Extending Conceptual DFT to Fourth Order: From Quartic Curvature to Third-Order Fukui Response

We extend the framework of conceptual density functional theory (DFT) to include fourth-order energy derivatives. We present the series of quantum reactivity descriptors at this order for the canonical and the grand canonical ensembles. After introducing a direct computational methodology, we investigate and describe the quartic curvature λ and the third-order Fukui function <i>f</i>⁽³⁾(<b>r</b>). These higher-order descriptors capture nonlinear aspects of electronic reactivity that go beyond the conventional concepts of chemical potential, hardness, or Fukui functions. The global descriptor λ, obtained via finite-difference approximations of frontier orbital energies, quantifies the curvature of the chemical hardness and offers insight into the electronic (in)stability of molecular systems under charge perturbations. The local descriptor <i>f</i>⁽³⁾(<b>r</b>), derived as the third-order response of the electron density, reveals spatially resolved regions of charge-transfer sensitivity. We apply these descriptors to a series of push-pull organic molecules and open-shell systems, demonstrating their ability to highlight differences in internal charge transfer character and electronic delocalization. The results support the use of fourth-order conceptual DFT tools as chemically meaningful indicators of reactivity beyond the harmonic regime.