Condensed Fukui functions

Table 3 The net charges and condensed Fukui functions for 1,2,5-thiadiazole 1 with C2v symmetry <1997JMT67>...

A common simplification of the Fukui function is to condense its values to individual atoms in the molecule [33]. That is, through the use of a particular population analysis, one can determine the number of electrons associated with every atom in the molecule. The condensed Fukui functions is then determined... 

Similar to condensed Fukui function defined in terms of atomic charge, one can also define condensed-to-atom (k) Fukui function, by summing over contribution of all... 

Addition of radicals to a different unsaturated substrate is an important class of organic reactions. To understand its regiochemistry, one needs to examine the condensed Fukui function (f°) or atomic softness (.v°) for radical attack of the different potential sites within the reactant substrate. We consider now a simple problem summarized in Example 3. 

One advantage of this response of molecular fragment approach [24] to condensed Fukui functions is that Equations 18.21 through 18.24 are easily evaluated from the population analysis data that accompanies the output of most quantum chemistry calculations. 

There seems to be no mathematical reason to favor one formulation of the condensed Fukui function over the other. 

The condensed Fukui functions for propylene and BF3 are given below. 

For propylene, the condensed Fukui function not only predicts that the electrophilic attack occurs on one of the doubly bonded carbon atoms, but it also predicts that there is a preference for the terminal carbon, in accordance with Markonikov s rule. Nucleophilic attack is predicted at the boron atom in BF3. 

In these examples, the condensed Fukui functions were computed using Hirsh-feld population analysis [26], which is unique among the commonly employed population analysis methods, because the same results are obtained from the response of molecular fragment and the fragment of molecular response approaches [24]. There are other arguments in favor of the Hirshfeld scheme too [27,28], many of them based on the tendency for the atom-condensed Hirshfeld Fukui functions to be nonnegative [25,29,30]. Nonetheless, condensed Fukui functions maybe computed using any population analysis method common methods... 

It is to be noted that/(r) is normalized to unity. Due to discontinuity problem in the number of electrons [13] in atoms and molecules, the right- and left-hand side derivatives at a fixed number of electrons introduces the concepts of EF for nucleophilic and electrophilic attack, respectively. Introducing the finite difference approximation and the concept of atom condensed Fukui function (CFF) [14], the working equations are... 

In a finite difference approximation, the condensed Fukui function [14] of an atom, say x, in a molecule with N electrons is defined as... 

The relative Michael-acceptor abilities of a variety of substituted aromatic and aliphatic nitroalkenes have been elucidated by computational methods. Several global and local reactivity indices were evaluated with the incorporation of the natural charge obtained from natural bond orbital (NBO) analysis. Natural charges at the carbon atom to the NO2 group and the condensed Fukui functions derived by this method were found to be consistent with the reactivity.187... 

The three functions f, fk, and refer to an electrophile, a nucleophile, and a radical. They are the sensitivity, to a small change in the number of electrons, of the electron density in the LUMO, in the HOMO, and in a kind of average HOMO/ LIJMO half-occupied orbital. Practical implementations of these condensed Fukui functions are the condensed-to-atom forms of Yang and Mortier [155] ... 

Here q SN) is the electron population (not the charge) on atom k, etc. (see below). Note that/ is just the average of/j. and /k. The condensed Fukui functions measure the sensitivity to a small change in the number of electrons of the electron density at atom k in the LUMO (/ "), in the HOMO (fk ), and in a kind of intermediate orbital (/j° ) they provide an indication of the reactivity of atom k as an electrophile (reactivity toward nucleophiles), as a nucleophile (reactivity toward electrophiles), and as a free radical (reactivity toward radicals). 

In a study of the reaction of alkynes with hydrogen isocyanide the condensed Fukui function was combined with the overall or global softness to try to rationalize the regioselectivity of attack on the triple bond [153] ... 

The NICS of each ring, as a criterion of aromaticity, has been used to explain the stability order of benzo[/)]thio-phene and its isomer. The results indicate that the benzene ring is aromatic in all the systems. The five-membered ring of benzo[. ]thiophene is also aromatic, whereas in benzo[r]thiophene it is nonaromatic. This could be an explanation of the stability of the former molecule. The MOS and the condensed Fukui functions derived from the electronic-structure calculations explain the expected electrophilic substitution of these compounds. The theoretical structure, ionization energies, order of aromaticity, stability, and reactivity are in good agreement with the experimental results <2003T6415>. 

