Can one provide a quantitative measure of "chemical hardness"? A 1983 JACS paper by Robert Parr and Ralph Pearson suggested that for a specific atom or molecule it could be defined as the second derivative of the ground state energy with respect to the number of electrons N
or the discrete version
Aside: Physicists can relate eta to the charge compressibility.
At the end of their article they point out the fascinating fact (to me) that eta is equal to the Hubbard U. [I think this is just the case for an open shell system, i.e. where N is such that there is one electron in the HOMO.]
The argument that Parr and Pearson use to just the HSAB principle from their definition of hardness was not very clear to me. Perhaps it is clearer in an earlier paper of Klopman that they cite. I was hoping to see something like a simple proof [based on an asymmetric 2 site Hubbard model?] that the acid-base bonding is a maximum when the difference between the U on the two sites is minimal? But, maybe this is not possible...
A recent J Phys. Chem. article by James Reed on the subject considers some of the complexities of the subject.