An Occam's razor approach to chemical hardness: lex parsimoniae

The term "chemical hardness" refers to the resistance to deformation of the electronic density of a system; the greater this resistance, the "harder" the system. Polarizability, a physical property, is an inverse measure of resistance to deformation and thus should be inversely related to hardness. This is indeed generally accepted. Hardness has been postulated to be the second derivative of a system's energy with respect to its number of electrons, despite the fact that this involves the differentiation of a noncontinuous function. This second derivative is typically approximated as the difference between the ionization energy I and the electron affinity A of the ground-state system, which results in ambiguity in that many molecules do not form stable negative ions. For atoms, the quantity I - A does vary approximately inversely with polarizability, but this is only because the electron affinity is usually relatively low and ionization energy is known to be inversely related to polarizability for atoms. However, molecular polarizability depends primarily upon volume, and so does not show an acceptable inverse correlation with I - A. Since both hardness and polarizability refer to the same property of a system-its resistance to deformation of the electronic density, we propose that the reciprocal of polarizability be taken to be a measure of hardness. We show that polarizabilities that are not known can be estimated quite accurately in terms of the average local ionization energies on the atomic or molecular surfaces and, for molecules, their volumes.