The Functional Atomic Chemical Hardness

Among the various ways of chemical hardness (//) formulations, here will be considered that one that connects the global hardness of an electronic system to its global softness (5) by the inverse relation (4.196)  

Quantum Nanochemistry—Volume II Quantum Atoms and Periodicity 

Since the chemical hardness describes the electronic cloud s inertia to deform under the nuclear (or applied weak electrical field) the global softness accounts for such deformations. The global softness is then expressed for atomic systems through assuming the forms (Putz et al., 2003,2012b,c)  

The last expression was obtained making use of the above rule of decomposition adapted for the present situation ( =4). Therefore, the global softness provides the global hardness with expression  

However, to be situated in the same type of approximation as for the atomic polarizability was made, the non-local contribution to the global softness, the term, has to be neglected. In practice this approximation is extended to neglect the term ZAP instead of AP only. This way, the approximation was performed in the hardness formula not in the softness one. This extension will be useful in applying this kind of approximation to another model for ehemical hardness derivation. Finally, the hardness derived from softness displays the simple formula  

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