The dual delocalization descriptor: first- and second-order responses of bond-resolved electron sharing

This work introduces a dual delocalization descriptor (DDD) defined through the first-order (f(1)AB) and second-order (f(2)AB) responses of the interatomic delocalization index to variations in the total electron number. The DDD quantifies the response of interatomic electron sharing to variations in the total electron number, providing a bond-resolved measure of electron redistribution upon electron addition or removal. Positive (negative) values indicate a net increase (decrease) in interatomic electron sharing, as determined by the relative contributions of electron-attachment and electron-removal channels. An explicit comparison of occupation-number formulations demonstrates that bilinear forms (Ángyán-Loos-Mayer type) yield well-defined, finite response functions that decouple inter-orbital and self-delocalization contributions. By contrast, square-root functionals (Müller type) do not admit a well-defined one-sided derivative with respect to electron number, making their response formalisms non-differentiable at integer electron numbers within the PPLB ensemble framework. This establishes functional differentiability with respect to electron number as a key requirement for constructing bond-level derivative descriptors. The analytical DDD is shown to consistently recover bond-resolved finite-difference trends across a series of aromatic systems while relying exclusively on the electronic properties of the neutral reference state.