Fukui electron density indices

Electronegativity, Hardness, Softness and the Fukui Function Electron Density Reactivity Indexes... 

An electronic parameter that often correlates with metabolic rates is the electrophilic (or nucleophilic) superdelocalizability. This quantity is a reactivity index formulated by Fukui and colleagues as an orbital-weighted electron density.145 The total electrophilic superdelocalizability, 2SE, summed over all atoms in a molecule, exhibits a parallelism with the hydrophobic parameter, log P, in several series of compounds such as PAHs and aliphatic amines, where it is probably approximating molecular volume. 

Now, in the case of the finite differences approximation to the derivative in the second equality of Eq. (9), because of the local dependence on the position within the molecule, instead of using f(r) directly, it is more simple to condense its values around each atomic site into a single value that characterizes the atom in the molecule. This can be done by first condensing the electronic density to the charge of each atom in the molecule, and differentiating afterwards with respect to the total number of electrons in the system [30]. Thus, the finite differences approximation leads to three indexes known as the condensed fukui functions. 

Abstract. The development of theories for interpreting the course of chemical reactions is one of the most important achievements of theoretical chemistry in the twentieth century. I selected the paper by Fukui et al. from 1952, proposing the frontier electron density as the reactivity index for the orientation of electrophilic substitution reactions. This paper may be regarded as a bridge between an older reactivity theory, the electronic theory of organic chemistry, and new ones predicting the stereochemical courses of reactions such as frontier orbital theory and the Woodward-Hoffmann rule. 

In many instances, the relaxation terms, <]>, can be ignored, and the frontier molecular orbital theory as first put forth by Fukui - is recovered. For particular molecules, the identification reactive sites using f(t) has been successful.A convenient way to visualize the reactive sites in a molecule using the above reactivity index is to first display an isosurface of the electron density that just encloses the van der JKials volumes of the individual atoms in the molecule. Typically, the value of this isosurface is between 0.002 and 0.005. Next, the values of the reactivity index are mapped upon this surface and color coded from blue (zero) to ted (most positive). 

Density functional theory (DFT) provides an efficient method to include correlation energy in electronic structure calculations, namely the Kohn-Sham method 1 in addition, it constitutes a solid support to reactivity models.2 DFT framework has been used to formalize empirical reactivity descriptors, such as electronegativity,3 hardness4 and electrophilicity index.5 The frontier orbital theory was generalized by the introduction of Fukui function,6 and new reactivity parameters have also been proposed.7,8 Moreover, relationships between those parameters have been found, and general methods to relate new quantities exist.9... 

Characterizes molecular stmcture also by their propensity to engage chemical reactivity, with specific frontier measure appropriately modeled by the Fukui function (the density to number of electrons derivatives) as well as the chemical hardness iteratively evolution among adducts in a complex chemical interaction or through various chemical interactions channels (so explaining by quantum chemical reactivity principles the Le Chatelier-Braim principle of reactive equilibrium and bonding, or the electronic delocalization by novel sharing index, just to name few preeminent applications) ... 

To conclude, we have shown that, despite the complexity of their structural and electronic properties, the reactivity of small metallic clusters can in many cases be understood in simple terms with the use of both global reactivity indexes (ionization potentials, electron affinities, and chemical hardness, which provide a measure of the global cluster reactivity) and local Fukui functions (which allow the identification of special reactive sites within a cluster). All of them can be easily evaluated within the framework of Density Functional Theory. 

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