Global hardness orbitals

Now, by its very definition, the global hardness rj is a measure of the HOMO-LUMO gap of a compound. Consequently, it seems reasonable to assume that the AEE AE should scale with the global hardness. The <(ag/r)3> term describes the p-orbital expansion or contraction as electrons are added or removed from the shielded nucleus. Using effective atomic numbers Znp and Slater rules ... 

In order to understand the importance of frontier orbitals in chemical reactivity, Berkowitz [213] studied the frontier-controlled reactions within the purview of density functional theory. It is evident that the directional characteristics of frontier orbitals determine the extent of charge transfer, and soft-soft interactions are frontier-con-trolled. A somewhat similar analysis showed that charge transfer would be facilitated at a place where the difference in local softness of two partners is large [87], It may be noted that Fukui function is obtainable from local softness but the reverse is not true. On the other hand, local hardness suffers from the drawback of ambiguity [87], which allows one to even consider it to be equal to global hardness without disturbing their... 

The global hardness of a compound has been related to the energy gap between the frontier orbitals [53]. Since the global hardness reflects the hardness of the constituent atoms, we expect a smaller gap for a series of homologous compounds containing softer elements. 

The equations 7, 8 and 9 fail to operate when the HOMO - LUMO energy gap becomes too small and do not consider the influence on chemical properties of other orbitals, besides the HOMO and LUMO s. Moreover, it is not possible to study the site selectivity of a chemical species considering only absolute hardness other than space-dependent (local) versions of hardness/softness concepts [5]. Thus in addition to the global definition of r) and S, the local hardness [4] and local softness [5] have been introduced as follows ... 

While the electronegativity and the absolute hardness are global properties of the system, the reaction between two molecules depends on the properties of the involved orbitals. In order to measure the chemical reactivity of a particular orbital in a molecule, different local variables, such as orbital softness (sq) and Fukui (fpolarization functions (no), can be computed through equations 24, 34, 36. 

Thus, one can see that within the framework provided by density functional theory, the basic equations for the description of a chemical event, Eqs. (4) and (7), may be expressed in terms of basic variables such as the chemical potential (electronegativity), the chemical hardness and the fukui function (frontier orbitals). In fact, through this approach one may introduce a coherent quantitative language of hardness and softness functions which are nonlocal, local, and global [29]. The global softness is given by... 

As we have already observed above, within the hardness (interaction) representation (see Tables 1, 3) the FF indices provide important weighting factors in combination formulas which express the global CS and potentials in terms of the local properties, relevant for the resolution in question. The FF expressions from the EE equations are invalid in the MO resolution, since no equalization of the orbital potentials can take place, due to obvious constraints on the MO occupations in the Hartree-Fock (HF) theory [61]. Moreover, standard chain-rule transformations of derivatives are not applicable in the MO resolution since some of the derivatives involved are not properly defined. Various approaches to the local FF, f(f), have been proposed e.g., those expressing f(f), in terms of the frontier orbital densities [11, 25], or the spin densities [38]. Also the finite difference estimates of the chemical potential (electronegativity) and hardness have been proposed in the MO and Kohn-Sham theories for various electron configurations [10, 11, 19, 52, 61b, 62, 63]. 

The present approach may be complemented with other works in which also input electronegativity in Eqs. (4.344)-(4.346) is expressed in the same context of DFT softness kernel, with various systematic forms in terms of the atomic valence shell Slater quantities as effective charge and orbital exponent (Putz, 2006). Equally, since the present approach strongly relies on associated chemical hardness, local-to-global hierarchies may be... 

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