Revisiting the chemical reactivity indices as the state function derivatives. The role of classical chemical hardness

The chemical reactivity indices as the equilibrium state-function derivatives are revisited. They are obtained in terms of the central moments (fluctuation formulas). To analyze the role of the chemical hardness introduced by Pearson [J. Am. Chem. Soc. 105, 7512 (1983)], the relations between the derivatives up to the third-order and the central moments are obtained. As shown, the chemical hardness and the chemical potential are really the principal indices of the chemical reactivity theory. It is clear from the results presented here that the chemical hardness is not the derivative of the Mulliken chemical potential (this means also not the second derivative of the energy at zero-temperature limit). The conventional quadratic dependence of energy, observed at finite temperature, reduces to linear dependence on the electron number at zero-temperature limit. The chemical hardness plays a double role in the admixture of ionic states to the reference neutral state energy: it determines the amplitude of the admixture and regulates the damping of its thermal factor.