Computing the Unconstrained Local Hardness

The unconstrained local hardness is calculated using an exact equation in the Kohn-Sham density functional theory. Several numerical difficulties are encountered, mostly related to the exponential asymptotic divergence of the unconstrained local hardness. The unconstrained local hardness 46shows shell structure. Somewhat unexpectedly, it takes negative values in certain regions. The unconstrained local hardness is calculated using an exact equation in the Kohn-Sham density functional theory. Several numerical difficulties are encountered, mostly related to the exponential asymptotic divergence of the unconstrained local hardness. The unconstrained local hardness shows shell structure. Somewhat unexpectedly, it takes negative values in certain regions. Many of the greatest successes in the density functional theory (DFT) of chemical reactivity are tied to the chemical hardness. Prior to the seminal 1983 paper by R. G. Parr and R. G. Pearson, the chemical hardness was only an empirical, qualitative measure of chemical reactivity preferences. The mathematical theory of the hard soft acid base (HSAB) principle is, however, incomplete unless one can also consider the local HSAB principle. Specifically, given a molecule with two reactive sites of differing local hardness, the hard reagents will prefer to react at the site with the greater local hardness and soft reagents will prefer to react at the site with smaller local hardness.