Based on functional expansion methods recently developed, the total energy of an atomic and molecular system is expanded in terms of its chemical potential (electronegativity), hardness, softness, Fukui function, response function, and so on, up to the second order. Expansions are given within the four basic ensembles in density functional theory : the canonical, grand canonical, isomorphic, and grand isomorphic ensembles. The formulas obtained provide explicit relations among the important quantities related to the chemical reactivity of a system, and they provide a means to calculate one quantity when others are known. Furthermore, a favorable viewpoint of the maximum hardness principle follows from these formulas. At fixed electron number, external potential and chemical potential, the global hardness should go to a maximum as the total energy approaches a minimum. Arguments for the second-order truncation are given.