The Softness of Excited States

A finite-temperature DFT is, in effect, an excited-state theory. As the temperature increases, more and more excited states are mixed in with the ground state. The excited states of greatest chemical importance are the electronic ones, including the ionized states M+ and M. Excited vibrational states will also play a role since the equilibrium electron density function, which determines all the properties, depends on the mix of electron density functions for all the excited states, as well as the ground state. 

The problem is that the Hohenberg-Kohn theorem is strictly for ground 

The search for a workable DFT of excited states is an important and active field at the present time. There has been little done, however, on the concepts of chemical potential and hardness, which is our main concern. If we assume that Equation (4.1) is valid, we can draw some conclusions about p and p for excited states. 

There is no change in the electronic chemical potential, but the hardness of the excited state is less than that of the ground state. 

This conclusion can be drawn in another way. Write the reactions 

---