Long-Range Local-to-Global Behavior

Given the form of the linear response kernel of Eq. (4.198) (Ayers, 2001 Sablon et al., 2010 Yang et al., 2012) one has the problem to formulate the local form of the softness kernel 5(r,r ) fulfilling the Berkowitz-Ghosh-Parr relationship (4.197) (Berkowitz et al., 1985 Berkowitz Parr, 1988) within conceptual DFT. To this aim one starts with rearranging the Eq. (4.198) as an integral form along the chemical reaction path (Putz Chattaraj, 2013) 

Observe, for instance, that when integrating both sides of Eq. (4.200) with respect to r, one may use the fundamental DFT normalization constraint (4.168) that along the integration result of Eq. (4.198) leaves with the useful constrain 

Actually, the entire present development stays under the valence or long-range regime of electrons in atoms and molecules in various forms and approximations of conceptual DFT. The minus sign in Eq. (4.203) agrees with the opposite phenomenological behavior in density and potential variation, as provided by Poisson equation—for instance (Putz et al., 2005), and is in accordance with alternative derivation based on ehemical action principle and virial theorem (Putz, 2009a). 

one uses the integral linking the hardness and softness kernels through the reciprocity relation of Eq. (4.189) without the chemical hardness factor ( 2 vs. 1/2 ), see Eq. (4.158) of Section 4.2.3.3 as well as the discussion of Eq. (4.252) in Section 4.5 

Quantum Nanochemistry— Volume II Quantum Atoms and Periodicity 

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