Hyperhardness and hypersoftness of atoms and their ions

The theory of reactivity based on cDFT has been supplemented with the new method of calculation of the atomic and local indices. With the use of previously derived relationship of the electron density gradient to the softness kernel and to the linear response function we deliver analytical scheme to obtain significant reactivity indices - the electron density derivatives: local softness s(r), local hypersoftness [∂s(r)/∂N] for atoms. The global hyperhardness and the global derivative [∂S/∂N] have also been obtained. The local derivatives have been applied in the analysis of responses of atoms to perturbation by an external potential ∆v(r) by the alchemical approach. The vital role of the local softness has been confirmed; the potential role of the hypersoftness has been indicated for reactions in a confinement.