Higher order relationships in the canonical ensemble for local reactivity indices

In this paper, higher order relations derived from the canonical ensemble N,νr⃗ are presented in the framework of Density Functional Theory (DFT), with a focus on the Fukui function [f(r⃗)], dual descriptor [f2(r⃗)], and the function t(r⃗) introduced by Fuentealba and Parr. These relationships extend existing theoretical models, providing a more detailed understanding of local and non-local chemical reactivity. The equations developed establish direct connections between higher order reactivity indices, such as hyperhardness (γ), the fourth order energy function f3r⃗, and non-local response functions, offering new insights into the activation and deactivation of molecular systems under external perturbations. These findings are particularly relevant in the study of activation processes by the presence of n bodies generating external perturbations. The results highlight the fundamental role of higher-order kernels in describing changes in reactivity, contributing to the development of a more refined theoretical framework in conceptual DFT.