Nonlinear electronic responses, Fukui functions and hardnesses as functionals of the ground-state electronic density

The exact equations for the variations of the electronic density induced by an adiabatic external potential are derived to an arbitrary perturbation order in the framework of density-functional-theory. The formal solutions of these density perturbation equations are given and the exact relations between the electronic response functions and the Hohenberg–Kohn functional are derived. Using these relations, the static nonlinear electronic response functions are constructed from the linear one. Nonconserving electron number perturbations are also directly included in the formalism to all perturbation orders. In this way the well-known results of the density-functional reactivity theory are generalized beyond the first and the second-order. This makes it possible to derive the exact relations between the Hohenberg–Kohn functional and the linear and nonlinear Fukui responses, the nonlinear Fukui functions and the nonlinear hardnesses. These relations allow us to reformulate all the derivatives of the electronic energy relative to the external potential and to the particle number in terms of the linear response kernel and in terms of the linear Fukui function. The formalism is applied to the Thomas–Fermi–Dirac-λ von Weiszäcker model of the Hohenberg–Kohn functional.