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 f(3)(r). 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 f(3)(r), 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.