Revisiting Pearson’s HSAB Principle: The Hardness of Cations and Anions from Atomic Polarizabilities

The hard–soft acid–base (HSAB) principle is among the most useful tools to rationalize chemical reactivity, establishing that hard acids tend to associate with hard bases, while soft acids tend to associate with soft bases. The hard/soft nature of acids and bases can be quantified using absolute hardness (η), which was defined by Pearson as η = (I – A)/2, where I and A are the ionization potential and electron affinity, respectively. We found that this definition results in anomalies for some ions across the periodic table (i.e., La3+ and Ce3+ have significantly different η values of 15.4 and 8.4 eV, respectively, but similar coordination chemistry). This prompted us to propose an alternative definition of η based on the electric dipole polarizability (α) and the radius of the ion, estimated from the maxima of the radial distribution functions (⟨r⟩) obtained with relativistic CCSD(T) calculations: η = ⟨r⟩2/α. We show that this simple relationship affords coherent η values for cations across the periodic table, overcoming the anomalies observed with Pearson’s definition. Furthermore, this definition allows calculating η values for anions and other species for which electron affinities are not readily available, offering a useful tool to rationalize chemical reactivity across different areas of chemistry.