First Order Variation in Charge and Potential

Further improvement in our systematic applies when also the potential variation is taken into account. Such picture is particularly meaningful for assigning reactivity indices based on external potential direct influence (Ayers Parr, 2001). 

Under the first order in charge and potential variation, the dififerential EN expansion primarily looks like (Putz, 2006)  

Since the functional derivative of the with respect to the potential V r) may be determined from the fundamental DFT relation (4.173) the Eq. (4.272) rewrites as  

Nevertheless, when performing the integration to get, for instance, the absolute electronegativity of (4.253) the path integral over SV r) is involved. This can be solved between the adiabatic (F(r) = 0) and vertical V r) = ct 0 limits. Such treatment corresponds with the physical picture in which an electron can be added to the chemical system from infinity due to its electronegativity (Putz et al, 2003). This approach is consistent also with/P and 

However, in performing the potential path integral of (4.275) another assumption was made, namely to consider both L(r) and p(r) as independently of V(r) as far as they do not pose an explicit dependence on it. Instead, the new introduced U(r)-dependent quantities, b and of Eqs. (4.276) and (4.277), respectively, were considered independenf of as before was the case in Eq. (4.223). 

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