Accuracy of the microsolvation–continuum approach in computing the pKa and the free energies of formation of phosphate species in aqueous solution

First principles density functional theory (Perdew–Burke–Ernzerhof) calculations have been used to compute the hydration properties, aqueous-phase acid dissociation constants (pKa) and Gibbs free energies of formation of small polyphosphates in aqueous solution. The effect of the hydrated environment has been simulated through a hybrid microsolvation–continuum approach, where the phosphate species are simulated as microsolvated solutes, while the remainder of the bulk solvent is treated as a dielectric continuum using the COSMO solvation model. The solvation free energies of orthophosphates and pyrophosphates have been computed applying monomer and cluster thermodynamic cycles, and using the geometries optimised in the gas-phase as well as in the COSMO environment. The results indicate that the simple polarisable continuum or microsolvation–continuum models are unable to compute accurate free energies of solvation for charged species like phosphates. The calculation of the pKa shows that the computed values of acid dissociation constants are critically dependent on the number of water molecules nH2O included in the hydrated phosphate clusters. The optimal number nH2O is determined from the minimum value of the “incremental” water binding free energy associated with the process of adding a water molecule to a micro-solvated phosphate species. Analysis of the effect of nH2O on the free energies of orthophosphate condensation reactions shows that can vary by tenths of kcal mol−1, depending on the particular choice of nH2O for the monomeric and dimeric species. We discuss a methodology for the determination of nH2O; for the orthophosphates the “incremental” binding energy approach is used to determine nH2O, whereas for the polyphosphates the number of explicit water molecules is simply equal to the effective charge of these anions. The application of this method to compute the free energy of formation of pyro- and tri-phosphates gives generally good agreement with the available experimental data.