Hardness and excitation energy

The concept of the ensemble Kohn-Sham hardness is introduced. It is shown that the first excitation energy can be given by the Kohn-Sham hardness (i.e. the energy difference of the ground-state lowest unoccupied and highest occupied levels) plus an extra term coming from the partial derivative of the ensemble exchange-correlation energy with respect to the weighting factorw in the limitw → 0. It is proposed that the first excitation energy can be used as a reactivity index instead of the hardness.