Fukui functions

A reaction will in general involve a change in the electron density, which may be quantified in terms of the Fukui function. ... 

The Fukui function indicates the change in the electron density at a given position when the number of electrons is changed. We may define two finite difference versions of the function, corresponding to addition or removal of an electron. 

In the frozen MO approximation the last terms are zero and the Fukui functions are given directly by the contributions from the HOMO and LUMO. The preferred site of attack is therefore at the atom(s) with the largest MO coefficients in the HOMO/LUMO, in exact agreement with FMO theory. The Fukui function(s) may be considered as the equivalent (or generalization) of FMO methods within Density Functional Theory (Chapter 6). 

Besides the already mentioned Fukui function, there are a couple of other commonly used concepts which can be connected with Density Functional Theory (Chapter 6). The electronic chemical potential p is given as the first derivative of the energy with respect to the number of electrons, which in a finite difference version is given as half the sum of the ionization potential and the electron affinity. Except for a difference in sign, this is exactly the Mulliken definition of electronegativity. ... 

Gazquez JL, Vela A, Galvan M (1987) Fukui Function, Electronegativity and Hardness in the Kohn-Sham Theory. 66 79-98... 

Yang, W., and R. G. Parr. 1985. Hardness, softness and the Fukui function in the electronic theory of metals and clusters. Proc. Natl. Acad. Sci. USA 82, 6723. 

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

The chemical reactivity of 1,3,4-thiadiazole 1 was predicted using DFT by calculating the net atomic charges and the Fukui functions /+,/ , and f° (Table 2). 

Quantum mechanical approaches have been successfully used to predict hydrogen abstraction potentials and likely sites of metabolism of drug molecules [78-81]. AMI, Fukui functions, and density functional theory calculations could identify potential sites of metabolism. Activation energies for hydrogen abstraction were calculated by Olsen et al. [81] to be below 80 kj/mol, suggesting most CH groups can be metabolized which particular one depends on steric accessibility and intrinsic reactivities. 

Moreover, Berkowitz and Parr [39] have shown that the static density response may be expressed in terms of the softness kernel s(f,r/), the global softness S and the Fukui functions /(r) as follows ... 

Next, introduce definition (67) of the Fukui function to obtain... 

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. 

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... 

For the second derivative of the electronic density with respect to the number of electrons, the dual descriptor, one can proceed as in the case of the energy. That is, the Fukui function using the Heaviside function [25] is written as... 

From the interpretation given to the Fukui function, one can note that the sign of the dual descriptor is very important to characterize the reactivity of a site within a molecule toward a nucleophilic or an electrophilic attack [29,30]. That is, if A/(r) > 0, then the site is favored for a nucleophilic attack, whereas if A/(r) < 0, then the site may be favored for an electrophilic attack. 

Now, the Fukui function is closely related to the frontier orbitals. This can be seen from Equations 2.29 and 2.30, together with Equation 2.42, because if one determines the electron densities of the iV0 — 1- and the N() + 1-electron systems with the orbitals set corresponding to the Mj-electron system, then... 

On the other hand, the calculation of the Fukui function with the HF frontier orbitals is, in general, qualitatively very similar to the one obtained through KS orbitals. However, there may be cases where the absence of correlation effects in HF may lead to large differences with respect to the KS description. 

These generalized Fukui functions are local reactivity indexes, as they depend on the coordinates r where they are determined, in other words their values vary from one point to another within the molecule. It is clear that these quantities can be used to know how the charge or (and) spin densities respond when there are charge or (and) spin transfer to the reacting molecule. 

There is no more research on the analysis or applications of SP-DFT generalized Fukui functions, per se. Instead, condensed-to-atoms SP-DFT Fukui function schemes have been developed and applied to different chemical reactivity problems. In these schemes, the information of the Fukui functions is condensed on an atomic position. In addition, the Fukui function/ (r) is related with the extension of global to local spin-donicity and spin-philicity, defined as [20]... 

Fukui Function and Local Softness as Reactivity Descriptors... 

In order to understand the detailed reaction mechanism such as the regio-selectivity, apart from the global properties, local reactivity parameters are necessary for differentiating the reactive behavior of atoms forming a molecule. The Fukui function [10] if) and local softness [11] t.v) are two of the most commonly used local reactivity parameters. 

The Fukui function is primarily associated with the response of the density function of a system to a change in number of electrons (N) under the constraint of a constant external potential [v(r)]. To probe the more global reactivity, indicators in the grand canonical ensemble are often obtained by replacing derivatives with respect to N, by derivatives with respect to the chemical potential /x. As a consequence, in the grand canonical ensemble, the local softness sir) replaces the Fukui function/(r). Both quantities are thus mutually related and can be written as follows ... 

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 ... 

Using one-electron orbital picture, Fukui functions can be approximately defined as... 

For studying reactivity at the atomic level, however, a more convenient way of calculating the /(r) functions at atomic resolution is used. The condensed-to-atom Fukui functions for an atom k in a molecule are expressed as [13]... 

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