Semiempirical Philosophy

In the last chapter, the full formalism of Hartree-Fock theory was developed. While this theory is impressive as a physical and mathematical construct, it has several limitations in a practical sense. Particularly during the early days of computational chemistry, when computational power was minimal, carrying out HF calculations without any further approximations, even for small systems with small basis sets, was a challenging task. 

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The parametrization of a given implementation serves to determine optimum parameter values by calibrating against suitable reference data. The most widely used methods (see Section 21.2) adhere to the semiempirical philosophy and attempt to reproduce experiment. However, if reliable experimental reference data are not available, accurate theoretical data (e.g. from high-level ab initio calculations) are now generally considered acceptable as substitutes for experimental data. The quality of semiempirical results is strongly influenced by the effort put into the parametrization. 

Originally, there have been two basic strategies for parametrization. Approximate MO methods aim at reproducing ab initio MO calculations with the same minimal basis set (MBS), whereas semiempirical MO methods attempt to reproduce experimental data. Nowadays the limitations of MBS ab initio calculations are well known and the predominant feeling is that approximate MO methods would not be useful enough in practice even if they would exactly mimic MBS ab initio calculations. Hence with the exception of PRDDO [18], current parametrizations usually adhere to the semiempirical philosophy and employ experimental reference data (or possibly, accurate high-level theoretical predictions as substitutes for experimental data see Section III.E). 

Historically, two philosophies began to emerge at this stage with respect to how best to make further progress. The first philosophy might be summed up as follows The HF equations are very powerful but still, after all, chemically flawed. Thus, other approximations that may be introduced to simplify their solution, and possibly at the same time improve their accuracy (by some sort of parameterization to reproduce key experimental quantities), are well justified. Many computational chemists continue to be guided by this philosophy today, and it underlies the motivation for so-called semiempirical MO theories, which are discussed in detail in the next chapter. 

SRP, a term first coined by Rossi and Truhlar (1995), stands for specific reaction (or range) parameters . An SRP model is one where the standard parameters of a semiempirical model are adjusted so as to foster better performance on a particular problem or class of problems. In a sense, the SRP concept represents completion of a full circle in the philosophy of semiempirical modeling. It tacitly recognizes the generally robust character of some underlying semiempirical model, and proceeds from there to optimize that model for a particular system of interest. In application, then, SRP models are similar to the very first semiempirical models, which also tended to be developed on an ad hoc, problem-specific basis. The difference, however, is that the early models typically were developed essentially from scratch, while SRP models may be viewed as perturbations of more general models. 

Figure 13.5 An application of a hybrid MO/MO philosophy to the indicated RNA trimer proceeds using correlated levels of electronic structure theory for various tautomers and protonation states of the central base pair, this parr then representing the small system in the MO/MO analog of Eq. (13.6), and semiempirical theory for both die small system and the frozen-geometry larger system... 

In conclusion, we share the philosophy H. F. Schaefer III expressed in 1979 (which we believe is still valid in 1986, and very likely to be for the foreseeable future) We have been convinced for about five years that ab initio electronic structure calculations should not even attempt (except for the very simplest systems) to predict the entire potential energy surface . Since the success of a semiempirical method stems from the judicious combination of theory and experiment, we present a brief survey of the main theoretical methods in the remainder of this section. 

Jeffry L. Ramsey holds a B.A. and an M.S. in chemistry and a Ph.D. in the conceptual foundations of science from the University of Chicago. He was assistant professor of philosophy at Oregon State University, Corvallis, and is now at Smith College. His research focuses on how scientists view questions about theory construction, explanation, and reduction when they are faced with problems that are insoluble in practice or in principle (e.g., in the area of semiempirical models). He is also interested in questions of conceptual analysis as they arise in the chemical sciences (e.g., the case of shape). His essays have appeared in Philosophy of Science, Studies in History and Philosophy of Science, and Synthese, among other journals. 

A straight forward application of approximation IV to calculate W (r) maps is quite exacting, because the calculation of the potential contribution due to the couple distributions xt li 1S time consuming when directly performed on the Slater functions. This fact clashes with the basic philosophy of semiempirical methods, which is to sacrifice some reliability to speed up the calculations. It has been shown40) that expansion of each Slater-type orbital into three Gaussian functions (3G expansion41)) gives a substantial improvement of the computational times of W (r), without an appreciable reduction in the quality of the results. 

Within the realm of ab initio methods one should distinguish two different approaches. In the calibrated approach, favored by Pople and coworkers, the full exaa equations of the ab initio method are used without approximation. The basis set is fixed in a semiempirical way, however, by calibrating calculations on a variety of molecules. The error in any new application of the method is estimated based on the average error obtained, compared with experimental data, on the calibrating molecules. This is different in philosophy from the converged approach favored by chemical physicists interested in small molecules. In the latter approach, a sequence of calculations with improving basis sets is done on one molecule until convergence is reached. The error in the calculation is estimated from the sensitivity of the result to further refinements in the basis set. Clearly the calibrated method is the only one that is practical for routine use in computational chemistry. Con-... 

A question of philosophy arises concerning the molecular properties to be predicted by semiempirical treatments. On the one hand, the molecular orbital NDO methods were designed to mimic minimum basis set ab initio self-consistent field (SCF) property calculations parameters were chosen accordingly. Although internally consistent, this procedure is limited in accuracy by the minimum basis ab initio SCF results themselves. An alternative approach is to fit, and predia, measured physical properties. This is not internally consistent because experimental values cannot, in principle, be obtained from self-consistent field molecular orbital theory, regardless of basis set, because of the lack of elearon correlation. Furthermore, experimental values of properties obtained at room temperature cannot be equated to those calculated for 0° K. Nonetheless, the relatively high level of accuracy that can be achieved makes such an approach useful, and that is why it has been pursued by Dewar and co-workers in their series of M(odified)NDO methods. 

There is a great variety of approaches to fuel cell performance modeling. The simplest approach used in system simulations deals with the semiempirical polarization curves of the cell or stack under investigation. Such curves are obtained by fitting a simple analytical model equation to measured data. This philosophy is very useful in the optimization of FC systems with numerous peripheral components (blowers. 

The parameterization process allows assignment of specific values to quantities not directly available from experiment and difficult to estimate accurately a priori. As important as the theoretical model in semiempirical methodology is, often the philosophy of parameterization that was used to derive the parameters for each element from the experimental data is at least as important. This is certainly the case with AMI and the closely related PM3 method. These two methods employ the same theoretical model, but differ only by the approach taken in parameterization. This resulted in two very different chemical models. Parameterizations may be based on either experimental data or the results of more complete HF calculations. AMI, along with most other currently used methods, depended on experimental data. This has significant advantages. First, any theoretical inadequacy of higher level ab initio methods are circumvented. Second, relaxation of the parameters in such a way as to reproduce experiment handles in one step a multitude of chemical factors not directly includable or includable only at vast expense (i.e., dynamic electron correlation) in the model. The procedure and philosophy used in AMI and the other Dewar-style methods (except PM3) will be discussed below (see also Parameterization of Semiempirical MO Methods). 

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