On hardness and electronegativity equalization in chemical reactivity theory

Chemical Reactivity Theory (CRT) contains reactivity indices defined as first and second derivatives of ground-state properties with respect to electron number such as the electronegativity and the hardness. This necessitates use of the Perdew, Parr, Levy, and Balduz (PPLB) version of noninteger density-functional theory (NIDFT) to provide a basis for CRT in DFT. However, the PPLB NIDFT yields ground-state properties which are piecewise linear continuous functions of number, yielding vanishing hardness and staircase electronegativities which do not admit electronegativity equalization. To overcome these difficulties, in the present paper we modify the relationship between CRT and DFT, basing the former on our previously formulated "atoms" in "molecules" theory (AIMT) but retaining the PPLB NIDFT. We recapture electronegativity equalization through the agency of a uniquely defined reactivity potential. We demonstrate that a positive definite hardness matrix can be defined which controls the minimum cost to the AIMT energy functional of internal fluctuations of the electron numbers of the parts of a system.