Electron-Number Response of QTAIM Delocalization Indices: A Differential Decomposition into Population and Orbital-Relaxation Contributions

This work establishes a differential framework for analyzing the electron-number response of QTAIM delocalization indices within the piecewise-linear ensemble formalism of conceptual density functional theory (C-DFT). The resulting first-order response is decomposed into three distinct contributions: a population term arising from occupation number changes, an orbital relaxation term, and a geometric domain relaxation term. The latter follows from the application of the Reynolds transport theorem to QTAIM zero-flux domains and accounts for the displacement of atomic boundaries. The geometric contribution is expressed in terms of the normal velocity of the zero-flux surface. Assuming fixed QTAIM domains, the geometric contribution vanishes by construction, leading to a consistent operational partition of the delocalization index variation into population and orbital relaxation components. Applications to aromatic systems show that the electron-number response of the delocalization index is governed by the interplay between occupation number and orbital relaxation effects, whose relative importance depends sensitively on both the molecular environment and the direction of the electron-number change.