Methods of Searching for Optimal

Mathematical Methods of Searching for Optimal Catalytic Materials 

Many properties and conditions influence the capability of materials to serve as catalysts for a particular reaction. First, the composition of the material is important, identified by the chemical elements or compounds present, and by their respective mass or molar fractions in the material. Furthermore, the structure and texture of the material, which are determined mainly by the preparation method employed, play an important role. The composition and structure of the bulk and the surface are interconnected with further properties of the material, such as acidity, basicity, redox potential, and electronic conductivity these properties are usually denoted as descriptors (cf. Klanner, 2004 Klanner et al, 2004 Farrusseng et al, 2005). Finally, the catalytic performance of a material does not depend solely on the material itself, but also on the conditions to which that material is exposed in the reaction, at a particular temperature, partial pressures of the reactants, total pressure, and space velocity of the feed. The descriptors of the material and the reaction conditions together are called input variables in the sequel. 

In a catalytic experiment, a certain number of catalytic materials is tested, chosen from a very comprehensive set of potential materials. The number of materials to be examined is affected by the following aspects  

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Development of methods of searching for the optimal level of a characteristic thermodynamic function of the system C(xext). Interpretation of the proposed methods on the basis of a thermodynamic tree. 

Optimization should be viewed as a tool to aid in decision making. Its purpose is to aid in the selection of better values for the decisions that can be made by a person in solving a problem. To formulate an optimization problem, one must resolve three issues. First, one must have a representation of the artifact that can be used to determine how the artifac t performs in response to the decisions one makes. This representation may be a mathematical model or the artifact itself. Second, one must have a way to evaluate the performance—an objective function—which is used to compare alternative solutions. Third, one must have a method to search for the improvement. This section concentrates on the third issue, the methods one might use. The first two items are difficult ones, but discussing them at length is outside the scope of this sec tion. 

Prior to the advent of high-speed computers, methods of optimization were limited primarily to analytical methods, that is, methods of calculating a potential extremum were based on using the necessary conditions and analytical derivatives as well as values of the objective function. Modem computers have made possible iterative, or numerical, methods that search for an extremum by using function and sometimes derivative values of fix) at a sequence of trial points x1, x2,. 

While these polymerizations are generally more involved than chemical or electrochemical oxidative methods, well-defined monomers and highly selective transition metal catalysts yield polymers virtually free of (3-coupling defects. Hence, the ability to obtain structurally superior materials has greatly fueled the search for optimized condensation reactions. 

Optimization techniques may be classified as parametric statistical methods and nonparametric search methods. Parametric statistical methods, usually employed for optimization, are full factorial designs, half factorial designs, simplex designs, and Lagrangian multiple regression analysis [21]. Parametric methods are best suited for formula optimization in the early stages of product development. Constraint analysis, described previously, is used to simplify the testing protocol and the analysis of experimental results. 

The constrained optimization procedure, originally developed from the simplex method and first described by Box, is ideally suited to model refinement (.8). It is a search method that searches for the minimum of a multidimensional function within given intervals. It possesses all the advantages of search methods, among them that calculation of derivatives is not necessary, a test to assure the independence of variables can be omitted, and diverse variables can be easily included. These are exactly the requirements of model refinement where bond lengths, bond angles, torsion angles, and other parameters are used within experimentally defined limits. 

Now that structures could be optimized, some questions arose concerning the conformer searches. In version 3.1 of the SPARTAN software package, the conformer searching algorithm employed the method of Osawa for rings. While it is quite clear from theory how the search should proceed when applied to a monocyclic system, the implementation for bicyclic systems in unclear. 

J. Pillardy, L. Piela, Smoothing techniques of global optimization distance scaling method in searches for most stable Lennard-Jones atomic clusters, J. Comp. Chem. 19 (1998)... 

The semi-empirical molecular orbital calculation software MOPAC in the CAChe Work System for Windows ver. 6.01 (Fujitsu, Inc.) was used in all of calculations for optimization of geometry by the Eigenvector Following method, for search of potential energies of various geometries of intermediates by use of the program with Optimized map, for search of the reaction path from the reactants to the products via the transition state by calculation of the intrinsic reaction coordinate (IRC) [10]. 

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