Long-Range Chemical Softness

For practical implementation we provide further softness and chemical hardness dependence on the electronic density and potential, alternatively to Eq. (4.226). To this aim we use a local-nonlocal ansatz for softness kernel within the present long-range/shape function approximation of Eq. (4.207), namely as (Garza Robles, 1993 Putz, 2009a) 

For the local hardness determination one may use the identity, somehow similar with that of Eq. (4.196), yet here at the complete local level, namely 

Equation (4.230) can be proved within long-range density interaction by starting from Eq. (4.205) its integral over r coordinate is taken first, then, by using the basic DFT constraint (4.168) and kemel-to-local softness integral relationship (4.213) one arrives at the 1/2-chemcial hardness modified form of Eq. (4.191), as 

This way, Eq. (4.230) allows the local chemical hardness in the density-potential formulation  

Note that Eq. (4.232) can be rearranged so that it can be compared with Eq. (4.211) that is (Putz Chattaraj, 2013) 

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Electrophilicity Long Range Chemical Softness Formulation.227... 

ELECTROPHILICITY LONG RANGE CHEMICAL SOFTNESS FORMULATION... 

The long-range consistency of local chemical hardness (4.211) with softness kernel (4.210) may be proved also by checking the key softness hierarchy condition in passing from kernel to local dependency ... 

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