On the equvilence between the maximum hardness principle (MHP) and the maximum entropy principle (MEP)

Inspired by Parr’s definition of the hardness functional for the ground state of an electron system with electron density ρ (J. Phys. Chem. 1993, 97, 3939-3940), we maximize Shannon’s information entropy S ρ = - ∫ ρ r lnρ r dr , subject to the constraint of minimum total electronic energy E ρ = μN - H ρ , where μ is the chemical potential, N is the total number of electrons, and H[ρ] is the haedness functional. Then we obtain electron density ρ r = exp - λ δ H ρ δ ρ r + 1 with the maximum entropy S max [ ρ ] = λ ∫ δ H ρ δ ρ r exp - λ δ H ρ δ ρ r + 1 dr + N , where λ is the undetermined Lagrange multiplier with a positive value to make sure the convergence of electron density ρ, which implies that Shannon’s information entropy S is a maximum just when the hardness H is a maximum, and in turn the total electronic energy E is a minimum.