Spin-Resolved Nonlocal Second-Order Fukui Index for Open-Shell Systems from Delocalization Index Response

This work presents the first analytical derivation of the spin-resolved nonlocal Second-Order Fukui Index (SOFI) for open-shell molecular systems. The approach is based on the linear response of interatomic delocalization indices (δ AB ), providing a direct connection between frontier molecular orbital characteristics and real-space electron-sharing redistribution. The methodology is applied to substituted phenoxyl radicals to quantify how electron-donating (EDG) and electron-withdrawing (EWG) groups modulate α- and β-spin channels. The results provide a detailed map of spin-dependent electronic redistribution upon electron addition or removal. Spin-resolved SOFI indices are shown to be effective nonlocal descriptors for rationalizing radical reactivity. This study establishes a theoretical framework for the quantitative investigation of open-shell systems and the influence of substituents on spin-dependent electronic structure.