Hardness kernel definition

Nalcwajski recently proposed the Recursive Combination Rules [32] for the calculation of molecular hardness. Although derived along a different path, these are essentially equivalent with the EEM formalism. To demonstrate this, we proceed as follows. From the definition of the hardness kernel, we may write (Eqs. (7) and (18)) ... 

An advantage of the Combination Rules is, that one is not restricted to only one definition for the calculation of the hardness kernels. Other, more refined approximations can be used (e.g. the Mataga-Nishimoto [34] formula or the formulas by Pariser and Ohno) which may enhance the accuracy. This is not the case in EEM, at least when we want to retain its internal consistency. For larger systems, EEM has the advantage that it enables the direct calculation of global and local properties while the Combination Rules have to be applied recursively. Also EEM is directly applicable to the solid state. 

Then Harbola et al. and Ghosh gave a more extensive definition based on the hardness kernel q(r, r )... 

The purpose of this work is to start from the basic equations of density functional theory to describe the changes in the energy associated with the transition from one ground-state to another, in terms of different sets of variables. In this process one will find the natural definitions of the hardness and softness kernels, the local hardness, the local softness, the global hardness and the global softness [23]. Then, we will proceed to establish their relation with ionization potentials and electron affinities, in order to confirm their behavior as a measure of chemical hardness or softness [14, 24]. Finally, this theoretical framework will be used to analyze the maximum hardness and the HSAB principles. 

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Geerlings De Proft, 2008 Cardenas et al., 2009 Senet, 1996). One may obtain a condensed-to-atom variant and also for the electrophilic, nucleophilic and radical attacks in the usual way. Moreover, the inverse of tu(r,r ) may generate a hierarchy of nucleophilicity kernel. Unlike the previous formulations, the overall treatment here is general and analytic with hardly any bearing on the explicit form of E(N). The traditional operational definition of local softness and hardness contain the same potential information and they should be interpreted as the local abundance or concentration of their corresponding global properties. 

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