A Mulliken population analysis was used to estimate the condensed reactivity indexes. In Table 62, the absolute values for the condensed Fukui function for electrophilic attack are shown for the relevant atoms in the heterocyclic compounds. 

For benzo [, ]thiophene, S and C-3 are the most reactive sites. Note the large condensed Fukui function on the S-atom of benzo[ ]thiophene. The results concur with the experimental information concerning the reactivity and stability of these systems. The principle of maximum hardness establishes that the system would be more stable if the global hardness, related to the FIOMO-LUMO gap, was a maximum. The FIOMO-LUMO gap correlates well with the expected stability of these molecules. This is an indication of the possibility to use hardness as a criterion of stability. 

The quantity/ is called the condensed Fukui function.It has a single value for each atom, k, in the molecule, and is not otherwise a function of position. The qkS are net charges on the atoms. In the last equation 4 is simply the square of the atom coefficient in the HOMO. It is also the frontier orbital density in FMO theory. It is the easiest to calculate, since we only need the wave function for the HOMO, which can often be found, at least roughly, from HMO theory. 

Senthilkumar, L. and Kolandaivel, P. (2005) Study of effective hardness and condensed Fukui functions using AIM, ab initio, and DFTmethods. Mol. Phys., 103, 547-556. 

The expression for condensed Fukui functions for the zth atom in a molecule can be obtained by considering finite difference approximation and Mulliken s population... 

How the reactivity of a particular site changes during a chemical reaction has been studied [152], Ab initio SCF calculations [152] of condensed Fukui functions at various sites at different stages of a dissociation reaction have been performed. The variation of these reactivity indices along the reaction path is consistent with chemical intuition. 

This says that /(r) is the functional derivative (section 7.2.3.2, The Kohn-Sham equations) of the chemical potential with respect to the external potential (i.e. the potential caused by the nuclear framework), at constant electron number and that it is also the derivative of the electron density with respect to electron number at constant external potential. The second equality shows /(r) to be the sensitivity of p(r) to a change in N, at constant geometry. A change in electron density should be primarily electron withdrawal from or addition to the HOMO or LUMO, the frontier orbitals of Fukui [114], hence the name bestowed on the function by Parr and Yang. Since p(r) varies from point to point in a molecule, so does the Fukui function. Parr and Yang argue that a large value of fix) at a site favors reactivity at that site, but to apply the concept to specific reactions they define three Fukui functions ( condensed Fukui functions [80]) ... 

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. 

This approximation establishes that the strongest bond in a molecule is the one formed by the adjacent atoms with the smallest values of the condensed fukui function, and that the weakest bond is the one formed by the adjacent atoms with the largest values of the condensed fukui function. Note that since the condensed fukui functions are different for nucleophilic, electrophilic, and free radical attacks, the weakest bond in a molecule, which may be associated with the most reactive site (this one may be either of the two atoms forming the bond or the bond itself), may be a different one, depending on the type of attack, in agreement... 

It is important to mention that if one makes use of the experimental values of I and A in Eq. (11), to determine the hardnesses in Eq. (39), and one makes use of molecular orbital theory to determine the values of the condensed fukui function, then, if one sets Ng = 1, one finds that this expression provides the correct trends, and reasonable estimates of the bond energies of a wide variety of molecular systems [16]. 

In previous publications, it had already been inferred [8,9,21-26,43], that the larger the fukui function, the greater the reactivity, and this statement had already been successfully used to explain several aspects of the chemical reactivity of different systems. The present approach allows one to understand that this may be due to the fact that the sites with the largest values of the appropriate condensed fukui function may be associated with the weakest bonds, and with those reaction paths with the smallest activation energy barriers. 

There has also been considerable interest in the theoretical analysis of the relative reactivity of the position on the indole ring. One of the quantities that can be calculated is the condensed Fukui function [f ], which, in the context of density function theory, provides a measure of respruise to an approaching electrophile [8]. For indole, the 1-, 2-, and 3-positi(Mis are calculated as 0.08, 0.05, and 0.18, consistent with the observed preference for substitution at C-3 [9]. 

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