Chemical reactivity and the shape function

This quantity is trivially computed from the Fukui function, / (r) [78-80], and the shape function, and it has a simple interpretation the shape Fukui function measures where the relative abundance of electrons increases or decreases when electrons are added to (or removed from) a system. In our experience, plotting r) often provides a simpler and easier way to interpret picture of chemical reactivity than the Fukui function itself. Perhaps this is because ofr) is the local density approximation (LDA) to the Fukui function [81]. Since the numerator in Equation 19.27, / (r) — olr), is the post-LDA correction to the Fukui function [81], the shape... 

Knowing the shape of a substance s molecules is a key to understanding its physical and chemical behavior. One of the most important and far-reaching effects of molecular shape is molecular polarity, which can influence melting and boiling points, solubility, chemical reactivity, and even biological function. 

The paper of Parr and Bartolotti is prescient in many ways [1], It defines the shape function and describes its meaning. It notes the previously stated link to Levy s constrained search. It establishes the importance of the shape function in resolving ambiguous functional derivatives in the DFT approach to chemical reactivity—the subdiscipline of DFT that Parr has recently begun to call chemical DFT [6-9]. Indeed, until the recent resurgence of interest in the shape function, the Parr-Bartolotti paper was usually cited because of its elegant and incisive analysis of the electronic chemical potential [10],... 

Equations 19.17 and 19.20 provided the foundation for further progress on the shape function-based perspective on chemical DFT. The first extension, by Baekelandt et al. [27], laid out the mathematical structure associated with these new pictures and introduced new reactivity indicators. This paper reveals that the isomorphic ensemble provides a particularly useful approach to the hardness picture of chemical reactivity, and allows one to define a local hardness indicator,... 

These relationships link variations in the shape function to the reactivity descriptors commonly used in DFT and provide explicit methods for describing chemical reactivity in terms of the shape function yielding conceptual shape function theory . Thus, it is clearly seen that the shape function not only determines all the physical properties of an isolated molecule but, since reactivity descriptors can be explicitly constructed from the shape function, also its chemical properties. As the resulting equations are not always that simple to apply at first sight, we pass in next paragraph to some pragmatic procedures for extracting descriptors from the shape function. 

Ayers, P Cedillo, A. The Shape Function. Chemical Reactivity Theory A Density Functionl View. Chattaraj, P. K., Ed. Taylor and Francis, Boca Raton, 2009, p. 269. 

There are well-defined criteria for this "reversible" system in terms of peak separation, wave shape, etc. and the maximum current scales inversely with the square root of the scan rate. The half-wave potential of a "reversible" redox process may readily be obtained from the voltammogram. If, however, the electron transfer produces a species that is chemically reactive on the experimental time scale, then the return wave is missing and the peak potential shifts as a function of the kinetics of the follow-up processes. The peak is not as well defined, and without a proper return wave it is now not straightforward to obtain thermodynamic half-wave potentials from the trace of such an irreversible system. Furthermore, if a disk electrode is used of micrometer-dimensions, then hemispherical diffusion now takes place and a sigmoidal current-potential curve is obtained [Fig. 4(b)]. 

The field of stereochemistry serves as a unifying theme for the expanded definition and diversification of chemistry. The consequences of molecular and macromolecular shape and topology are central to issues of chemical reactivity, physical properties, and biological function. With that view, the importance of stereochemistry had never been greater, and it is hoped that this series will provide a forum for documentation of significant advances in all of these subdisciplines of chemistry. 

In turbulent reactive flows, the chemical species and temperature fluctuate in time and space. As a result, any variable can be decomposed in its mean and fluctuation. In Reynolds-averaged Navier-Stokes (RANS) simulations, only the means of the variables are computed. Therefore, a method to obtain a turbulent database (containing the means of species, temperature, etc.) from the laminar data is needed. In this work, the mean variables are calculated by PDF-averaging their laminar values with an assumed shape PDF function. For details the reader is referred to Refs. [16, 17]. In the combustion model, transport equations for the mean and variances of the mixture fraction and the progress variable and the mean mass fraction of NO are solved. More details about this turbulent implementation of the flamelet combustion model can also be found in Ref. [20],... 

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