Reactivity kernels, the normal modes of chemical reactivity, and the hardness and softness spectra

Chemical reactivity theory provides a basis for predicting the reactive proclivities of molecular or condensed systems. The frontier-orbital concepts of Fukui, as generalized by Parr and collaborators within the framework of density-functional theory, were developed further by us in a previous paper (I) [J. Chem. Phys. 101, 8988 (1994)]. Nevertheless, five aspects of the theory still require further development; the reactivities are defined as local responses to global stimuli instead of nonlocal responses to local stimuli; there are ambiguities associated with the existence of energy bands in condensed systems; the theory is static and does not properly incorporate the internal dynamics of the reacting systems; the theory focuses on responses without a corresponding definition of chemical stimuli; and no connection is made with the potential energy surface and the reaction pathway. In the present paper, we concentrate on gapless systems extended in at least one dimension, metals, semimetals, and insulators. We show that taking as measures of chemical reactivity the reactivity kernels of Nalewajski [J. Chem. Phys. 78, 6112 (1983); 81, 2088 (1984); 89, 2831 (1985)] and of Berkowitz and Parr [J. Chem. Phys. 88, 2554 (1988)] in the form introduced by the latter meets the first two requirements. We find an explicit expression for the softness kernel which is a natural generalization of that found for the local softness in I. The response functions entering it are calculable via the Kohn-Sham formalism. As recognized by Nalewajski and Koniński [Z. Naturforsch. 42a, 451 (1987)], there is an infinite set of modes of chemical reactivity with corresponding spectra of softness and hardness eigenvalues. The softness spectrum has an upper bound corresponding to the most reactive mode or set of degenerate modes [Z. Naturforsh. 42a, 451 (1987)] and a vanishing lower bound. The softness kernel itself can be expressed simply in terms of the static dielectric function. We generalize the nuclear softness introduced in I to a nuclear softness kernel analogous to the electronic softness kernel and find a simple expression for it in terms of the dielectric function. The present theory provides a basis for the remaining developments needed.