Bivariate Electronegativity-Chemical Hardness Relationship

Emplo5dng the Kohn-Sham equation for the (reactivity) equalized chemical potential (minus electronegativity) eigenvalue, among atomic or molecular one-electronic orbitals, recalling for instance (3.65) and (3.298)-(3.300), 

For proper characterization of the chemical systems the quantum Ehrenfest version of the fundamental Newtonian law, linking the observed force with the minus of the potential gradient F = -VF, is here written for chemical reactions modeled throughout the charge transfer along the reaction path (Putz, 2008a)  

combining the last two equations projected on the reaction path we successively obtain  

At this point, taking for the natural differential the finite correspondence in what regarding the chemical potential formal (absolute) definition. 

The remaining issue is to identify the chemical-potential related force meaning in above equation. In this respect, by considering recognizing the electronegativity-chemical potential relationship, we have that the electronegativity energy equation becomes  

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