Fukui function types

It is important to mention that the chemical potential and the hardness, p, and 17, are global-type response functions that characterize the molecule as a whole, while the electronic density p(r), the Fukui function fir), and the dual descriptor A/(r) are local-type response functions whose values depend upon the position within the molecule. 

Once again, due to the discontinuity of the electron density with respect to N, finite difference approximation leads to three types of Fukui function for a system, namely (l)/+(r) for nucleophilic attack measured by the electron density change following addition of an electron, (2)/ (r) for electrophilic attack measured by the electron density change upon removal of an electron, and (3)/°(r) for radical attack approximated as the average of both previous terms. They are defined as follows ... 

Evaluation of the only appropriate Fukui function is required for investigating an intramolecular reaction, as local softness is merely scaling of Fukui function (as shown in Equation 12.7), and does not alter the intramolecular reactivity trend. For this type, one needs to evaluate the proper Fukui functions (/+ or / ) for the different potential sites of the substrate. For example, the Fukui function values for the C and O atoms of H2CO, shown above, predicts that O atom should be the preferred site for an electrophilic attack, whereas C atom will be open to a nucleophilic attack. Atomic Fukui function for electrophilic attack (fc ) for the ring carbon atoms has been used to study the directing ability of substituents in electrophilic substitution reaction of monosubstituted benzene [23]. In some cases, it was shown that relative electrophilicity (f+/f ) or nucleophilicity (/ /f+) indices provide better intramolecular reactivity trend [23]. For example, basicity of substituted anilines could be explained successfully using relative nucleophilicity index ( / /f 1) [23]. Note however that these parameters are not able to differentiate the preferred site of protonation in benzene derivatives, determined from the absolute proton affinities [24],... 

Perhaps the most successful application of Fukui function and local softness is in the elucidation of the region-selective behavior of different types of pericyclic reactions including the 1,3-dipolar cycloadditions (13DC), Diels-Alder reactions, etc. These reactions can be represented as shown in Scheme 12.4. Considering the concerted approach of the two reactants A and B, there are two possible modes of addition as shown in Pathway-I and Pathway-II. 

From the relation between Fukui function and local softness, electrophilic and nucleophilic local softnesses can be computed. Donor and acceptor sites can also be identified by large values of both types of local softnesses in addition, it can be used to compare sites of different molecules and to identity which one is softer or harder. The elec-trophilicity index can also be extended to a local context,21 and a comparison of the electrophilicity of sites in different species can be made. 

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... 

The applications of local quantities start with the use of the Fukui function in the frontier-electron theory of chemical reactivity within a density functional framework [19]. In this approach there are three different types of Fukui functions, viz.,... 

Although sophisticated electronic structure methods may be able to accurately predict a molecular structure or the outcome of a chemical reaction, the results are often hard to rationalize. Generalizing the results to other similar systems therefore becomes difficult. Qualitative theories, on the other hand, are unable to provide accurate results but they may be useful for gaining insight, for example why a certain reaction is favoured over another. They also provide a link to many concepts used by experimentalists. Frontier molecular orbital theory considers the interaction of the orbitals of the reactants and attempts to predict relative reactivities by second-order perturbation theory. It may also be considered as a simplified version of the Fukui function, which considered how easily the total electron density can be distorted. The Woodward-Hoffmann rules allow a rationalization of the stereochemistry of certain types of reactions, while the more general qualitative orbital interaction model can often rationalize the preference for certain molecular structures over other possible arrangements. 

The fact that features in the total electron density are closely related to the shapes of the HOMO and LUMO provides a much better rationale than the perturbation derivation as to why FMO theory works as well as it does. It should be noted, however, that improvements in the wave function do not lead to better performance of the FMO method. Indeed, the use of MOs from semi-empirical methods usually works better than data from ab initio wave functions. Furthermore, it should be kept in mind that only the HOMO orbital converges to a specific shape and energy as the basis set is improved in an ab initio calculation the LUMO is normally determined by the most diffuse functions in the basis. The Fukui functions, on the other hand, can be calculated for any type of wave function. 

Owing to a discontinuity in the derivative in Eq. (17) for integral values of N, three different types of Fukui functions can be defined by applying the finite difference and frozen core approximations as follows [261, 290] ... 

Before attempting to accurately compute a quantity it is useful to know something about its qualitative behavior. Here we consider some results of this type for the Fukui function. 

The Fukui function successfully predicts relative site reactivities for most chemical systems. As such it provides a method for understanding and categorizing chemical reactions. More importantly, the Fukui function can be used to predict what the products of a given reaction will be. As computing the Fukui function becomes faster and easier, its predictive ability might be routinely used to winnow the list of potentially useful reagents, catalysts, etc. before performing the types of experiments or calculations necessary to fully characterize a chemical reaction. This predictive ability renders the Fukui function an important tool of the chemist. 

To describe the site selectivity and reactivity of an atom in a molecule, it is required to condense the values of fir) and s(r) around each atomic site. Thus, for an atom A in a molecule, depending on the type of electron transfer, one can define three different types of Fukui functions as proposed by Yang and Mortier ... 

In the difference term one recognizes a fluctuation term, f(r) the deviation of the Fukui function from the average electron density per electron, multiplied by the functional derivative of E with respect to a(r) at constant number of electrons. This quantity (5E/5o(i))ig is a response function of the type (5E/5v(r))j. mentioned in the Introduction and measures the sensitivity of the system s energy to variations in the shape factor. Its evaluation seems to be far from trivial but it is possible to get already an idea of what might be factors of importance in this response function. Adopting an orbital formalism and using a Koopmans type approximation one arrives at an approximate expression... 

W. Langenaeker, M. de Decker and P. Geerlings. Quantum chemical study of the Fukui function as a reactivity index Probing the acidity of bridging hydroxyls in zeolite type model systems. J. Mol. Struct. (THEOCHEM) 207, 1990, 115. 

One of the stumbling blocks in the evaluation of this type of derivatives with respect to the number of electrons N is that the E = E(N) curve for isolated systems (atoms, molecules. ..) shows discontinuities at the integer values of N (the problem of the existence of such a function will be left out (see ref [25] for a discussion in depth) necessitating in the case of the Fukui fbnction and local softness the introduction of right and left hand derivatives f" and f... 

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