Theory of non-local (pair site) reactivity from model static-density response functions

Activation is a fundamental and well-known concept in chemistry. It may be qualitatively defined as an increase in the chemical reactivity pattern of a molecule at a given site k when the system is locally perturbed at a different site l, say. This external perturbation arise from a localized molecular rearrangement, a substitution, a selective solvation or simply by the approach of a reagent of variable hardness. This work presents a theoretical approach intending to quantify this activation concept in the density functional framework. This is done here by first calculating the fluctuation of the electron density at a given site k for the ground state of the isolated substrate (static reactivity model) and then incorporating the substrate and model electrophile reagents in a spatial disposition related to a virtual transition structure for the parent system. This perturbation is assumed representable by local changes in the external potential. It is shown that a local approximation to the softness kernel s(r, r′) yields a simple expression for the fluctuation of the electron density δρ(r_k), which shows that this change becomes proportional to the variation of an effective potential δu(r_k), containing the information on the variation in the chemical potential and the external perturbing potential at site k; the proportionality constant being the local softness s⁰(r_k) at that site. The strong local approximation made to the kernel s(r, r′) causes the second reactivity site (l) to implicitly appear in the formulation through the changes in the electronic chemical potential term. It is shown that the introduction of a less restrictive approach to the linear response function, obtained from a model Kohn-Sham one-electron density matrix, leads to the same result. Non-locality is therefore self-contained in the electronic chemical potential contribution to the modified potential, and may be associated with an intramolecular charge transfer between the active sites of the ambident nucleophilic/electrophilic substrate, promoted by the presence of the reagents. The resulting formulation of pair-site reactivity is illustrated for the electrophilic attack on the CN^- ion by different model electrophile agents of variable hardness. It is shown that correct reactivity indexes are obtained only when the topology of the transition structure is used as a vantage point to perturb the CN^- ion. The calculations were performed at both density functional theory and ab-initio Hartree-Fock levels. The results show that the proposed model is independent of the method used to obtain ρ(r).