Higher order derivatives for nuclear indexes in the framework of density functional theory

General and exact relationships for higher order derivatives of the nuclear Fukui function with respect to the electron number at constant external potential have been explicitly derived in the framework of the four Legendre transformed ensembles of density functional theory. These relationships complement and extend to nuclear reactivities some developments [F. De Proft, S. Liu, and P. Geerlings, J. Chem. Phys. 108, 7549 (1998)] and recent advances [E. Chamorro, R. Contreras, and P. Fuentealba, J. Chem. Phys. 113, 10861 (2000)] found for the high order electron responses introduced in the framework of a nonlocal (pair-site) reactivity formalism.