Ionic radii of hydrated sodium cation from QTAIM

The sodium cation is ubiquitous in aqueous chemistry and biological systems. Yet, in spite of numerous studies, the (average) distance between the sodium cation and its water ligands, and the corresponding ionic radii, are still controversial. Recent experimental values in solution are notably smaller than those from previous X-ray studies and ab initio molecular dynamics. Here we adopt a "bottom-up" approach of obtaining these distances from quantum chemistry calculations [full MP2 with the 6-31++G(d,p) and cc-pVTZ basis-sets] of gas-phase Na⁺(H₂O)_n clusters, as a function of the sodium coordination number (CN = 2-6). The bulk limit is obtained by the polarizable continuum model, which acts to increase the interatomic distances at small CN, but has a diminishing effect as the CN increases. This extends the CN dependence of the sodium-water distances from crystal structures (CN = 4-12) to lower CN values, revealing a switch between two power laws, having a small exponent at small CNs and a larger one at large CNs. We utilize Bader's theory of atoms in molecules to bisect the Na⁺-O distances into Na⁺ and water radii. Contrary to common wisdom, the water radius is not constant, decreasing even more than that of Na⁺ as the CN decreases. We also find that the electron density at the bond critical point increases exponentially as the sodium radius decreases.