Structure and Spectroscopy of Hydrated Sodium Ions at Different Temperatures and the Cluster Stability Rules

The sodium cation plays an important role in several physiological processes. Understanding its solvation may help understanding ion selectivity in sodium channels that are pivotal for nerve impulses. This paper presents a thorough investigation of over 75 isomers of gas-phase Na(+)(H2O)(n=1-8) clusters, whose optimized structures, energies, and (harmonic) vibrational frequencies were computed quantum mechanically at the full MP2/6-31++G(d,p) level of theory. From these data, we have calculated the temperature effects on the cluster thermodynamic functions, and thus the equilibrium Boltzmann distribution for each n. For a selected number of isomers, we have corrected the calculations for basis set superposition error (BSSE) to obtain accurate clustering energies, in excellent agreement with experiment. The computed clusters are overwhelmingly 4-coordinated, as opposed to bulk liquid water, where sodium cations are believed to be mostly 5- or 6-coordinated. To explain this, we suggest the "cluster stability rules", a set of coordination-number-dependent hydrogen-bond (HB) strengths that can be obtained using a single BSSE correction. Assuming additivity and transferability, these reproduce the relative stability of most of our computed isomers. These rules enable us to elucidate the trends in HB strengths, outlining the major determinants of cluster stability. For n = 4 and 5, we have also performed anharmonic vibrational calculations (VPT2) to compare with available photodissociation infrared spectra of these gas-phase clusters. The comparison suggests that the experiments actually monitor a mixture of predominantly 3-coordinated isomers, which is quite remote from the computed Boltzmann distribution, particularly at low temperatures. Surprisingly, for these experiments, water evaporation pathways can rationalize the non-equilibrium isomer distribution. The equilibrium isomer distribution is, in turn, rationalized by the entropy of internal rotations of "dangling" water molecules.