Relationships between the third-order reactivity indicators in chemical density-functional theory

Relationships between third-order reactivity indicators in the closed system [N, v(r)], open system [mu, v(r)], and density [rho(r)] pictures are derived. Our method of derivation unifies and extends known results. Among the relationships is a link between the third-order response of the energy to changes in the density and the quadratic response of the density to changes in external potential. This provides a link between hyperpolarizability and the system's sensitivity to changes in electron density. The dual descriptor is a unifying feature of many of the formulas we derive.