Chemical softness in model electronic systems: Dependence on temperature and chemical potential

Discrete and continuous model systems are studied in order to understand the behavior of average softness, for equilibrium states in a grand canonical ensemble, when the bath parameters are changed. While the former model seems to be appropriate for describing an isolated system, the latter could take care of atoms in a molecular framework. Three, four, and five level discrete models, and quadratic- and exponential-type continuous models were chosen for studying the dependence of average number of particles, average softness, average electronic, and Helmholtz free energies on temperature and chemical potential. For the quadratic model, the zero temperature limit of the average softness is the absolute softness of density functional theory. In the other models it generally goes to zero at 0 K, but for some specific values of mu. The high temperature limit of it is zero in all the studied cases. When the average number of particles is an integer or half integer in the discrete model at low temperature, there are local extrema in equilibrium softness and the maximum hardness principle is shown to be valid.