and other related quantities are discontinuous, the corresponding left and right derivatives being different. In view of this discontinuity, use of the derivative notation is best understood with an underlying finite-difference mathematical definition , where e.g. the right derivative is obtained by evaluating E(N+1) -E(N). This interpretation is particularly important for second derivatives such as the chemical hardness in eqn (17). Since the chemical hardness describes the change in the first derivative at an integer electron number N, it will be zero or infinite at N and have no physical meaning unless interpreted in the above finitedifference manner. The PPLB condition is thus the foundation for the discussion of derivatives. It leads to the identification of the left and right chemical potentials with the ionization energy I and the electron affinity A, respectively. This identification was used to establish the physical meaning of the HOMO and LUMO orbital energies as the density-functional prediction of -I and -A, respectively, associated with the functional approximation used. (2.5.4) Liu :