannot be obtained for any realistic system. 

As with the QM level, the MM level utilized can also affect decisively the outcome of the calculation. However, the choice of an appropriate MM level has some complications that were absent in the case of the choice of the QM level described earlier. The first of these complications is that there is no clear hierarchy of MM methods. In contrast with QM methods, all MM methods have similar computational costs, and the differences between them are concentrated mostly in the type of system for the which they are parameterized. 

A hierarchy of computational models is available to simulate dispersed gas-liquid-solid flows in three-phase slurry and fluidized bed reactors [84] continuum (Euler-Euler) method, discrete particle/bubble (Euler-Lagrange) method, or front tracking/capturing methods. While every method has its own... 

Wild DJ, Blankley J. Comparison of 2D fingerprint types and hierarchy level selection methods for structural grouping using Ward s clustering. J Chem Inf Comput Sci 2000 40 155-62. 

Similarly to catalysis, the properties of these composite materials are also determined by a hierarchy of structures on very different length/time scales. Therefore, linking mesoscale molecular models and continuum descriptions is relevant for their understanding and optimization. Together with advanced synthesis methods and functional testing, it is thus necessary also to develop new improved computational methods to provide an understanding of materials properties and to assist in the development of new functional materials. 

The A-representability constraints presented in this chapter can also be applied to computational methods based on the variational optimization of the reduced density matrix subject to necessary conditions for A-representability. Because of their hierarchical structure, the (g, R) conditions are also directly applicable to computational approaches based on the contracted Schrodinger equation. For example, consider the (2, 4) contracted Schrodinger equation. Requiring that the reconstmcted 4-matrix in the (2, 4) contracted Schrodinger equation satisfies the (4, 4) conditions is sufficient to ensure that the 2-matrix satisfies the rather stringent (2, 4) conditions. Conversely, if the 2-matrix does not satisfy the (2, 4) conditions, then it is impossible to construct a 4-matrix that is consistent with this 2-matrix and also satisfies the (4, 4) conditions. It seems that the (g, R) conditions provide important constraints for maintaining consistency at different levels of the contracted Schrodinger equation hierarchy. 

A second fundamental classification of quantum chemistry calculations can be made according to the quantity that is being calculated. Our introduction to DFT in the previous sections has emphasized that in DFT the aim is to compute the electron density, not the electron wave function. There are many methods, however, where the object of the calculation is to compute the full electron wave function. These wave-function-based methods hold a crucial advantage over DFT calculations in that there is a well-defined hierarchy of methods that, given infinite computer time, can converge to the exact solution of the Schrodinger equation. We cannot do justice to the breadth of this field in just a few paragraphs, but several excellent introductory texts are available... 

Terzopoulos (1986) has shown how to speed up the computation using multi-grid relaxation methods. He uses a hierarchy of multi-resolution grids where data propagates upward as well as downward through the hierarchy. The use of a multi-resolution pyramid allows information to propagate faster over larger distances. 

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