Natural internal coordinate method

Kremer s proposal, like those of Giiell and Fischbaum, stands midway between patents and grants.24 One of the objections raised by The Economist to Kremer s proposal is that the government would end up paying more than the consumer surplus and the welfare loss (triangles ABC and pp1 A in Figure 2.1). In the face of this situation, the new method would be clearly detrimental because it would force a tax increase to pay for the patent. Furthermore, the international nature of patents would require the existence of some kind of international coordination, not an easy task to accomplish. We are dealing, therefore, with a theoretical proposal that needs further contributions. 

As introduced in the previous section, class and sectional methods are based on a discretization of the internal coordinate so that the GPBE becomes a set of macroscopic balances in state space. Indeed, the fineness of the discretization will be dictated by the accuracy needed in the approximation of the integrals and derivative terms appearing in the GPBE. As has already been anticipated, the methods differ according to the number of internal coordinates used in the description and depend on the nature of the internal coordinates. Therefore, in what follows, we will discuss separately the univariate, bivariate, and multivariate PBE, and the use of these methods for the solution of the KE. 

In developing an ab initio PES for a chemical reaction in terms of a single reaction path potential, we first located the stationary points on the PES, then followed the reaction path by the steepest descent or distinguished coordinate methods, then chose coordinates, natural collision coordinates or internals, and finally expressed the PES as a series expansion about the... 

In the adiabatic bend approximation (ABA) for the same reaction,18 the three radial coordinates are explicitly treated while an adiabatic approximation was used for the three angles. These reduced dimensional studies are dynamically approximate in nature, but nevertheless can provide important information characterizing polyatomic reactions, and they have been reviewed extensively by Clary,19 and Bowman and Schatz.20 However, quantitative determination of reaction probabilities, cross-sections and thermal reaction rates, and their relation to the internal states of the reactants would require explicit treatment of five or the full six degrees-of-freedom in these four-atom reactions, which TI methods could not handle. Other approximate quantum approaches such as the negative imaginary potential method16,21 and mixed classical and quantum time-dependent method have also been used.22... 

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