ASPECTS OF THE SOFTNESS AND HARDNESS CONCEPTS OF DENSITY-FUNCTIONAL THEORY

In the density-functional theory of the ground state of an electronic system there arise the concepts of softness, hardness, local softness, and local hardness. Definitions of these quantities are reviewed, and then local softness and local hardness are discussed in some detail. The local softness of a species the derivative (s(r) = [partial derivative rho(r)/partial derivative mu]nu(r), is a measure of the chemical reactivity of a site in the molecule. From it can be obtained the total global absolute softness in the sense of Pearson and a normalized chemical reactivity index of frontier type. Several formulas for s(r) are obtained, including new fluctuation formulas, and its determinative role in chemisorption, catalysis, and frontier-controlled charge-transfer processes is briefly discussed. Local hardness is a corresponding appropriately defined functional derivative eta(r) = [delta-mu/delta-rho(r)]nu(r). Difficulties associated with ambiguities in this definition are discussed and resolved. It is concluded that for most purposes the best working formula for local hardness is eta(r) = integral eta(r, r')f(r') dr', where eta(r, r') is the hardness kernel; eta(r, r') = delta-2F[rho]/delta-rho(r)delta-rho(r'), where F[rho] is the usual Hohenbetg-Kohn functional and f(r) is the Fukui function. With this definition, eta(r) = eta, a constant which is the global hardness. Just as the chemical potential equalizes in the ground state, so does the hardness. It is demonstrated that hardness can be taken to be an average of orbital contributions.