Evaluation of hardness in Kohn-Sham theory with local density and generalized gradient approximations

By making use of Janak's interpolation in the Kohn-Sham method, together with an explicit and differentiable functional of the density for the exchange-correlation energy, like LDA or GGA, the second derivative of the energy with respect to the number of electrons, N, is evaluated from the derivative of the highest occupied molecular orbital (HOMO) with respect to N (left derivative), or from the derivative of the lowest unoccupied molecular orbital (LUMO) with respect to N (right derivative). Both cases lead to expressions that may be evaluated from a single post-SCF calculation that avoids the use of the frontier orbitals eigenvalues. The results obtained for two test sets of molecules and for several atoms indicate that both provide a better description than the LUMO-HOMO eigenvalue difference and that the left derivative seems to lead to more confident results than the right derivative.