Basic computational considerations

Applying computational techniques to chemical problems first requires a careful choice of the theoretical method. Basic knowledge of the capabilities and the drawbacks of the various methods is an absolute necessity. However, as no practical chemist can be expected to be well versed in the language and fine details of computational theory, we approach the subject by briefly (and by no means completely) reminding the reader of the underlying concepts of particular methods and of their often less well-documented limitations. 

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If wavefunctions are calculated using semiempirical methods that assume zero overlap between atomic orbitals (AOs) on different atomic centers, then a MuUiken population analysis [84, 85] can be applied to the calculated TD to yield transition monopoles distributed over each atomic center. Such an approach has proven to be effective [82] an advantage being that the interaction between distributed monopoles can be computed considerably faster then that between TDCs. At the same time, the basic topology of the donor-acceptor... 

The basic requirements for the Mephisto model was satisfactory accuracy, that means prediction of amplitude, position and phase relation between the various signals, and short computation times, typically a few minutes for the simulation of a whole Cscan, compatible with an intensive use. These a priori contradictory characteristics have been contented by means of appropriate approximations based on physical considerations. 

At Its most basic level separating the total strain of a structure into its components is a qualita tive exercise For example a computer drawn model of the eclipsed conformation of butane using ideal bond angles and bond distances (Figure 3 8) reveals that two pairs of hydrogens are separated by a distance of only 175 pm a value considerably smaller than the sum of their van der Waals radii (2 X 120 pm = 240 pm) Thus this conformation is destabilized not only by the torsional strain associ ated with its eclipsed bonds but also by van der Waals strain... 

Method of Continuity (Homotopy) In the case of n equations in n unknowns, when n is large, determining the approximate solution may involve considerable eftoid. In such a case the method of continuity is admirably suited for use on digital computers. It consists basically of the introduction of an extra variable into the n equations... 

Time available It takes considerable time to set up a simulation case. After ensuring that a basic configuration has run successfully, it is easy and quick to carry out additional computer simulations with changed boundary conditions. Experiments, on the other hand, are generally more time-consuming. 

Although the basic mechanisms are generally agreed on, the difficult part of the model development is to provide the model with the rate constants, physical properties and other model parameters needed for computation. For copolymerizations, there is only meager data available, particularly for cross-termination rate constants and Trommsdorff effects. In the development of our computer model, the considerable data available on relative homopolymerization rates of various monomers, relative propagation rates in copolymerization, and decomposition rates of many initiators were used. They were combined with various assumptions regarding Trommsdorff effects, cross termination constants and initiator efficiencies, to come up with a computer model flexible enough to treat quantitatively the polymerization processes of interest to us. 

The remainder of this text attempts to establish a rational framework within which many of these questions can be attacked. We will see that there is often considerable freedom of choice available in terms of the type of reactor and reaction conditions that will accomplish a given task. The development of an optimum processing scheme or even of an optimum reactor configuration and mode of operation requires a number of complex calculations that often involve iterative numerical calculations. Consequently machine computation is used extensively in industrial situations to simplify the optimization task. Nonetheless, we have deliberately chosen to present the concepts used in reactor design calculations in a framework that insofar as possible permits analytical solutions in order to divorce the basic concepts from the mass of detail associated with machine computation. 

Simplified mathematical models These models typically begin with the basic conservation equations of the first principle models but make simplifying assumptions (typically related to similarity theory) to reduce the problem to the solution of (simultaneous) ordinary differential equations. In the verification process, such models must also address the relevant physical phenomenon as well as be validated for the application being considered. Such models are typically easily solved on a computer with typically less user interaction than required for the solution of PDEs. Simplified mathematical models may also be used as screening tools to identify the most important release scenarios however, other modeling approaches should be considered only if they address and have been validated for the important aspects of the scenario under consideration. 

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