RELATIONSHIP BETWEEN ENERGY AND HARDNESS DIFFERENCES

It is shown that the energy change from one ground state to another is approximately given by DELTAE = NDELTAmu -1/2N2 DELTAeta, where N is the total number of electrons, DELTAmu is the change in the chemical potential, and DELTAeta is the change in the global hardness. Through this expression it is established that, under conditions of constant chemical potential, when a system evolves toward a state of greater hardness, its stability increases, and when it evolves toward a state of lower hardness, its stability decreases. Additionally, this expression is used to find a relationship between the derivatives of the electronic energy, the chemical potential, and the hardness with respect to the reaction coordinate, in order to show that under conditions of constant chemical potential, the hardness is a maximum where the electronic energy is a minimum and that the hardness is a minimum where the electronic energy is a maximum. The same conclusion is derived for the case in which the chemical potential is not constant, but it has a maximum or a minimum at the same point of the electronic energy maximum or minimum. The numerical results presented for several diatomic molecules show that indeed, when DELTAmu almost-equal-to 0 the change in the energy is directly proportional to minus the change in the hardness; however the proportionality constant is far from (1/2)N2, indicating that the relationship between the electronic density and the Fukui function used in the present work should be improved to obtain the adequate proportionality constant.