Fukui function finite difference approximations

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 ... 

The Fukui functions generalize the concept of frontier orbitals by including the relaxation of the orbital upon the net addition or removal of one electron. Because the number of electrons of an isolated system can only change by discrete integer number, the derivative in Equation 24.37 is not properly defined. Only the finite difference approximation of Equation 24.37 allows to define these Fukui functions (noted here by capital letters) F1 (r)... 

In addition to popular finite difference approximations of Fukui functions of an isolated system (Equation 24.38), at least six other different Fukui functions can be defined as responses to a potential. These later concepts do not depend on a net... 

Finite Difference Approximations of Fukui Functions as Potential Derivatives... 

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... 

Because of the discontinuity of this derivative, a backward Fukui function f"(r) and a forward Fukui function f (r) are defined, corresponding to local descriptors for electrophilic and nucleophilic attack, respectively. In terms of the finite difference approximation, both functions can be written as ... 

The electronic density provides the site reactivity information of ionic systems, while the Fukui function is better suited when dealing with neutral species. Owing to the discontinuity of the derivative of (4), two different Fukui functions can be defined by applying the finite difference approximation 8... 

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... 

In a finite differences approximation the fukui function becomes [9]... 

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. 

Michelak et al. [45] have computed the Fukui function using a variety of the methods previously described as well as some more sophisticated approximate approaches. Specifically, they considered the finite difference approximation to the derivative with IsN = 1 and AY = 0.01, a modified finite difference formula in which only terms linear in orbital changes and occupation number changes are retained, the frozen core approximation (Eqs. 47, 48) and an approximation to Eq. (46). 

Among the other derivatives, mention may be made of the concept of Fukui func-tion, denoting the derivative of the electron density, viz.,/(r) = (dp(r)ldN), which, within the finite difference approximation, represents the frontier orbitals—the HOMO and the LUMO. Nncleophilic and electrophilic attacks have thereby been rationalized in terms of these two approximations of the Fukui function. ... 

In 2 computational aspects are discussed, with the assessment of DFT methods ( 2.1) in the evaluation of (a) ionization energies and electron affinities, and via eqn. (11), finite difference estimates of electronegativities and hardnesses ( 2.1.1) Mid (b) of dipole and quadrupole moments ( 2.1.2).In the final paragraph 2.2 a problem at the borderline between computational and conceptual DFT is tackled the evaluation of Fukui fimctions "beyond" the finite difference approximation. In 3 conceptual DFT is discussed, where in 3.1 attention is paid to the evaluation and/or use of DFT based concepts as such the shape factor and the local softness as Molecular Similarity indicators ( 3.1.1), and the nuclear Fukui function ( 3.1.2). In the final part of this Section ( 3.2) the role of DFT based concepts in various principles is discussed. The influence of solvent on the acidity of alkylalcohols is discussed within the framework of Sanderson s Electron ativity Equalization Principle [30] ( 3.2.1). The Hard and Soft Acids and Bases Principle and Pearson s Maximum Hardness Principle [31] are used as the guiding prindples in the study of the cycloaddition reactions of HNC to alkenes and aligmes. 

Garza el al. presented a detailed comparison of frozen core and finite difference approximations for generalized Fukui functions and chemical potentials for the molecules NH3, H2O, HCOOH and CH2. They concluded that the former approximation usually gives similar results to the latter... 

It is clear from Eqs. (31a) and (31b) that/(r) is the DPT analogue of the frontier orbital regioselectivity for nucleophiUc/ (r) and electrophilic/ (r) attack and it is a restatement of frrMitier molecular orbital (FMO) theory. However, the last statement may not be true because Fukui function includes the effect of electron correlation and orbital relaxation that are a priori neglected in FMO theory. Flurchik and Bartorotti [124] noticed considerable differences between HOMO/ LUMO density and Fukui function value. Chattaraj et al. [125] and Pacios et al. [126, 127] proposed gradient approximation for Fukui function to avoid the calculation of metastable anion to calculate Fukui function using finite difference approximation ... 

We predict bitumen fragment adsorption configurations by using global softness and Fukui functions. Using the finite difference approximation, the global softness is evaluated as 5 = 1/(1P - EA), where IP is the ionization potential and EA is the electron affinity. In this work, for a neutral system that contains N electrons, we obtain... 

In the finite difference approximation, the condensed Fukui functions of atom X in a molecule that contains N electrons are defined in Equations 14.30 and 14.31. 

The Fukui function is approximated by the electron density difference using the finite difference method,... 

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] ... 

The discontinuity in the above derivative provides three different varieties of the Fukui function. One may further use finite difference and frozen core approximations to obtain the following definitions ... 

Note thaty(r) describes the change in electron density at point r in space with respect to the variation in the total number of electrons N, at constant external potential (i.e., at fixed molecular geometry). There are several approximate formulas to evaluate the condensed to atom Fukui function. A three-point finite difference approach is normally used [24, 61], albeit it may be also readily obtained by single-point calculations, using the frozen core approximation. The former approach will be used herein. Using the normalization to unity condition [57, 58] ... 

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