Fukui function, nuclear

In a second approach of the reactivity, one fragment A is represented by its electronic density and the other, B, by some reactivity probe of A. In the usual approach, which permits to define chemical hardness, softness, Fukui functions, etc., the probe is simply a change in the total number of electrons of A. [5,6,8] More realistic probes are an electrostatic potential cf>, a pseudopotential (as in Equation 24.102), or an electric field E. For instance, let us consider a homogeneous electric field E applied to a fragment A. How does this field modify the intermolecular forces in A Again, the Hellman-Feynman theorem [22,23] tells us that for an instantaneous nuclear configuration, the force on each atom changes by... 

Finally, the mixed second derivatives define the nuclear Fukui function (NFF) indices ... 

The HF results generated for representative polyatomic molecules have used the /V-derivatives estimated by finite differences, while the -derivatives have been calculated analytically, by standard methods of quantum chemistry. We have examined the effects of the electronic and nuclear relaxations on specific charge sensitivities used in the theory of chemical reactivity, e.g., the hardness, softness, and Fukui function descriptors. New concepts of the GFFs and related softnesses, which include the effects of molecular electronic and/or nuclear relaxations, have also been introduced. 

This issue was addressed in [2] for electron-transfer reactivities. The nuclear Fukui functions and softnesses defined there are valid for gapless systems. The analysis begins with nuclear forces specified through the Feynman-Hellmann theorem [48],... 

From these definitions, it follows immediately that the nuclear Fukui function is... 

It was pointed out in [2,3] that nuclear-configuration changes define chemical reactions so that nuclear reactivities should be defined and set on equal footing with the corresponding electronic reactivities. Thus nuclear Fukui functions , Eq. (59), and nuclear softnesses a Eq. (60), were defined in [2], and explicit Kohn-Sham expressions were found for them, Eqs. (61H64), as reviewed in Sect. 5. These are electron-transfer reactivities and are valid only for extended systems, leaving open the question of nuclear electron-transfer reactivities for localized systems and nuclear isoelectronic reactivities for all systems. 

Relationship to Fukui Function and Nuclear Fukui Function... 

Fa is the force exerted on nucleus a by the electron cloud. The reactivity index is also a nuclear Fukui function. Although it is completely general. Equation (6.27) has been applied mainly to surface atoms. 

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]) ... 

It should be noted that according to the definition adopted here the Fukui functions restrict the description of reactivity to its electronic aspects. A more complete picture is gained by inclusion of nuclear displacement, see for example [31-32] and literature cited in these papers. 

It is shown that Density Functional Theory offers both a conceptual and a computational tool for chemists in relating electronic structure of atoms and molecules to their properties both as isolated systems and upon interaction. The computational performance of DFT in the calculation of typical DFT quantities such as electronegativity and hardness and in the ev uation of atomic electronic affinities and molecular dipole and quadrupole momCTits is assessed. DFT concepts are discussed as such (a non finite difference evaluation of the electronic Fukui function, local softness and its use in similarity analysis of peptideisosteres and the nuclear Fukui function as a indicator of nuclear rearrangemCTits upon reaction) and in the context of principles (EEM, MHP, HSAB) for a variety of reactions involving the influence of solvent on the acidity of alcohols and the addition of HNC to dipolarophiles. 

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. 

The resulting derivative, called the nuclear Fukui function, can then be written as o., 2 (35)... 

In Table 5 the results are given for the nuclear Fukui function, and the shift in equilibrium distance AR and harmonic vibrational frequency Av upon changing the number of electrons for the above mentioned series, all results being obtained within a DFT scheme using a B3PW91 potential and the AVTZ basis. 

Although further work on polyatomic molecules is now in progress it can already be concluded that for an experimenal chemist electronic and nuclear Fukui fimctions offer complementary information on a molecule s fate upon attack by a nucleophile or electrophile. Whereas the electronic Fukui function indicates the preferential site of attack (site selectivity - see also 3.2.2.), the nuclear Fukui function foresees nuclear rearrangements (bond compression or extension) at the reactive site. 

These fm-ces have been used by Cohen [23] to define a column vector of the Nuclear Fukui Function (NFF) ... 

A final set of reactivity indices constitute the so-called nuclear reactivity indices. As can be seen from eqn (26), the Fukui function measures the electron density response due to the change of the number of electrons of the system. However, despite the fact that the electron density determines all ground state properties of an atomic or a molecular system, the response of the nuelei due to this perturbation remains unknown a reponse kernel is needed to translate electron density changes in external potential changes. Cohen et at. circumvented this problem by introducing a nuclear Fukui function, which they defined as the Hellmann-Feynman force F acting on the nuclei due to a perturbation in the number of electrons at a constant external potential ... 

This implies that the nuclear Fukui function can also be interpreted in terms of the contributions to binding (//r) < 0) and antibinding (/ (r) < 0) regions in the moleeule within the framework of Berlin s theorem. Illustrative examples and applications of this function have been performed. Also... 

In such cases, the characterization of the nuclear response can be reduced to the contribution analysis of the and fNs(y) Fukui functions to the... 

Fig. 6 Isosurface maps for the Berlin local function (see text), electronic / r) and nuclear Fukui functions for formadehyde optimized at the B3LYP/6-311+G(d,p) level of theory both atthe singlet ground state and the triplet excited state. The isosurface value for/." "(r) is 0.5. For the remaining maps, a value of 0.005 for the isosurface has been used. The positive regions are presented in blue and the negative regions in red. (Reproduced with permission from J. Chem. Phys., 2005, 123, 084104. Copyright 2005 American Institute of Physics).

So far O Eq. 16.60 has been applied successfully for the calculation of static and dynamic molecular polarizabilities (Carmona-Espindola et al. 2010 Flores-Moreno and Koster 2008 Shedge et al. 2010) using local and gradient-corrected functionals as well as for the calculation of Fukui functions (Flores-Moreno 2010 Flores-Moreno et al. 2008). Calculation of second derivatives with respect to nuclear displacements is currently under development. 

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