Local vs. Global

In their now classic monograph [1], Wooodward and Hoffmann concentrate on three basic types of no mechanism reaction Electrocyclic reactions -notably polyene cyclizations, cycloadditions, and sigmatropic rearrangements. These three reaction types will be taken up in this and the next two chapters from the viewpoint of Orbital Correspondence Analysis in Maximum Symmetry (OCAMS) [2, 3, 4], the formalism of which follows naturally from that developed in Chapter 4. The similarities to the original WH-LHA approach [5, 6], and the points at which OCAMS departs from it, will be illustrated. In addition, a few related concepts, such as allowedness and forbiddenness , global vs. local symmetry, and concertedness and synchronicity , will be taken up where appropriate. 

FIGURE 18.2 Prediction of percentage of wheat in validation set samples using GLOBAL vs. LOCAL equations. 

Thomas L G (2004). Are we all global now Local vs. foreign sources of corporate competence the case of the Japanese pharmaceutical industry. Strut. Manag. J. 25 865-886. 

This representation is referred to the open representation since the electrons and spin number can vary freely over a series of molecules at a well-defined chemical potential. In analogy to the previously written perturbation expansions, one can now perform a Taylor series expansion of H p.j, is,v,vs around changes in these four variables. In this expansion, the global and local softness indices will emerge in a natural way. These global softnessess can again be expressed in a matrix-vector notation as... 

Table 4.2 classifies the catalytic reactions of Table 4.1 on the basis of reaction type and shows their global r vs O behaviour. In Chapter 6 we will provide simple rules which enable one to rationalize and predict the behaviour of any given catalytic reaction both at the local and at the global level. Table 4.2, which classifies the 58 catalytic reactions of Table 4.1 on the basis of reaction type, also provides two additional pieces of information for each reaction ... 

It is a recent discovery, presented in Chapter 6, that behind the apparent chaos of Table 4.2 there are some simple regularities which enable one to predict the r vs local and global behaviour on the basis of the open-circuit... 

In sections 4.5.5 and 4.5.6 we have seen how the catalyst potential and work function affect the rates of catalytic reactions. We discussed in particular the r vs O dependence first at the local level (i.e. for small, e.g. 0.1-0.2 eV variations in eUwRand O) and then at the global level (i.e. for eUWRand variations as wide as the experimentally available eUWR range, typically 1.5-2 eV). 

The crucial task remains of examining to what extent it can also describe the effect of promotion, electrochemical or classical, on catalytic reaction kinetics. More specifically we will examine to what extent it can predict the four main types of global r vs O dependence and all the associated local and global electrochemical and chemical promotional rules. 

The macroscopic property of interest, e.g., heat of vaporization, is represented in terms of some subset of the computed quantities on the right side of Eq. (3.7). The latter are measures of various aspects of a molecule s interactive behavior, with all but surface area being defined in terms of the electrostatic potential computed on the molecular surface. Vs max and Fs min, the most positive and most negative values of V(r) on the surface, are site-specific they indicate the tendencies and most favorable locations for nucleophilic and electrophilic interactions. In contrast, II, a ot and v are statistically-based global quantities, which are defined in terms of the entire molecular surface. II is a measure of local polarity, °fot indicates the degree of variability of the potential on the surface, and v is a measure of the electrostatic balance between the positive and negative regions of V(r) (Murray et al. 1994 Murray and Politzer 1994). 

Q2 How can we cope with the highly nonlinear model with respect to the quality of its solution (/.e., local vs. global) ... 

In a mixture-of-experts model (Jacobs et al., 1991), different expert networks were assigned to tackle sub-tasks of training cases, and an extra gating network was used to decide which of the experts should determine the output. The model discovered a suitable decomposition of the input space as part of the learning process. Later the model was further extended (Jordan Jacobs, 1994) into a hierarchical system with a tree structure. In molecular applications, cascaded networks-where outputs of some networks become the inputs of others-were used to improve performance (Rost Sander, 1994). Multiple neural network modules may run in parallel in order to scale up the system (Wu et al., 1995). More than one network can also be used to extract different (e.g., local vs. global) features (Mahadevan Ghosh, 1994). 

With two parameters, we can plot contour lines of z on a graph of x vs. xi and hence get a visual representation of the behavior of z. For example. Figure 1.13 shows a schematic of a contour plot for a function that exhibits a local minimum of <30 at about (4,13) and a global minimum of <10 at about (15,19). Contour plots are useful... 

Balaban, A.T. (1994a). Local vs. Global (i.e. Atomic versus Molecular) Numerical Modeling of Molecular Graphs. J.Chem.Inf.Comput.ScL, 34,398-402. 

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